Fokker-Planck equation in the presence of a uniform magnetic field

Energy Technology Data Exchange (ETDEWEB)

Dong, Chao, E-mail: chaodong@iphy.ac.cn [Center for Plasma Theory and Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Department of Nuclear Engineering, Seoul National University, Seoul 151-744 (Korea, Republic of); Zhang, Wenlu [Center for Plasma Theory and Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Li, Ding, E-mail: dli@ustc.edu.cn [Center for Plasma Theory and Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Department of Modern Physics, University of Science and Technology of China, Anhui Hefei 230026 (China)

2016-08-15

The Fokker-Planck equation in the presence of a uniform magnetic field is derived which has the same form as the case of no magnetic field but with different Fokker-Planck coefficients. The coefficients are calculated explicitly within the binary collision model, which are free from infinite sums of Bessel functions. They can be used to investigate relaxation and transport phenomena conveniently. The kinetic equation is also manipulated into the Landau form from which it is straightforward to compare with previous results and prove the conservation laws.

Application of Fokker-Planck equation in positron diffusion trapping model

International Nuclear Information System (INIS)

Bartosova, I.; Ballo, P.

2015-01-01

This paper is a theoretical prelude to future work involving positron diffusion in solids for the purpose of positron annihilation lifetime spectroscopy (PALS). PALS is a powerful tool used to study defects present in materials. However, the behavior of positrons in solids is a process hard to describe. Various models have been established to undertake this task. Our preliminary model is based on the Diffusion Trapping Model (DTM) described by partial differential Fokker-Planck equation and is solved via time discretization. Fokker-Planck equation describes the time evolution of the probability density function of velocity of a particle under the influence of various forces. (authors)

The Fokker-Planck equation for coupled Brown-Néel-rotation.

Science.gov (United States)

Weizenecker, Jürgen

2018-01-22

Calculating the dynamic properties of magnetization of single-domain particles is of great importance for the tomographic imaging modality known as magnetic particle imaging (MPI). Although the assumption of instantaneous thermodynamic equilibrium (Langevin function) after application of time-dependent magnetic fields is sufficient for understanding the fundamental behavior, it is essential to consider the finite response times of magnetic particles for optimizing or analyzing various aspects, e.g. interpreting spectra, optimizing MPI sequences, developing new contrasts, and evaluating simplified models. The change in magnetization following the application of the fields is caused by two different movements: the geometric rotation of the particle and the rotation of magnetization with respect to the fixed particle axes. These individual rotations can be well described using the Langevin equations or the Fokker-Planck equation. However, because the two rotations generally exhibit interdependence, it is necessary to consider coupling between the two equations. This article shows how a coupled Fokker-Planck equation can be derived on the basis of coupled Langevin equations. Two physically equivalent Fokker-Planck equations are derived and transformed by means of an appropriate series expansion into a system of ordinary differential equations, which can be solved numerically. Finally, this system is also used to specify a system of differential equations for various limiting cases (Néel, Brown, uniaxial symmetry). Generally, the system exhibits a sparsely populated matrix and can therefore be handled well numerically.

The Fokker-Planck equation for coupled Brown-Néel-rotation

Science.gov (United States)

Weizenecker, Jürgen

2018-02-01

Calculating the dynamic properties of magnetization of single-domain particles is of great importance for the tomographic imaging modality known as magnetic particle imaging (MPI). Although the assumption of instantaneous thermodynamic equilibrium (Langevin function) after application of time-dependent magnetic fields is sufficient for understanding the fundamental behavior, it is essential to consider the finite response times of magnetic particles for optimizing or analyzing various aspects, e.g. interpreting spectra, optimizing MPI sequences, developing new contrasts, and evaluating simplified models. The change in magnetization following the application of the fields is caused by two different movements: the geometric rotation of the particle and the rotation of magnetization with respect to the fixed particle axes. These individual rotations can be well described using the Langevin equations or the Fokker-Planck equation. However, because the two rotations generally exhibit interdependence, it is necessary to consider coupling between the two equations. This article shows how a coupled Fokker-Planck equation can be derived on the basis of coupled Langevin equations. Two physically equivalent Fokker-Planck equations are derived and transformed by means of an appropriate series expansion into a system of ordinary differential equations, which can be solved numerically. Finally, this system is also used to specify a system of differential equations for various limiting cases (Néel, Brown, uniaxial symmetry). Generally, the system exhibits a sparsely populated matrix and can therefore be handled well numerically.

Integral propagator solvers for Vlasov-Fokker-Planck equations

International Nuclear Information System (INIS)

Donoso, J M; Rio, E del

2007-01-01

We briefly discuss the use of short-time integral propagators on solving the so-called Vlasov-Fokker-Planck equation for the dynamics of a distribution function. For this equation, the diffusion tensor is singular and the usual Gaussian representation of the short-time propagator is no longer valid. However, we prove that the path-integral approach on solving the equation is, in fact, reliable by means of our generalized propagator, which is obtained through the construction of an auxiliary solvable Fokker-Planck equation. The new representation of the grid-free advancing scheme describes the inherent cross- and self-diffusion processes, in both velocity and configuration spaces, in a natural manner, although these processes are not explicitly depicted in the differential equation. We also show that some splitting methods, as well as some finite-difference schemes, could fail in describing the aforementioned diffusion processes, governed in the whole phase space only by the velocity diffusion tensor. The short-time transition probability offers a stable and robust numerical algorithm that preserves the distribution positiveness and its norm, ensuring the smoothness of the evolving solution at any time step. (fast track communication)

An Adjoint-based Numerical Method for a class of nonlinear Fokker-Planck Equations

KAUST Repository

Festa, Adriano; Gomes, Diogo A.; Machado Velho, Roberto

2017-01-01

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer the properties of schemes for HJ equations to the FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

An Adjoint-based Numerical Method for a class of nonlinear Fokker-Planck Equations

KAUST Repository

Festa, Adriano

2017-03-22

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer the properties of schemes for HJ equations to the FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

Fokker-Planck equation in mirror research

International Nuclear Information System (INIS)

Post, R.F.

1983-01-01

Open confinement systems based on the magnetic mirror principle depend on the maintenance of particle distributions that may deviate substantially from Maxwellian distributions. Mirror research has therefore from the beginning relied on theoretical predictions of non-equilibrium rate processes obtained from solutions to the Fokker-Planck equation. The F-P equation plays three roles: Design of experiments, creation of classical standards against which to compare experiment, and predictions concerning mirror based fusion power systems. Analytical and computational approaches to solving the F-P equation for mirror systems will be reviewed, together with results and examples that apply to specific mirror systems, such as the tandem mirror

Hypersonic expansion of the Fokker--Planck equation

International Nuclear Information System (INIS)

Fernandez-Feria, R.

1989-01-01

A systematic study of the hypersonic limit of a heavy species diluted in a much lighter gas is made via the Fokker--Planck equation governing its velocity distribution function. In particular, two different hypersonic expansions of the Fokker--Planck equation are considered, differing from each other in the momentum equation of the heavy gas used as the basis of the expansion: in the first of them, the pressure tensor is neglected in that equation while, in the second expansion, the pressure tensor term is retained. The expansions are valid when the light gas Mach number is O(1) or larger and the difference between the mean velocities of light and heavy components is small compared to the light gas thermal speed. They can be applied away from regions where the spatial gradient of the distribution function is very large, but it is not restricted with respect to the temporal derivative of the distribution function. The hydrodynamic equations corresponding to the lowest order of both expansions constitute two different hypersonic closures of the moment equations. For the subsequent orders in the expansions, closed sets of moment equations (hydrodynamic equations) are given. Special emphasis is made on the order of magnitude of the errors of the lowest-order hydrodynamic quantities. It is shown that if the heat flux vanishes initially, these errors are smaller than one might have expected from the ordinary scaling of the hypersonic closure. Also it is found that the normal solution of both expansions is a Gaussian distribution at the lowest order

Maximum Path Information and Fokker Planck Equation

Science.gov (United States)

Li, Wei; Wang A., Q.; LeMehaute, A.

2008-04-01

We present a rigorous method to derive the nonlinear Fokker-Planck (FP) equation of anomalous diffusion directly from a generalization of the principle of least action of Maupertuis proposed by Wang [Chaos, Solitons & Fractals 23 (2005) 1253] for smooth or quasi-smooth irregular dynamics evolving in Markovian process. The FP equation obtained may take two different but equivalent forms. It was also found that the diffusion constant may depend on both q (the index of Tsallis entropy [J. Stat. Phys. 52 (1988) 479] and the time t.

Invalidity of the spectral Fokker-Planck equation forCauchy noise driven Langevin equation

DEFF Research Database (Denmark)

Ditlevsen, Ove Dalager

2004-01-01

-called alpha-stable noise (or Levy noise) the Fokker-Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. In stead it has been attempted to formulate an equation for the characteristic function (the Fourier transform...

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

International Nuclear Information System (INIS)

Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

2009-01-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

Energy Technology Data Exchange (ETDEWEB)

Densmore, Jeffery D [Los Alamos National Laboratory; Warsa, James S [Los Alamos National Laboratory; Lowrie, Robert B [Los Alamos National Laboratory; Morel, Jim E [TEXAS A& M UNIV

2008-01-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

Science.gov (United States)

Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

2009-09-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Chaotic universe dynamics using a Fokker-Planck equation

International Nuclear Information System (INIS)

Coule, D.H.; Olynyk, K.O.

1987-07-01

A Fokker-Planck equation that accounts for fluctuations in field and its conjugate momentum is solved numerically for the case of a λ phi 4 potential. Although the amount of inflation agrees closely with that expected classically, in certain cases (large initial fields or large dispersions),the ''slow rolling'' approximation appears invalid. In such cases inflation would stop prematurely before possibly restarting. 18 refs., 2 figs

A Fokker-Planck-Landau collision equation solver on two-dimensional velocity grid and its application to particle-in-cell simulation

Energy Technology Data Exchange (ETDEWEB)

Yoon, E. S.; Chang, C. S., E-mail: cschang@pppl.gov [Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Korea Advanced Institute of Science and Technology, Yuseong-gu, DaeJeon 305-701 (Korea, Republic of)

2014-03-15

An approximate two-dimensional solver of the nonlinear Fokker-Planck-Landau collision operator has been developed using the assumption that the particle probability distribution function is independent of gyroangle in the limit of strong magnetic field. The isotropic one-dimensional scheme developed for nonlinear Fokker-Planck-Landau equation by Buet and Cordier [J. Comput. Phys. 179, 43 (2002)] and for linear Fokker-Planck-Landau equation by Chang and Cooper [J. Comput. Phys. 6, 1 (1970)] have been modified and extended to two-dimensional nonlinear equation. In addition, a method is suggested to apply the new velocity-grid based collision solver to Lagrangian particle-in-cell simulation by adjusting the weights of marker particles and is applied to a five dimensional particle-in-cell code to calculate the neoclassical ion thermal conductivity in a tokamak plasma. Error verifications show practical aspects of the present scheme for both grid-based and particle-based kinetic codes.

Computing generalized Langevin equations and generalized Fokker-Planck equations.

Science.gov (United States)

Darve, Eric; Solomon, Jose; Kia, Amirali

2009-07-07

The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.

The Fokker-Planck equation for ray dispersion in gyrotropic stratified media

NARCIS (Netherlands)

Golynski, S.M.

1984-01-01

The Hamilton equations of geometrical optics determine the rays of the relevant wave field in the short wavelength. We give a systematic derivation of the Fokker-Planck equation for the joint probability density of the position and unit direction vector of rays propagating in a gyrotropic stratified

Space distribution and energy straggling of charged particles via Fokker-Planck equation

International Nuclear Information System (INIS)

Manservisi, S.; Molinari, V.; Nespoli, A.

1996-01-01

The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-know Bohr of Bethe formula is found as a first-order approximation of the Fokker-Planck equation

Fractional Fokker-Planck equation and oscillatory behavior of cumulant moments

International Nuclear Information System (INIS)

Suzuki, N.; Biyajima, M.

2002-01-01

The Fokker-Planck equation is considered, which is connected to the birth and death process with immigration by the Poisson transform. The fractional derivative in time variable is introduced into the Fokker-Planck equation in order to investigate an origin of oscillatory behavior of cumulant moments. From its solution (the probability density function), the generating function (GF) for the corresponding probability distribution is derived. We consider the case when the GF reduces to that of the negative binomial distribution (NBD), if the fractional derivative is replaced to the ordinary one. The H j moment derived from the GF of the NBD decreases monotonically as the rank j increases. However, the H j moment derived in our approach oscillates, which is contrasted with the case of the NBD. Calculated H j moments are compared with those of charged multiplicities observed in pp-bar, e + e - , and e + p collisions. A phenomenological meaning of introducing the fractional derivative in time variable is discussed

Expansion of the relativistic Fokker-Planck equation including non-linear terms and a non-Maxwellian background

International Nuclear Information System (INIS)

Shkarofsky, I.P.

1997-01-01

The relativistic Fokker-Planck collision term in Braams and Karney [Phys. Fluids B 1, 1355 (1989)] is expanded using Cartesian tensors (equivalent to associated Legendre spherical harmonics) retaining all non-linear terms and an arbitrary zeroth order distribution background. Expressions are given for collision terms between all harmonics and the background distribution in terms of the j and y functions in Braams and Karney. The results reduce to Braams and Karney for the first order harmonic term with a Maxwellian background and to those given by Shkarofsky [Can. J. Phys. 41, 1753 (1963)] in the non-relativistic limit. Expressions for the energy and momentum transfer associated with relativistic Coulomb collisions are given. The fast two dimensional Fokker-Planck solver in Shoucri and Shkarofsky [Comput. Phys. Commun. 82, 287 (1994)] has been extended to include the second order harmonic term. copyright 1997 American Institute of Physics

Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

Directory of Open Access Journals (Sweden)

Shao-Hong Yan

2014-01-01

Full Text Available The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

Fokker-Planck and quasilinear codes

International Nuclear Information System (INIS)

Karney, C.F.F.

1985-11-01

The interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term. Methods for solving this equation on a computer are discussed. 40 refs., 12 figs., 3 tabs

Numerical solution of modified fokker-planck equation with poissonian input

Czech Academy of Sciences Publication Activity Database

Náprstek, Jiří; Král, Radomil

2010-01-01

Roč. 17, 3/4 (2010), s. 251-268 ISSN 1802-1484 R&D Projects: GA AV ČR(CZ) IAA200710805; GA ČR(CZ) GA103/09/0094 Institutional research plan: CEZ:AV0Z20710524 Keywords : Fokker-Planck equation * poisson ian exciation * numerical solution * transition effects Subject RIV: JN - Civil Engineering

Green function of the double-fractional Fokker-Planck equation: Path integral and stochastic differential equations

Science.gov (United States)

Kleinert, H.; Zatloukal, V.

2013-11-01

The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

Solution of the Fokker-Planck equation for axially-channeled relativistic electrons

International Nuclear Information System (INIS)

Muralev, V.A.; Telegin, V.I.

1981-01-01

A method of the two dimensional kinetic equation of the Fokker-Planck type for axially-channeled electrons is proposed. This equation has been obtained recently by Beloshitsky and Kumakhov to describe the diffusion of channeling negative particles over the transverse energy and angular momentum. The results of computation of the dechanneling function of 1 GeV electrons in tungsten are presented. (author)

Single particle dynamics of many-body systems described by Vlasov-Fokker-Planck equations

International Nuclear Information System (INIS)

Frank, T.D.

2003-01-01

Using Langevin equations we describe the random walk of single particles that belong to particle systems satisfying Vlasov-Fokker-Planck equations. In doing so, we show that Haissinski distributions of bunched particles in electron storage rings can be derived from a particle dynamics model

Fermi-Dirac-Fokker-Planck equation : well-posedness & long-time asymptotics

OpenAIRE

Carrillo , José A.; Laurençot , Philippe; Rosado , Jesús

2009-01-01

International audience; A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a con...

Vectorized Fokker-Planck package for the CRAY-1

International Nuclear Information System (INIS)

McCoy, M.G.; Mirin, A.A.; Killeen, J.

1979-08-01

A program for the solution of the time-dependent, two dimensional, nonlinear, multi-species Fokker-Planck equation is described. The programming is written such that the loop structure is highly vectorizable on the CRAY FORTRAN Compiler. A brief discussion of the Fokker-Planck equation itself is followed by a description of the procedure developed to solve the equation efficiently. The Fokker-Planck equation is a second order partial differential equation whose coefficients depend upon moments of the distribution functions. Both the procedure for the calculation of these coefficients and the procedure for the time advancement of the equation itself must be done efficiently if significant overall time saving is to result. The coefficients are calculated in a series of nested loops, while time advancement is accomplished by a choice of either a splitting or an ADI technique. Overall, timing tests show that the vectorized CRAY program realizes up to a factor of 12 advantage over an optimized CDC-7600 program and up to a factor of 365 over a non-vectorized version of the same program on the CRAY

A purely Lagrangian method for the numerical integration of Fokker-Planck equations

International Nuclear Information System (INIS)

Combis, P.; Fronteau, J.

1986-01-01

A new numerical approach to Fokker-Planck equations is presented, in which the integration grid moves according to the solution of a differential system. The method is purely Lagrangian, the mean effect of the diffusion being inserted into the differential system itself

High energy ion range and deposited energy calculation using the Boltzmann-Fokker-Planck splitting of the Boltzmann transport equation

International Nuclear Information System (INIS)

Mozolevski, I.E.

2001-01-01

We consider the splitting of the straight-ahead Boltzmann transport equation in the Boltzmann-Fokker-Planck equation, decomposing the differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. The convergence of implantation profile, nuclear and electronic energy depositions, calculated from the Boltzmann-Fokker-Planck equation, to the respective exact distributions, calculated from Monte-Carlo method, was exanimate in a large-energy interval for various values of splitting parameter and for different ion-target mass relations. It is shown that for the universal potential there exists an optimal value of splitting parameter, for which range and deposited energy distributions, calculated from the Boltzmann-Fokker-Planck equation, accurately approximate the exact distributions and which minimizes the computational expenses

One-Dimensional Fokker-Planck Equation with Quadratically Nonlinear Quasilocal Drift

Science.gov (United States)

Shapovalov, A. V.

2018-04-01

The Fokker-Planck equation in one-dimensional spacetime with quadratically nonlinear nonlocal drift in the quasilocal approximation is reduced with the help of scaling of the coordinates and time to a partial differential equation with a third derivative in the spatial variable. Determining equations for the symmetries of the reduced equation are derived and the Lie symmetries are found. A group invariant solution having the form of a traveling wave is found. Within the framework of Adomian's iterative method, the first iterations of an approximate solution of the Cauchy problem are obtained. Two illustrative examples of exact solutions are found.

Cusping, transport and variance of solutions to generalized Fokker-Planck equations

Science.gov (United States)

Carnaffan, Sean; Kawai, Reiichiro

2017-06-01

We study properties of solutions to generalized Fokker-Planck equations through the lens of the probability density functions of anomalous diffusion processes. In particular, we examine solutions in terms of their cusping, travelling wave behaviours, and variance, within the framework of stochastic representations of generalized Fokker-Planck equations. We give our analysis in the cases of anomalous diffusion driven by the inverses of the stable, tempered stable and gamma subordinators, demonstrating the impact of changing the distribution of waiting times in the underlying anomalous diffusion model. We also analyse the cases where the underlying anomalous diffusion contains a Lévy jump component in the parent process, and when a diffusion process is time changed by an uninverted Lévy subordinator. On the whole, we present a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems. We discuss possible applications in physical experiments, including, with reference to specific examples, the potential for model misclassification and how combinations of our four criteria may be used to overcome this issue.

Time dependent solutions of the Fokker-Planck equation for fast fusion ions

International Nuclear Information System (INIS)

Gnavi, G.; Gratton, F.T.; Heyn, M.

1990-01-01

Approximate time dependent solutions for the Fokker-Planck equation for fast fusion ions from an isotropic, monoenergetic source are presented, for the problem of D - T - He 3 reactions. The equations include the effect of diffusion, which is particularly noticeable in the distribution of particles of lower energy and in the formation of a tail of particles with energy higher than that of the source. (Author)

Fermi-Dirac-Fokker-Planck equation: well-posedness and long-time asymptotics

OpenAIRE

Carrillo, José A.; Laurençot, Philippe; Rosado, Jesús

2008-01-01

A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asym...

Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes

International Nuclear Information System (INIS)

Morel, J.E.

1987-01-01

The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs

A Simple Map Between Fokker-Planck Equation and its Fractional form

International Nuclear Information System (INIS)

Zahran, M.A.; El-Shewy, E.K.

2008-01-01

A simple map between Fokker-Planck Equation (FPE) and its fractional form (FFPE), which recently formulates to describe sub diffusive processes, has been suggested. This connection based on a relation between k-orders for moments of ordinary time domain of FPE and the moments associated with fractional time domain of FFPE . Two classes of special interest of FFPE has been considered to outline this map

Modelling of thermal transport using Fokker-Planck equations in laser produced plasma

International Nuclear Information System (INIS)

Nakarmi, J.J.; Jha, L.N.

1996-12-01

The kinetic equation with Fokker-Planck collision term has been presented to obtain the distribution function in the corona of inertial confinement fusion, in the presence of the self generated magnetic field. The resulting distribution has non-local form with the convolution in Maxwellian. An expression for thermal flux with self generated magnetic field is obtained. (author). 22 refs

Efficient positive, conservative, Maxwellian preserving and implicit difference schemes for the 1-D isotropic Fokker-Planck-Landau equation; Schemas positifs, implicites, conservant l'energie et les etats d'equilibre pour l'equation de Fokker-Planck-Landau isotrope

Energy Technology Data Exchange (ETDEWEB)

Buet, Ch. [CEA Bruyeres-le-Chatel, Dept. Sciences de la Simulation et de l' Information, Service Numerique Environnement et Constantes, 91 (France); Le Thanh, K.C. [CEA Bruyeres-le-Chatel, Service Physique des Plasmas et Electromagnetisme, 91 (France). Dept. de Physique Theorique et Appliquee

2008-07-01

The aim of this paper is to describe the discretization of the Fokker-Planck-Landau (FPL) collision term in the isotropic case, which models the self-collision for the electrons when they are totally isotropized by heavy particles background such as ions. The discussion focuses on schemes, which could preserve positivity, mass, energy and Maxwellian equilibrium. The Chang and Cooper method is widely used by plasma's physicists for the FPL equation (and for Fokker-Planck type equations). We present a new variant that is both positive and conservative contrary to the existing one's. We propose also a non Chang and Cooper 'type scheme on non-uniform grid, which is also both positive, conservative and equilibrium state preserving contrary to existing one's. The case of Coulombian potential is emphasized. We address also the problem of the time discretization. In particular we show how to recast some implicit methods to get band diagonal system and to solve it by direct method with a linear cost. (authors)

Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium.

Science.gov (United States)

Ceccato, Alessandro; Frezzato, Diego

2018-02-14

The chemical Langevin equation and the associated chemical Fokker-Planck equation are well-known continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailed-balanced kinetic schemes. An illustration is given for a model case.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field

Science.gov (United States)

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-01

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

From the nonlinear Fokker-Planck equation to the Vlasov description and back: Confined interacting particles with drag

Science.gov (United States)

Plastino, A. R.; Curado, E. M. F.; Nobre, F. D.; Tsallis, C.

2018-02-01

Nonlinear Fokker-Planck equations endowed with power-law diffusion terms have proven to be valuable tools for the study of diverse complex systems in physics, biology, and other fields. The nonlinearity appearing in these evolution equations can be interpreted as providing an effective description of a system of particles interacting via short-range forces while performing overdamped motion under the effect of an external confining potential. This point of view has been recently applied to the study of thermodynamical features of interacting vortices in type II superconductors. In the present work we explore an embedding of the nonlinear Fokker-Planck equation within a Vlasov equation, thus incorporating inertial effects to the concomitant particle dynamics. Exact time-dependent solutions of the q -Gaussian form (with compact support) are obtained for the Vlasov equation in the case of quadratic confining potentials.

Fokker-Planck-Rosenbluth-type equations for self-gravitating systems in the 1PN approximation

International Nuclear Information System (INIS)

Ramos-Caro, Javier; Gonzalez, Guillermo A

2008-01-01

We present two formulations of Fokker-Planck-Rosenbluth-type (FPR) equations for many-particle self-gravitating systems, with first-order relativistic corrections in the post-Newtonian approach (1PN). The first starts from a covariant Fokker-Planck equation for a simple gas, introduced recently by Chacon-Acosta and Kremer (2007 Phys. Rev. E 76 021201). The second derivation is based on the establishment of an 1PN-BBGKY hierarchy, developed systematically from the 1PN microscopic law of force and using the Klimontovich-Dupree (KD) method. We close the hierarchy by the introduction of a two-point correlation function that describes adequately the relaxation process. This picture reveals an aspect that is not considered in the first formulation: the contribution of ternary correlation patterns to the diffusion coefficients, as a consequence of the nature of 1PN interaction. Both formulations can be considered as a generalization of the equation derived by Rezania and Sobouti (2000 Astron. Astrophys. 354 1110), to stellar systems where the relativistic effects of gravitation play a significant role

FOKKER-PLANCK ANALYSIS OF TRANSVERSE COLLECTIVE INSTABILITIES IN ELECTRON STORAGE RINGS

Energy Technology Data Exchange (ETDEWEB)

Lindberg, R. R.

2017-06-25

We analyze single bunch transverse instabilities due to wakefields using a Fokker-Planck model. We expand on the work of Suzuki [1], writing out the linear matrix equation including chromaticity, both dipolar and quadrupolar transverse wakefields, and the effects of damping and diffusion due to the synchrotron radiation. The eigenvalues and eigenvectors determine the collective stability of the beam, and we show that the predicted threshold current for transverse instability and the profile of the unstable agree well with tracking simulations. In particular, we find that predicting collective stability for high energy electron beams at moderate to large values of chromaticity requires the full Fokker-Planck analysis to properly account for the effects of damping and diffusion due to synchrotron radiation.

Reduced Fokker-Planck models for fast particle distribution across a transition layer of disparate plasma temperatures

Science.gov (United States)

Tang, Xian-Zhu; Berk, H. L.; Guo, Zehua; McDevitt, C. J.

2014-03-01

Across a transition layer of disparate plasma temperatures, the high energy tail of the plasma distribution can have appreciable deviations from the local Maxwellian distribution due to the Knudson layer effect. The Fokker-Planck equation for the tail particle population can be simplified in a series of practically useful limiting cases. The first is the approximation of background Maxwellian distribution for linearizing the collision operator. The second is the supra-thermal particle speed ordering of vTi ≪ v ≪ vTe for the tail ions and vTi ≪ vTe ≪ v for the tail electrons. Keeping both the collisional drag and energy scattering is essential for the collision operator to produce a Maxwellian tail distribution. The Fokker-Planck model for following the tail ion distribution for a given background plasma profile is explicitly worked out for systems of one spatial dimension, in both slab and spherical geometry. A third simplification is an expansion of the tail particle distribution using the spherical harmonics, which are eigenfunctions of the pitch angle scattering operator. This produces a set of coupled Fokker-Planck equations that contain energy-dependent spatial diffusion terms in two coordinates (position and energy), which originate from pitch angle scattering in the original Fokker-Planck equation. It is shown that the well-known diffusive Fokker-Planck model is a poor approximation of the two-mode truncation model, which itself has fundamental deficiency compared with the three-mode truncation model. The cause is the lack of even-symmetry representation in pitch dependence in the two-mode truncation model.

New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators

Directory of Open Access Journals (Sweden)

Hassan Kamil Jassim

2015-01-01

Full Text Available We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations. Illustrative examples are given to show the accuracy and reliable results.

Some exact solutions for a unidimensional fokker-planck equation by using lie symmetries

Directory of Open Access Journals (Sweden)

Hugo Hernán Ortíz-Álvarez

2015-01-01

Full Text Available The Fokker Planck equation appears in the study of diffusion phenomena, stochastics processes and quantum and classical mechanics. A particular case fromthis equation, ut − uxx − xux − u=0, is examined by the Lie group method approach. From the invariant condition it was possible to obtain the infinitesimal generators or vectors associated to this equation, identifying the corresponding symmetry groups. Exact solution were found for each one of this generators and new solution were constructed by using symmetry properties.

Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation

Science.gov (United States)

Sorokin, M. V.; Dubinko, V. I.; Borodin, V. A.

2017-01-01

The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.

A nonlinear Fokker-Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics

Science.gov (United States)

Marin, D.; Ribeiro, M. A.; Ribeiro, H. V.; Lenzi, E. K.

2018-07-01

We investigate the solutions for a set of coupled nonlinear Fokker-Planck equations coupled by the diffusion coefficient in presence of external forces. The coupling by the diffusion coefficient implies that the diffusion of each species is influenced by the other and vice versa due to this term, which represents an interaction among them. The solutions for the stationary case are given in terms of the Tsallis distributions, when arbitrary external forces are considered. We also use the Tsallis distributions to obtain a time dependent solution for a linear external force. The results obtained from this analysis show a rich class of behavior related to anomalous diffusion, which can be characterized by compact or long-tailed distributions.

Stationary solution of the Fokker-Planck equation for linearly coupled motion in an electron storage ring

International Nuclear Information System (INIS)

Chao, A.W.; Lee, M.J.

1975-09-01

Effects upon longitudinal bunch shape in a storage ring due to linear and nonlinear potential can be calculated by finding the stationary solution to the Fokker-Planck equation for the particle distribution. Effects upon transverse bunch shape of a stored electron beam due to photon emissions and damping can be calculated by this method. It has been found that this method can also be used for a case in which the transverse modes of oscillation are coupled to the energy deviation δ. Examples of lattice elements which produce linear coupling between these oscillations are skew quadrupole magnets and solenoid magnets. For the linearly coupled case the stationary solution has been found to be given by exp (ΣΣA/sub ij/ x/sub i/x/sub j/) with x/sub i/ the canonical variables (x,p/sub x/, y, p/sub y/, δ, p/sub δ/) and A /sub ij/ some constants. The solution for the values of A /sub ij/'s will be described in this report. It will be shown that this solution can be expressed in a compact form. For simple cases, this form of solution leads directly to analytic expressions for the values of A /sub ij/'s and the bunch shape can be calculated by integrating the distribution function over some of the coordinates; for the more complex cases, it can be conveniently adapted as an algorithm for numerical evaluation. 16 refs

Analytic solution of the two-dimensional Fokker-Planck equation governing stochastic ion heating by a lower hybrid wave

International Nuclear Information System (INIS)

Malescio, G.

1981-04-01

The two-dimensional Fokker-Planck equation describing the ion motion in a coherent lower hybrid wave above the stochasticity threshold is analytically solved. An expression is given for the steady state power dissipation

Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes

International Nuclear Information System (INIS)

Ortega J, R.; Valle G, E. del

2003-01-01

There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S 4 with expansions of the dispersion cross sections until P 3 order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

KAUST Repository

ARNOLD, ANTON

2012-11-01

We consider the linear WignerFokkerPlanck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for FokkerPlanck type operators in certain weighted L 2-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate. © 2012 World Scientific Publishing Company.

Quantal Brownian Motion from RPA dynamics: The master and Fokker-Planck equations

International Nuclear Information System (INIS)

Yannouleas, C.

1984-05-01

From the purely quantal RPA description of the damped harmonic oscillator and of the corresponding Brownian Motion within the full space (phonon subspace plus reservoir), a master equation (as well as a Fokker-Planck equation) for the reduced density matrix (for the reduced Wigner function, respectively) within the phonon subspace is extracted. The RPA master equation agrees with the master equation derived by the time-dependent perturbative approaches which utilize Tamm-Dancoff Hilbert spaces and invoke the rotating wave approximation. Since the RPA yields a full, as well as a contracted description, it can account for both the kinetic and the unperturbed oscillator momenta. The RPA description of the quantal Brownian Motion contrasts with the descriptions provided by the time perturbative approaches whether they invoke or not the rotating wave approximation. The RPA description also contrasts with the phenomenological phase space quantization. (orig.)

Multi-dimensional Fokker-Planck equation analysis using the modified finite element method

Czech Academy of Sciences Publication Activity Database

Náprstek, Jiří; Král, Radomil

2016-01-01

Roč. 744, č. 1 (2016), č. článku 012177. ISSN 1742-6588. [International Conference on Motion and Vibration Control (MOVIC 2016) /13./ and International Conference on Recent Advances in Structural Dynamics (RASD 2016) /12./. Southampton, 04.07.2016-06.07.2016] R&D Projects: GA ČR(CZ) GP14-34467P; GA ČR(CZ) GA15-01035S Institutional support: RVO:68378297 Keywords : Fokker-Planck equation * finite element method * single degree of freedom systems (SDOF) Subject RIV: JM - Building Engineering http://iopscience.iop.org/article/10.1088/1742-6596/744/1/012177

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

International Nuclear Information System (INIS)

Balazs, N.L.

1978-01-01

In the classical theory of Brownian motion the Langevin equation can be considered as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, the Langevin equation can be reinterpreted as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta one can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. One may associate, according to Weyl, functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions the streaming can be expressed as a continuity equation of the Wigner function. It is found that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation. (Auth.)

Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes; Solucion de la ecuacion de transporte de Boltzmann-Fokker-Planck usando esquemas nodales exponenciales

Energy Technology Data Exchange (ETDEWEB)

Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx

2003-07-01

There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

Numerical Resolution of N-dimensional Fokker-Planck stochastic equations

International Nuclear Information System (INIS)

Garcia-Olivares, R. A.; Munoz Roldan, A.

1992-01-01

This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs

A Fokker-Planck treatment of stochastic particle motion within the framework of a fully coupled 6-dimensional formalism for electron-positron storage rings including classical spin motion in linear approximation

International Nuclear Information System (INIS)

Barber, D.P.; Heinemann, K.; Mais, H.; Ripken, G.

1991-12-01

In the following report we investigate stochastic particle motion in electron-positron storage ring in the framework of a Fokker-Planck treatment. The motion is described by using the canonical variables χ, p χ , z, p z , σ = s - cxt, p σ = ΔE/E 0 of the fully six-dimensional formalism. Thus synchrotron- and betatron-oscillations are treated simultaneously taking into account all kinds of coupling (synchro-betatron coupling and the coupling of the betatron oscillations by skew quadrupoles and solenoids). In order to set up the Fokker-Planck equation, action-angle variables of the linear coupled motion are introduced. The averaged dimensions of the bunch, resulting from radiation damping of the synchro-betatron oscillations and from an excitation of these oscillations by quantum fluctuations, are calculated by solving the Fokker-Planck equation. The surfaces of constant density in the six-dimensional phase space, given by six-dimensional ellipsoids, are determined. It is shown that the motion of such an ellipsoid under the influence of external fields can be described by six generating orbit vectors which may be combined into a six-dimenional matrix B(s). This 'bunch-shape matrix', B(s), contains complete information about the configuration of the bunch. Classical spin diffusion in linear approximation has also been included so that the dependence of the polarization vector on the orbital phase space coordinates can be studied and another derivation of the linearized depolarization time obtained. (orig.)

On the linear discrepancy model and risky shifts in group behavior: a nonlinear Fokker-Planck perspective

International Nuclear Information System (INIS)

Frank, T D

2009-01-01

Using a nonlinear Fokker-Planck perspective we re-formulate the linear discrepancy model proposed by Boster and colleagues that describes the emergence of risky shifts during group decision making. Analytical expressions for the stationary case are derived and risky shifts are obtained by Monte Carlo simulations. Striking similarities with the Kuramoto model for group synchronization are pointed out

On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks

KAUST Repository

Annunziato, Mario

2014-09-01

In the framework of stochastic processes, the connection between the dynamic programming scheme given by the Hamilton-Jacobi-Bellman equation and a recently proposed control approach based on the Fokker-Planck equation is discussed. Under appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the FokkerPlanck formalism allows considering a larger class of objectives. To illustrate the connection between the two control strategies, the cases of an Itō stochastic process and of a piecewise-deterministic process are considered.

Hypocoercivity for linear kinetic equations conserving mass

KAUST Repository

Dolbeault, Jean; Mouhot, Clé ment; Schmeiser, Christian

2015-01-01

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

Hypocoercivity for linear kinetic equations conserving mass

KAUST Repository

Dolbeault, Jean

2015-02-03

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

Simulating transient dynamics of the time-dependent time fractional Fokker-Planck systems

Science.gov (United States)

Kang, Yan-Mei

2016-09-01

For a physically realistic type of time-dependent time fractional Fokker-Planck (FP) equation, derived as the continuous limit of the continuous time random walk with time-modulated Boltzmann jumping weight, a semi-analytic iteration scheme based on the truncated (generalized) Fourier series is presented to simulate the resultant transient dynamics when the external time modulation is a piece-wise constant signal. At first, the iteration scheme is demonstrated with a simple time-dependent time fractional FP equation on finite interval with two absorbing boundaries, and then it is generalized to the more general time-dependent Smoluchowski-type time fractional Fokker-Planck equation. The numerical examples verify the efficiency and accuracy of the iteration method, and some novel dynamical phenomena including polarized motion orientations and periodic response death are discussed.

FIFPC, a fast ion Fokker--Planck code

International Nuclear Information System (INIS)

Fowler, R.H.; Callen, J.D.; Rome, J.A.; Smith, J.

1976-07-01

A computer code is described which solves the Fokker--Planck equation for the velocity space distribution of fast ions injected into a tokamak plasma. The numerical techniques are described and use of the code is outlined. The program is written in FORTRAN IV and is modularized in order to provide greater flexibility to the user. A program listing is provided and the results of sample cases are presented

Efficient statistically accurate algorithms for the Fokker-Planck equation in large dimensions

Science.gov (United States)

Chen, Nan; Majda, Andrew J.

2018-02-01

Solving the Fokker-Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computationally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O (100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6

Massively parallel Fokker-Planck calculations

International Nuclear Information System (INIS)

Mirin, A.A.

1990-01-01

This paper reports that the Fokker-Planck package FPPAC, which solves the complete nonlinear multispecies Fokker-Planck collision operator for a plasma in two-dimensional velocity space, has been rewritten for the Connection Machine 2. This has involved allocation of variables either to the front end or the CM2, minimization of data flow, and replacement of Cray-optimized algorithms with ones suitable for a massively parallel architecture. Calculations have been carried out on various Connection Machines throughout the country. Results and timings on these machines have been compared to each other and to those on the static memory Cray-2. For large problem size, the Connection Machine 2 is found to be cost-efficient

Theoretical background and implementation of the finite element method for multi-dimensional Fokker-Planck equation analysis

Czech Academy of Sciences Publication Activity Database

Král, Radomil; Náprstek, Jiří

2017-01-01

Roč. 113, November (2017), s. 54-75 ISSN 0965-9978 R&D Projects: GA ČR(CZ) GP14-34467P; GA ČR(CZ) GA15-01035S Institutional support: RVO:68378297 Keywords : Fokker-Planck equation * finite element method * simplex element * multi-dimensional problem * non-symmetric operator Subject RIV: JM - Building Engineering OBOR OECD: Mechanical engineering Impact factor: 3.000, year: 2016 https://www.sciencedirect.com/science/ article /pii/S0965997817301904

Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries

CERN Document Server

Nier, Francis

2018-01-01

This article is concerned with the maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic laplacian.

Massively parallel Fokker-Planck code ALLAp

International Nuclear Information System (INIS)

Batishcheva, A.A.; Krasheninnikov, S.I.; Craddock, G.G.; Djordjevic, V.

1996-01-01

The recently developed for workstations Fokker-Planck code ALLA simulates the temporal evolution of 1V, 2V and 1D2V collisional edge plasmas. In this work we present the results of code parallelization on the CRI T3D massively parallel platform (ALLAp version). Simultaneously we benchmark the 1D2V parallel vesion against an analytic self-similar solution of the collisional kinetic equation. This test is not trivial as it demands a very strong spatial temperature and density variation within the simulation domain. (orig.)

Fokker-Planck modeling of pitting corrosion in underground pipelines

Energy Technology Data Exchange (ETDEWEB)

Camacho, Eliana Nogueira [Risco Ambiental Engenharia, Rio de Janeiro, RJ (Brazil); Melo, Paulo F. Frutuoso e [Coordenacao dos Programas de Pos-Graduacao de Engenharia (COPPE/UFRJ), Rio de Janeiro, RJ (Brazil). Programa de Engenharia Nuclear; Saldanha, Pedro Luiz C. [Comissao Nacional de Energia Nuclear (CGRC/CNEN), Rio de Janeiro, RJ (Brazil). Coordenacao Geral de Reatores e Ciclo do Combustivel; Silva, Edson de Pinho da [Universidade Federal Rural do Rio de Janeiro (UFRRJ), Seropedica, RJ (Brazil). Dept. of Physics

2011-07-01

Full text: The stochastic nature of pitting corrosion has been recognized since the 1930s. It has been learned that this damage retains no memory of its past. Instead, the future state is determined only by the knowledge of its present state. This Markovian property that underlies the stochastic process governing pitting corrosion has been explored as a discrete Markovian process by many authors since the beginning of the 1990s for underground pipelines of the oil and gas industries and nuclear power plants. Corrosion is a genuine continuous time and space state Markovian process, so to model it as a discrete time and/or state space is an approximation to the problem. Markovian chains approaches, with an increasing number of states, could involve a large number of parameters, the transition rates between states, to be experimentally determined. Besides, such an increase in the number of states produces matrices with huge dimensions leading to time-consuming computational solutions. Recent approaches involving Markovian discrete process have overcome those difficulties but, on the other hand, a large number of soil and pipe stochastic variables have to be known. In this work we propose a continuous time and space state approach to the evolution of pit corrosion depths in underground pipelines. In order to illustrate the application of the model for defect depth growth a combination of real life data and Monte Carlo simulation was used. The process is described by a Fokker-Planck equation. The Fokker-Planck equation is completely determined by the knowledge of two functions known as the drift and diffusion coefficients. In this work we also show that those functions can be estimated from corrosion depth data from in-line inspections. Some particular forms of drift and diffusion coefficients lead to particular Fokker-Planck equations for which analytical solutions are known, as is the case for the Wiener process, the Ornstein-Uhlenbeck process and the Brownian motion

Steady state solution of the Fokker-Planck equation combined with unidirectional quasilinear diffusion under detailed balance conditions

International Nuclear Information System (INIS)

Hizanidis, K.

1984-04-01

The relativistic collisional Fokker-Planck equation combined with an externally imposed unidirectional quasilinear (rf) diffusion is solved for arbitrary values of rf diffusion coefficient under conditions of detailed balance of the staionary joint distribution involved. The detailed balance condition imposes a restriction on the functional form of the quasilinear diffusion coefficient which might be associated with the existence of a saturated spectrum of fluctuation in a quasilinearly rf-driven plasma

Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma

Science.gov (United States)

Bakhtiyari-Ramezani, M.; Mahmoodi, J.; Alinejad, N.

2015-11-01

In the fusion devices, ions, H atoms, and H2 molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H2 molecules, and desorption of the recombined H2 molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.

Numerical Resolution of N-dimensional Fokker-Planck stochastic equations; Resolucion Numerica de Ecuaciones Estocasticas de tipo Fokker-Planck en Varias Dimensiones

Energy Technology Data Exchange (ETDEWEB)

Garcia-Olivares, R A; Munoz Roldan, A

1992-07-01

This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs.

An approximate factorization procedure for solving nine-point elliptic difference equations. Application for a fast 2-D relativistic Fokker-Planck solver

Energy Technology Data Exchange (ETDEWEB)

Peysson, Y. [Association Euratom-CEA, CEA Grenoble, 38 (France). Dept. de Recherches sur la Fusion Controlee; Choucri, M. [Centre Canadien de Fusion Magnetique, Varennes, PQ (Canada)

1997-09-01

A full implicit numerical procedure based on the use of a nine-point difference operator is presented to solve the two dimensional (2{sub D}) relativistic Fokker-Planck equation for the current drive problem and synergetic effects between the lower hybrid and the electron cyclotron waves in tokamaks. As compared to the standard approach based on the use of a five-point difference operator [M. Shoucri, I. Shkarofsky, Comput. Phys. Comm. 82 (1994) 287], the convergence rate towards the steady state solution may be significantly enhanced with no loss of accuracy on the distribution function. Moreover, it is shown that the numerical stability may be strongly improved without a large degradation of the CPU time consumption as in the five-point scheme, making this approach very attractive for a fast solution of the 2-D Fokker-Planck equation on a fine grid in conjunction with other numerical codes for realistic plasma simulations. This new algorithm, based on an approximate matrix factorization technique, may be applied to all numerical problems with large sets of equations which involve nine-point difference operators. (author) 21 refs.

An approximate factorization procedure for solving nine-point elliptic difference equations. Application for a fast 2-D relativistic Fokker-Planck solver

International Nuclear Information System (INIS)

Peysson, Y.

1997-09-01

A full implicit numerical procedure based on the use of a nine-point difference operator is presented to solve the two dimensional (2 D ) relativistic Fokker-Planck equation for the current drive problem and synergetic effects between the lower hybrid and the electron cyclotron waves in tokamaks. As compared to the standard approach based on the use of a five-point difference operator [M. Shoucri, I. Shkarofsky, Comput. Phys. Comm. 82 (1994) 287], the convergence rate towards the steady state solution may be significantly enhanced with no loss of accuracy on the distribution function. Moreover, it is shown that the numerical stability may be strongly improved without a large degradation of the CPU time consumption as in the five-point scheme, making this approach very attractive for a fast solution of the 2-D Fokker-Planck equation on a fine grid in conjunction with other numerical codes for realistic plasma simulations. This new algorithm, based on an approximate matrix factorization technique, may be applied to all numerical problems with large sets of equations which involve nine-point difference operators. (author)

Proof of the path integral representation of the nonlinear Fokker-Planck equation by means of Fourier series

International Nuclear Information System (INIS)

Dekker, H.

1978-01-01

The lagrangian for the action occurring in the path integral solution of the nonlinear Fokker-Planck equation with constant diffusion function is derived by means of a straightforward Fourier series analysis. In this manner the path between the prepoint and the postpoint in the short time propagator is not restricted a priori to the usually considered straight line. Earlier results by Graham, Stratonovich, Horsthemke and Back, and the author's are recovered and thus put on much safer ground. (Auth.)

A closed-form solution for the two-dimensional Fokker-Planck equation for electron transport in the range of Compton Effect

Energy Technology Data Exchange (ETDEWEB)

Rodriguez, B.D.A. [Universidade Federal Rio Grande do Sul, Programa de Pos-Graduacao em Engenharia Mecanica, Rua Portuguesa 218/304, 90650-12 Porto Alegre, RS (Brazil)], E-mail: barbara.arodriguez@gmail.com; Vilhena, M.T. [Universidade Federal Rio Grande do Sul, Departamento de Matematica Pura e Aplicada, Porto Alegre, RS (Brazil)], E-mail: vilhena@mat.ufrgs.br; Borges, V. [Universidade Federal Rio Grande do Sul, Programa de Pos-Graduacao em Engenharia Mecanica, Rua Portuguesa 218/304, 90650-12 Porto Alegre, RS (Brazil)], E-mail: borges@ufrgs.br; Hoff, G. [Pontificia Universidade Catolica do Rio Grande do Sul, Faculdade de Fisica, Porto Alegre, RS (Brazil)], E-mail: hoff@pucrs.br

2008-05-15

In this paper we solve the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation for charged particles in a rectangular domain. To construct the solution we begin applying the P{sub N} approximation in the angular variable and the Laplace Transform in the x-variable, thus obtaining a first order linear differential equation in y-variable, which the solution is straightforward. The angular flux of electrons and the parameters of the medium are used for the calculation of the energy deposited by the secondary electrons generated by Compton Effect. The remaining effects will not be taken into account. The results will be presented under absorbed energy form in several points of interested. We present numerical simulations and comparisons with results obtained by using Geant4 (version 8) program which applies the Monte Carlo's technique to low energy libraries for a two-dimensional problem assuming the screened Rutherford differential scattering cross-section.

A closed-form solution for the two-dimensional Fokker-Planck equation for electron transport in the range of Compton Effect

International Nuclear Information System (INIS)

Rodriguez, B.D.A.; Vilhena, M.T.; Borges, V.; Hoff, G.

2008-01-01

In this paper we solve the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation for charged particles in a rectangular domain. To construct the solution we begin applying the P N approximation in the angular variable and the Laplace Transform in the x-variable, thus obtaining a first order linear differential equation in y-variable, which the solution is straightforward. The angular flux of electrons and the parameters of the medium are used for the calculation of the energy deposited by the secondary electrons generated by Compton Effect. The remaining effects will not be taken into account. The results will be presented under absorbed energy form in several points of interested. We present numerical simulations and comparisons with results obtained by using Geant4 (version 8) program which applies the Monte Carlo's technique to low energy libraries for a two-dimensional problem assuming the screened Rutherford differential scattering cross-section

Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics

Science.gov (United States)

Colmenares, Pedro J.

2018-05-01

This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.

Studies of parallel algorithms for the solution of a Fokker-Planck equation

International Nuclear Information System (INIS)

Deck, D.; Samba, G.

1995-11-01

The study of laser-created plasmas often requires the use of a kinetic model rather than a hydrodynamic one. This model change occurs, for example, in the hot spot formation in an ICF experiment or during the relaxation of colliding plasmas. When the gradients scalelengths or the size of a given system are not small compared to the characteristic mean-free-path, we have to deal with non-equilibrium situations, which can be described by the distribution functions of every species in the system. We present here a numerical method in plane or spherical 1-D geometry, for the solution of a Fokker-Planck equation that describes the evolution of stich functions in the phase space. The size and the time scale of kinetic simulations require the use of Massively Parallel Computers (MPP). We have adopted a message-passing strategy using Parallel Virtual Machine (PVM)

Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum

Science.gov (United States)

Guarnieri, F.; Moon, W.; Wettlaufer, J. S.

2017-09-01

Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with a negative constant drift, described by a Fokker-Planck equation with a potential V (x ) =-[b ln(x ) +a x ] , for b >0 and a finance. The Bessel-like process we consider can be solved by seeking solutions through an expansion into a complete set of eigenfunctions. The associated imaginary-time Schrödinger equation exhibits a mix of discrete and continuous eigenvalue spectra, corresponding to the quantum Coulomb potential describing the bound states of the hydrogen atom. We present a technique to evaluate the normalization factor of the continuous spectrum of eigenfunctions that relies solely upon their asymptotic behavior. We demonstrate the technique by solving the Brownian motion problem and the Bessel process both with a constant negative drift. We conclude with a comparison to other analytical methods and with numerical solutions.

Finite element method analysis of Fokker-Plank equation in stationary and evolutionary versions

Czech Academy of Sciences Publication Activity Database

Náprstek, Jiří; Král, Radomil

2014-01-01

Roč. 72, June (2014), s. 28-38 ISSN 0965-9978 R&D Projects: GA ČR(CZ) GA103/09/0094; GA AV ČR(CZ) IAA200710902 Institutional support: RVO:68378297 Keywords : Fokker-Planck equation * numerical solution * transition effects * stochastic mechanics * probability density function * non-linear dynamic systems Subject RIV: JM - Building Engineering Impact factor: 1.402, year: 2014 http://www.sciencedirect.com/science/article/pii/S0965997813001142

Fokker-Planck description for the queue dynamics of large tick stocks

Science.gov (United States)

Garèche, A.; Disdier, G.; Kockelkoren, J.; Bouchaud, J.-P.

2013-09-01

Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. “Jump” events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.

Solution of the relativistic 2-D Fokker-Planck equation for LH current drive

International Nuclear Information System (INIS)

Hizanidis, K.; Hewett, D.W.; Bers, A.

1984-03-01

We solve numerically the steady-state two-dimensional relativistic Fokker-Planck equation with strong rf diffusion using spectra relevant to recent experiments in ALCATOR-C. The results (current generated, power dissipated, and the distribution of energetic electrons) are sensitive to the location of the spectrum in momentum space. Relativistic effects play an important role, especially for wide spectra. The dependence on the ionic charge number Z/sub i/ is also investigated. Particular attention is paid to the perpendicular temperature inside the resonant region and beyond, as well as to the angular energetic particle-temperature distribution, T/sub μ/, a function of the pitch angle parameter μ. The dependence of the perpendicular temperature on the location of the spectrum is also investigated analytically with a model based on the method of moments and the results compared with those found numerically

LLE-LLNL progress report on studies in nonlocal heat transport in spherical plasmas using the Fokker-Planck code SPARK

International Nuclear Information System (INIS)

Epperlein, E.M.

1992-01-01

Preliminary 1-D studies of nonlocal heat transport in spherical plasmas based on the Fokker-Planck code SPARK indicate significant levels of electron preheat and radial heat flux across a spherical heat sink surface kept at fixed temperature. However, the diffusive approximation to the Fokker-Planck equation is shown to be particularly sensitive to the nature of the inner surface boundary condition chosen. A suggested remedy is the inclusion of a target capsule in future simulations studies with SPARK

Fokker-Planck modeling of current penetration during electron cyclotron current drive

International Nuclear Information System (INIS)

Merkulov, A.; Westerhof, E.; Schueller, F. C.

2007-01-01

The current penetration during electron cyclotron current drive (ECCD) on the resistive time scale is studied with a Fokker-Planck simulation, which includes a model for the magnetic diffusion that determines the parallel electric field evolution. The existence of the synergy between the inductive electric field and EC driven current complicates the process of the current penetration and invalidates the standard method of calculation in which Ohm's law is simply approximated by j-j cd =σE. Here it is proposed to obtain at every time step a self-consistent approximation to the plasma resistivity from the Fokker-Planck code, which is then used in a concurrent calculation of the magnetic diffusion equation in order to obtain the inductive electric field at the next time step. A series of Fokker-Planck calculations including a self-consistent evolution of the inductive electric field has been performed. Both the ECCD power and the electron density have been varied, thus varying the well known nonlinearity parameter for ECCD P rf [MW/m -3 ]/n e 2 [10 19 m -3 ] [R. W. Harvey et al., Phys. Rev. Lett 62, 426 (1989)]. This parameter turns out also to be a good predictor of the synergetic effects. The results are then compared with the standard method of calculations of the current penetration using a transport code. At low values of the Harvey parameter, the standard method is in quantitative agreement with Fokker-Planck calculations. However, at high values of the Harvey parameter, synergy between ECCD and E parallel is found. In the case of cocurrent drive, this synergy leads to the generation of large amounts of nonthermal electrons and a concomitant increase of the electrical conductivity and current penetration time. In the case of countercurrent drive, the ECCD efficiency is suppressed by the synergy with E parallel while only a small amount of nonthermal electrons is produced

Generalized Fokker-Planck theory for electron and photon transport in biological tissues: application to radiotherapy.

Science.gov (United States)

Olbrant, Edgar; Frank, Martin

2010-12-01

In this paper, we study a deterministic method for particle transport in biological tissues. The method is specifically developed for dose calculations in cancer therapy and for radiological imaging. Generalized Fokker-Planck (GFP) theory [Leakeas and Larsen, Nucl. Sci. Eng. 137 (2001), pp. 236-250] has been developed to improve the Fokker-Planck (FP) equation in cases where scattering is forward-peaked and where there is a sufficient amount of large-angle scattering. We compare grid-based numerical solutions to FP and GFP in realistic medical applications. First, electron dose calculations in heterogeneous parts of the human body are performed. Therefore, accurate electron scattering cross sections are included and their incorporation into our model is extensively described. Second, we solve GFP approximations of the radiative transport equation to investigate reflectance and transmittance of light in biological tissues. All results are compared with either Monte Carlo or discrete-ordinates transport solutions.

SU-E-T-22: A Deterministic Solver of the Boltzmann-Fokker-Planck Equation for Dose Calculation

Energy Technology Data Exchange (ETDEWEB)

Hong, X; Gao, H [Shanghai Jiao Tong University, Shanghai, Shanghai (China); Paganetti, H [Massachusetts General Hospital, Boston, MA (United States)

2015-06-15

Purpose: The Boltzmann-Fokker-Planck equation (BFPE) accurately models the migration of photons/charged particles in tissues. While the Monte Carlo (MC) method is popular for solving BFPE in a statistical manner, we aim to develop a deterministic BFPE solver based on various state-of-art numerical acceleration techniques for rapid and accurate dose calculation. Methods: Our BFPE solver is based on the structured grid that is maximally parallelizable, with the discretization in energy, angle and space, and its cross section coefficients are derived or directly imported from the Geant4 database. The physical processes that are taken into account are Compton scattering, photoelectric effect, pair production for photons, and elastic scattering, ionization and bremsstrahlung for charged particles.While the spatial discretization is based on the diamond scheme, the angular discretization synergizes finite element method (FEM) and spherical harmonics (SH). Thus, SH is used to globally expand the scattering kernel and FFM is used to locally discretize the angular sphere. As a Result, this hybrid method (FEM-SH) is both accurate in dealing with forward-peaking scattering via FEM, and efficient for multi-energy-group computation via SH. In addition, FEM-SH enables the analytical integration in energy variable of delta scattering kernel for elastic scattering with reduced truncation error from the numerical integration based on the classic SH-based multi-energy-group method. Results: The accuracy of the proposed BFPE solver was benchmarked against Geant4 for photon dose calculation. In particular, FEM-SH had improved accuracy compared to FEM, while both were within 2% of the results obtained with Geant4. Conclusion: A deterministic solver of the Boltzmann-Fokker-Planck equation is developed for dose calculation, and benchmarked against Geant4. Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang

A transformed path integral approach for solution of the Fokker-Planck equation

Science.gov (United States)

Subramaniam, Gnana M.; Vedula, Prakash

2017-10-01

A novel path integral (PI) based method for solution of the Fokker-Planck equation is presented. The proposed method, termed the transformed path integral (TPI) method, utilizes a new formulation for the underlying short-time propagator to perform the evolution of the probability density function (PDF) in a transformed computational domain where a more accurate representation of the PDF can be ensured. The new formulation, based on a dynamic transformation of the original state space with the statistics of the PDF as parameters, preserves the non-negativity of the PDF and incorporates short-time properties of the underlying stochastic process. New update equations for the state PDF in a transformed space and the parameters of the transformation (including mean and covariance) that better accommodate nonlinearities in drift and non-Gaussian behavior in distributions are proposed (based on properties of the SDE). Owing to the choice of transformation considered, the proposed method maps a fixed grid in transformed space to a dynamically adaptive grid in the original state space. The TPI method, in contrast to conventional methods such as Monte Carlo simulations and fixed grid approaches, is able to better represent the distributions (especially the tail information) and better address challenges in processes with large diffusion, large drift and large concentration of PDF. Additionally, in the proposed TPI method, error bounds on the probability in the computational domain can be obtained using the Chebyshev's inequality. The benefits of the TPI method over conventional methods are illustrated through simulations of linear and nonlinear drift processes in one-dimensional and multidimensional state spaces. The effects of spatial and temporal grid resolutions as well as that of the diffusion coefficient on the error in the PDF are also characterized.

Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker-Planck equations

International Nuclear Information System (INIS)

Jumarie, Guy

2004-01-01

There are presently two different models of fractional Brownian motions available in the literature: the Riemann-Liouville fractional derivative of white noise on the one hand, and the complex-valued Brownian motion of order n defined by using a random walk in the complex plane, on the other hand. The paper provides a comparison between these two approaches, and in addition, takes this opportunity to contribute some complements. These two models are more or less equivalent on the theoretical standpoint for fractional order between 0 and 1/2, but their practical significances are quite different. Otherwise, for order larger than 1/2, the fractional derivative model has no counterpart in the complex plane. These differences are illustrated by an example drawn from mathematical finance. Taylor expansion of fractional order provides the expression of fractional difference in terms of finite difference, and this allows us to improve the derivation of Fokker-Planck equation and Kramers-Moyal expansion, and to get more insight in their relation with stochastic differential equations of fractional order. In the case of multi-fractal systems, the Fokker-Planck equation can be solved by using path integrals, and the fractional dynamic equations of the state moments of the stochastic system can be easily obtained. By combining fractional derivative and complex white noise of order n, one obtains a family of complex-valued fractional Brownian motions which exhibits long-range dependence. The conclusion outlines suggestions for further research, mainly regarding Lorentz transformation of fractional noises

Dynamic least-squares kernel density modeling of Fokker-Planck equations with application to neural population.

Science.gov (United States)

Shotorban, Babak

2010-04-01

The dynamic least-squares kernel density (LSQKD) model [C. Pantano and B. Shotorban, Phys. Rev. E 76, 066705 (2007)] is used to solve the Fokker-Planck equations. In this model the probability density function (PDF) is approximated by a linear combination of basis functions with unknown parameters whose governing equations are determined by a global least-squares approximation of the PDF in the phase space. In this work basis functions are set to be Gaussian for which the mean, variance, and covariances are governed by a set of partial differential equations (PDEs) or ordinary differential equations (ODEs) depending on what phase-space variables are approximated by Gaussian functions. Three sample problems of univariate double-well potential, bivariate bistable neurodynamical system [G. Deco and D. Martí, Phys. Rev. E 75, 031913 (2007)], and bivariate Brownian particles in a nonuniform gas are studied. The LSQKD is verified for these problems as its results are compared against the results of the method of characteristics in nondiffusive cases and the stochastic particle method in diffusive cases. For the double-well potential problem it is observed that for low to moderate diffusivity the dynamic LSQKD well predicts the stationary PDF for which there is an exact solution. A similar observation is made for the bistable neurodynamical system. In both these problems least-squares approximation is made on all phase-space variables resulting in a set of ODEs with time as the independent variable for the Gaussian function parameters. In the problem of Brownian particles in a nonuniform gas, this approximation is made only for the particle velocity variable leading to a set of PDEs with time and particle position as independent variables. Solving these PDEs, a very good performance by LSQKD is observed for a wide range of diffusivities.

A Fokker-Planck based kinetic model for diatomic rarefied gas flows

Science.gov (United States)

Gorji, M. Hossein; Jenny, Patrick

2013-06-01

A Fokker-Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker-Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau-Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

Automatic mesh refinement and parallel load balancing for Fokker-Planck-DSMC algorithm

Science.gov (United States)

Küchlin, Stephan; Jenny, Patrick

2018-06-01

Recently, a parallel Fokker-Planck-DSMC algorithm for rarefied gas flow simulation in complex domains at all Knudsen numbers was developed by the authors. Fokker-Planck-DSMC (FP-DSMC) is an augmentation of the classical DSMC algorithm, which mitigates the near-continuum deficiencies in terms of computational cost of pure DSMC. At each time step, based on a local Knudsen number criterion, the discrete DSMC collision operator is dynamically switched to the Fokker-Planck operator, which is based on the integration of continuous stochastic processes in time, and has fixed computational cost per particle, rather than per collision. In this contribution, we present an extension of the previous implementation with automatic local mesh refinement and parallel load-balancing. In particular, we show how the properties of discrete approximations to space-filling curves enable an efficient implementation. Exemplary numerical studies highlight the capabilities of the new code.

Fokker-Planck theory of electron cyclotron assisted startup and breakdown in Tokamaks

International Nuclear Information System (INIS)

Fidone, I.; Granata, G.

1993-04-01

The kinetic theory of plasma startup in a tokamak in the presence of electron cyclotron resonance heating is discussed. The linear theory of the X-mode and the upper-hybrid converted mode damping in low density and temperature plasmas are first reviewed. Then, the kinetic equation for the electron velocity distribution is considered, which is determined by the perpendicular electron cyclotron quasilinear diffusion operator, the parallel electric field, elastic and inelastic electron-neutral collisions and various losses. Two different time scales, namely the elastic electron-neutral collision time and the much longer ionization time, are identified. Thus a two time scale ordering procedure is legitimated for which the velocity distribution is determined by the quasilinear diffusion and the electron-neutral collision frequency; the ionization rate is computed using the Fokker-Planck solution for the electron velocity distribution

Linearized gyro-kinetic equation

International Nuclear Information System (INIS)

Catto, P.J.; Tsang, K.T.

1976-01-01

An ordering of the linearized Fokker-Planck equation is performed in which gyroradius corrections are retained to lowest order and the radial dependence appropriate for sheared magnetic fields is treated without resorting to a WKB technique. This description is shown to be necessary to obtain the proper radial dependence when the product of the poloidal wavenumber and the gyroradius is large (k rho much greater than 1). A like particle collision operator valid for arbitrary k rho also has been derived. In addition, neoclassical, drift, finite β (plasma pressure/magnetic pressure), and unperturbed toroidal electric field modifications are treated

Generalized study of the return to equilibrium of a particle in a plasma (Fokker-Planck formalism) (1961); Etude generale du retour a l'equilibre d'une particule au sein d'un plasma (formalisme de Fokker-Planck) (1961)

Energy Technology Data Exchange (ETDEWEB)

Salmon, J [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires; Faculte des Sciences de Caen, 14 (France)

1961-07-01

The author examines the problem of the return to equilibrium of a particle in a plasma and completely explains Fokker-Planck equation. After that, he studies the possibility of interpreting the return of the test particle to Maxwellian distribution, using the development - which is obtained. He discusses the validity limits of the Rosenbluth, MacDonald and Judd approximation. (author) [French] Examinant le probleme du retour a l'equilibre d'une particule test au sein d'un plasma en equilibre, l'auteur cherche a expliciter completement l'expression de l'operateur de Fokker-Planck. Il etudie ensuite les conditions de coherence, c'est-a-dire la possibilite pour le developpement obtenu de traduire le retour de la particule test a l'etat maxwellien et discute des limites de validite de la formule de 'Rosenbluth, Mac Donald et Judd'. (auteur)

New Fokker-Planck derivation of heavy gas models for neutron thermalization

International Nuclear Information System (INIS)

Larsen, E.W.; Williams, M.M.R.

1990-01-01

This paper is concerned with the derivation of new generalized heavy gas models for the infinite medium neutron energy spectrum equation. Our approach is general and can be used to derive improved Fokker-Planck approximations for other types of kinetic equations. In this paper we obtain two distinct heavy gas models, together with estimates for the corresponding errors. The models are shown in a special case to reduce to modified heavy gas models proposed earlier by Corngold (1962). The error estimates show that both of the new models should be more accurate than Corngold's modified heavy gas model, and that the first of the two new models should generally be more accurate than the second. (author)

Fokker-Planck transport in solid state accelerator concepts

International Nuclear Information System (INIS)

Newberger, B.; Tajima, T.

1989-01-01

Particle transport in a crystalline solid under channeling conditions is considered by means of a Fokker-Planck description. The model includes electron multiple scattering, radiation damping and an accelerating electric field. Analytic solutions have been obtained using a harmonic potential model to describe the channeling forces. These solutions will be described

Numerical resolution of the Fokker-Planck equation for non-lineal cases

International Nuclear Information System (INIS)

Mastropiero, Daniel

1997-01-01

The author resolves the Fokker-Plank equation of the statistical thermodynamics of irreversible processes for the flow of particles in a medium, considering two cases: a) The diffusion coefficient of the medium depends on concentration; and b) The medium is unhomogeneous, i.e. the diffusion coefficient varies with the position. Different cases are analyzed and compared

On the Fokker-Planck theory of electron three-body recombination

International Nuclear Information System (INIS)

Sayasov, Yu. S.

1977-01-01

The Fokker-Planck theory of electron three-body recombination based on the concept of electron diffusion along the energy scale in the excited hydrogen-like atoms formed in the recombining plasmas, is extended in several respects. 1) An universal formula for population distribution of the excited atoms in strongly ionized plasmas was found under a sole assumption, that the cross-sections for the inelastic atom-electron collisions are governed by the classical impulse approximation. 2) A general Fokker-Planck theory of the recombination in a slightly ionized, two-temperature plasmas was formulated. The recombination coefficients for such plasmas were shown to possess some peculiar properties in case the electronic temperature differs appreciable from the atomic one. A few limitations of the existing schemas for calculation of the recombination kinetics are briefly discussed. (orig.) [de

A New Fokker-Planck Approach for the Relaxation-driven Evolution of Galactic Nuclei

Science.gov (United States)

Vasiliev, Eugene

2017-10-01

We present an approach for simulating the collisional evolution of spherical isotropic stellar systems based on the one-dimensional Fokker-Planck equation. A novel aspect is that we use the phase volume as the argument of the distribution function instead of the traditionally used energy, which facilitates the solution. The publicly available code PhaseFlow implements a high-accuracy finite-element method for the Fokker-Planck equation, and can handle multiple-component systems, optionally with the central black hole and taking into account loss-cone effects and star formation. We discuss the energy balance in the general setting, and in application to the Bahcall-Wolf cusp around a central black hole, for which we derive a perturbative solution. We stress that the cusp is not a steady-state structure, but rather evolves in amplitude while retaining an approximately ρ \\propto {r}-7/4 density profile. Finally, we apply the method to the nuclear star cluster of the milky Way, and illustrate a possible evolutionary scenario in which a two-component system of lighter main-sequence stars and stellar-mass black holes develops a Bahcall-Wolf cusp in the heavier component and a weaker ρ \\propto {r}-3/2 cusp in the lighter, visible component, over the period of several Gyr. The present-day density profile is consistent with the recently detected mild cusp inside the central parsec, and is weakly sensitive to initial conditions.

A local analytic approach for the fast solution of the Fokker-Planck equation

International Nuclear Information System (INIS)

Sajjadi, S.G.; Nicholas, D.J.

1987-11-01

In this report we describe a method of obtaining a closed form for the Focker-Planck equation rendering it amenable to solution in time-step with a complete hydrodynamic treatment of a plasma. We present a local expression for the heat flux, by solving the Focker-Planck equation for electrons in one space and two velocity dimensions in the presence of a self consistent electronic field. (author)

Development of parallel Fokker-Planck code ALLAp

International Nuclear Information System (INIS)

Batishcheva, A.A.; Sigmar, D.J.; Koniges, A.E.

1996-01-01

We report on our ongoing development of the 3D Fokker-Planck code ALLA for a highly collisional scrape-off-layer (SOL) plasma. A SOL with strong gradients of density and temperature in the spatial dimension is modeled. Our method is based on a 3-D adaptive grid (in space, magnitude of the velocity, and cosine of the pitch angle) and a second order conservative scheme. Note that the grid size is typically 100 x 257 x 65 nodes. It was shown in our previous work that only these capabilities make it possible to benchmark a 3D code against a spatially-dependent self-similar solution of a kinetic equation with the Landau collision term. In the present work we show results of a more precise benchmarking against the exact solutions of the kinetic equation using a new parallel code ALLAp with an improved method of parallelization and a modified boundary condition at the plasma edge. We also report first results from the code parallelization using Message Passing Interface for a Massively Parallel CRI T3D platform. We evaluate the ALLAp code performance versus the number of T3D processors used and compare its efficiency against a Work/Data Sharing parallelization scheme and a workstation version

Supersymmetric quantum mechanics method for the Fokker-Planck equation with applications to protein folding dynamics

Science.gov (United States)

Polotto, Franciele; Drigo Filho, Elso; Chahine, Jorge; Oliveira, Ronaldo Junio de

2018-03-01

This work developed analytical methods to explore the kinetics of the time-dependent probability distributions over thermodynamic free energy profiles of protein folding and compared the results with simulation. The Fokker-Planck equation is mapped onto a Schrödinger-type equation due to the well-known solutions of the latter. Through a semi-analytical description, the supersymmetric quantum mechanics formalism is invoked and the time-dependent probability distributions are obtained with numerical calculations by using the variational method. A coarse-grained structure-based model of the two-state protein Tm CSP was simulated at a Cα level of resolution and the thermodynamics and kinetics were fully characterized. Analytical solutions from non-equilibrium conditions were obtained with the simulated double-well free energy potential and kinetic folding times were calculated. It was found that analytical folding time as a function of temperature agrees, quantitatively, with simulations and experiments from the literature of Tm CSP having the well-known 'U' shape of the Chevron Plots. The simple analytical model developed in this study has a potential to be used by theoreticians and experimentalists willing to explore, quantitatively, rates and the kinetic behavior of their system by informing the thermally activated barrier. The theory developed describes a stochastic process and, therefore, can be applied to a variety of biological as well as condensed-phase two-state systems.

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

Science.gov (United States)

Frank, T. D.

2008-02-01

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.

Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions

International Nuclear Information System (INIS)

Jumarie, Guy

2009-01-01

A probability distribution of fractional (or fractal) order is defined by the measure μ{dx} = p(x)(dx) α , 0 α (D x α h α )f(x) provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier's transform of fractional order using the Mittag-Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker-Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions.

Stationary solutions of linear stochastic delay differential equations: applications to biological systems.

Science.gov (United States)

Frank, T D; Beek, P J

2001-08-01

Recently, Küchler and Mensch [Stochastics Stochastics Rep. 40, 23 (1992)] derived exact stationary probability densities for linear stochastic delay differential equations. This paper presents an alternative derivation of these solutions by means of the Fokker-Planck approach introduced by Guillouzic [Phys. Rev. E 59, 3970 (1999); 61, 4906 (2000)]. Applications of this approach, which is argued to have greater generality, are discussed in the context of stochastic models for population growth and tracking movements.

An adaptive, implicit, conservative, 1D-2V multi-species Vlasov-Fokker-Planck multi-scale solver in planar geometry

Science.gov (United States)

Taitano, W. T.; Chacón, L.; Simakov, A. N.

2018-07-01

We consider a 1D-2V Vlasov-Fokker-Planck multi-species ionic description coupled to fluid electrons. We address temporal stiffness with implicit time stepping, suitably preconditioned. To address temperature disparity in time and space, we extend the conservative adaptive velocity-space discretization scheme proposed in [Taitano et al., J. Comput. Phys., 318, 391-420, (2016)] to a spatially inhomogeneous system. In this approach, we normalize the velocity-space coordinate to a temporally and spatially varying local characteristic speed per species. We explicitly consider the resulting inertial terms in the Vlasov equation, and derive a discrete formulation that conserves mass, momentum, and energy up to a prescribed nonlinear tolerance upon convergence. Our conservation strategy employs nonlinear constraints to enforce these properties discretely for both the Vlasov operator and the Fokker-Planck collision operator. Numerical examples of varying degrees of complexity, including shock-wave propagation, demonstrate the favorable efficiency and accuracy properties of the scheme.

A combination between Laplace transformation technique and numerical approximations to the Fokker-Planck equation solutions

International Nuclear Information System (INIS)

Monticelli, Cintia O.; Wortmann, Sergio; Segatto, Cynthia F.

2005-01-01

In this work is obtained a hybrid solution to the Fokker-Planck equation with energy dependency, very used in ion implantation problems. The main idea relies on the application of Laplace transform in the energy variable, and finite-difference in the spatial variable and in the angular variable. This procedure leads to a symbolic matrix problem for the transformed energy. To solve this system, is needed to do the Laplace inverse of the (sI+A) matrix, where s is a complex parameter, I is the identity matrix and A is a square matrix that was proceeded from the finite-difference in the spatial variable and in the angular variable. The matrix A is not defective, then is taken decomposition of A in a sum of two others matrices, where one is defective. It leads a iterative inversion method, similar the source fixed method combined with the diagonalization method, then is obtained the values to the angular flux. Hereafter we can to determine the energy deposited into the electronic system and in the nuclear system of the target. To comprove the results obtained, the simulation of implantation of B into Si at energies ranging from 1 KeV to 50 MeV was carried out and compared with the results by software SRIM2003. (author)

A fractional Fokker-Planck model for anomalous diffusion

Energy Technology Data Exchange (ETDEWEB)

Anderson, Johan, E-mail: anderson.johan@gmail.com [Department of Earth and Space Sciences, Chalmers University of Technology, SE-412 96 Göteborg (Sweden); Kim, Eun-jin [Department of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH (United Kingdom); Moradi, Sara [Ecole Polytechnique, CNRS UMR7648, LPP, F-91128 Palaiseau (France)

2014-12-15

In this paper, we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality of the stable Lévy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.

Collective motions and non-polynomial lagrangians in Fokker-Planck dynamics

International Nuclear Information System (INIS)

Spina, A.; Vucetich, H.

1986-04-01

A method based on the ideas of collective motion is applied to Fokker-Planck dynamics. The usual diagramatic techniques used in stochastic dynamics are enlarged in order to deal with the non-polynomial Lagrangians that appear in the theory. The technique is tested and applied to the case of a self-sustained sinusoidal oscillator whose statistical behaviour is well understood: the Poincare oscillator. (author)

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

Science.gov (United States)

Herda, Maxime; Rodrigues, L. Miguel

2018-03-01

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L^2 space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ → ∞ to the strongly collisional regime τ → 0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ -dependent constraint on δ ensuring exponential decay with explicit τ -dependent rates towards the stationary solution. In the strongly collisional limit τ → 0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L^2 space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

A Simple Stochastic Differential Equation with Discontinuous Drift

DEFF Research Database (Denmark)

Simonsen, Maria; Leth, John-Josef; Schiøler, Henrik

2013-01-01

In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. We apply two approaches: The Euler-Maruyama method and the Fokker-Planck equation and show that a candidate density function based on the Euler-Maruyama method approximates a candidate density...... function based on the stationary Fokker-Planck equation. Furthermore, we introduce a smooth function which approximates the discontinuous drift and apply the Euler-Maruyama method and the Fokker-Planck equation with this input. The point of departure for this work is a particular SDE with discontinuous...

On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks

KAUST Repository

Annunziato, Mario; Borzì , Alfio; Nobile, Fabio; Tempone, Raul

2014-01-01

appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the FokkerPlanck formalism allows considering a larger class of objectives. To illustrate the connection between the two control

Multi-diffusive nonlinear Fokker–Planck equation

International Nuclear Information System (INIS)

Ribeiro, Mauricio S; Casas, Gabriela A; Nobre, Fernando D

2017-01-01

Nonlinear Fokker–Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker–Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker–Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker–Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution. (paper)

Nonparametric estimates of drift and diffusion profiles via Fokker-Planck algebra.

Science.gov (United States)

Lund, Steven P; Hubbard, Joseph B; Halter, Michael

2014-11-06

Diffusion processes superimposed upon deterministic motion play a key role in understanding and controlling the transport of matter, energy, momentum, and even information in physics, chemistry, material science, biology, and communications technology. Given functions defining these random and deterministic components, the Fokker-Planck (FP) equation is often used to model these diffusive systems. Many methods exist for estimating the drift and diffusion profiles from one or more identifiable diffusive trajectories; however, when many identical entities diffuse simultaneously, it may not be possible to identify individual trajectories. Here we present a method capable of simultaneously providing nonparametric estimates for both drift and diffusion profiles from evolving density profiles, requiring only the validity of Langevin/FP dynamics. This algebraic FP manipulation provides a flexible and robust framework for estimating stationary drift and diffusion coefficient profiles, is not based on fluctuation theory or solved diffusion equations, and may facilitate predictions for many experimental systems. We illustrate this approach on experimental data obtained from a model lipid bilayer system exhibiting free diffusion and electric field induced drift. The wide range over which this approach provides accurate estimates for drift and diffusion profiles is demonstrated through simulation.

Fokker-Planck simulation study of Alfven eigenmode burst

International Nuclear Information System (INIS)

Todo, Y.; Watanabe, T.; Park, Hyoung-Bin; Sato, T.

2001-01-01

Recurrent bursts of toroidicity-induced Alfven eigenmodes (TAEs) are reproduced with a Fokker-Planck-magnetohydrodynamic simulation where a fast-ion source and slowing down are incorporated self-consistently. The bursts take place at regular time intervals and the behaviors of all the TAEs are synchronized. The fast-ion transport due to TAE activity spatially broadens the classical fast-ion distribution and significantly reduces its peak value. Only a small change of the distribution takes place with each burst, leading to loss of a small fraction of the fast ions. The system stays close to the marginal stability state established through the interplay of the fast-ion source, slowing down, and TAE activity. (author)

About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation; Sur les schemas positifs, conservant l'energie et les etats d'equilibre pour l'equation de Fokker-Planck-Landau isotrope

Energy Technology Data Exchange (ETDEWEB)

Buet, Ch. [CEA Bruyeres-le-Chatel, Dept. des Sciences de la Simulation et de l' Information, 91 (France); Le Thanh, K.C. [CEA Bruyeres-le-Chatel, Dept. de Physique Theorique et Appliquee, 91 (France)

2006-07-01

The aim of this paper is to describe the discretization of the Fokker-Planck-Landau (FPL) collision term in the isotropic case which models the self collision for the electrons when they are totally isotropized by heavy particles background such as ions. The discussion focus on schemes which could preserve positivity, mass, energy and Maxwellian equilibrium. First, we analyze in detail the popular Chang and Cooper method for this non-linear collision term: derivation, conservation and positivity properties. We show that some variants of this method, based on the drift-diffusion form of the FPL operator, could not be positive or could not conserve the energy. We present a new variant of the Chang and Cooper method derived from the Landau form that is both positive and conservative. We also propose two new alternatives and simpler schemes for the FPL operator which show that the Chang and Cooper method is not the only way to construct positive, energy conservative and equilibrium state preserving schemes for this operator. For all these schemes, we explain clearly the properties of conservation of the density and the energy, the positivity of the solution and the conservation of the equilibrium states, or their lack. The case of Maxwellian and Coulombian potentials are emphasized. (authors)

Quasilinear simulation of auroral kilometric radiation by a relativistic Fokker-Planck code

International Nuclear Information System (INIS)

Matsuda, Y.

1991-01-01

An intense terrestrial radiation called the auroral kilometric radiation (AKR) is believed to be generated by cyclotron maser instability. We study a quasilinear evolution of this instability by means of a two-dimensional relativistic Fokker-Planck code which treats waves and distributions self-consistently, including radiation loss and electron source and sink. We compare the distributions and wave amplitude with spacecraft observations to elucidate physical processes involved. 3 refs., 1 fig

High-field transport of electrons and radiative effects using coupled force-balance and Fokker-Planck equations beyond the relaxation-time approximation

International Nuclear Information System (INIS)

Huang, Danhong; Apostolova, T.; Alsing, P.M.; Cardimona, D.A.

2004-01-01

The dynamics of a many-electron system under both dc and infrared fields is separated into a center-of-mass and a relative motion. The first-order force-balance equation is employed for the slow center-of-mass motion of electrons, and the Fokker-Planck equation is used for the ultrafast relative scattering motion of degenerate electrons. This approach allows us to include the anisotropic energy-relaxation process which has been neglected in the energy-balance equation in the past. It also leads us to include the anisotropic coupling to the incident infrared field with different polarizations. Based on this model, the transport of electrons is explored under strong dc and infrared fields by going beyond the relaxation-time approximation. The anisotropic dependence of the electron distribution function on the parallel and perpendicular kinetic energies of electrons is displayed with respect to the dc field direction, and the effect of anisotropic coupling to an incident infrared field with polarizations parallel and perpendicular to the applied dc electric field is shown. The heating of electrons is more accurately described beyond the energy-balance equation with the inclusion of an anisotropic coupling to the infrared field. The drift velocity of electrons is found to increase with the amplitude of the infrared field due to a suppressed momentum-relaxation process (or frictional force) under parallel polarization but decreases with the amplitude due to an enhanced momentum-relaxation process under perpendicular polarization

An application of the ideas of collective motion to Fokker-Planck dynamics

International Nuclear Information System (INIS)

Spina, A.

1985-08-01

The implementation of ideas and techniques of field theory to statistical physics have proved invaluable both in deepening our understanding in this second area and as a powerful computational tool. In this paper we analyze some aspects of the application to Fokker-Planck dynamics for the case of self-sustained oscillators driven by white noise of a concept that has been found fruitful in quantum field theory, namely the collective coordinate method. (author)

Fully non-linear multi-species Fokker-Planck-Landau collisions for gyrokinetic particle-in-cell simulations of fusion plasma

Science.gov (United States)

Hager, Robert; Yoon, E. S.; Ku, S.; D'Azevedo, E. F.; Worley, P. H.; Chang, C. S.

2015-11-01

We describe the implementation, and application of a time-dependent, fully nonlinear multi-species Fokker-Planck-Landau collision operator based on the single-species work of Yoon and Chang [Phys. Plasmas 21, 032503 (2014)] in the full-function gyrokinetic particle-in-cell codes XGC1 [Ku et al., Nucl. Fusion 49, 115021 (2009)] and XGCa. XGC simulations include the pedestal and scrape-off layer, where significant deviations of the particle distribution function from a Maxwellian can occur. Thus, in order to describe collisional effects on neoclassical and turbulence physics accurately, the use of a non-linear collision operator is a necessity. Our collision operator is based on a finite volume method using the velocity-space distribution functions sampled from the marker particles. Since the same fine configuration space mesh is used for collisions and the Poisson solver, the workload due to collisions can be comparable to or larger than the workload due to particle motion. We demonstrate that computing time spent on collisions can be kept affordable by applying advanced parallelization strategies while conserving mass, momentum, and energy to reasonable accuracy. We also show results of production scale XGCa simulations in the H-mode pedestal and compare to conventional theory. Work supported by US DOE OFES and OASCR.

Application of the Fokker-Plank-Kolmogorov equation for affluence forecast to hydropower reservoirs (Betania Case)

International Nuclear Information System (INIS)

Dominguez Calle, Efrain Antonio

2004-01-01

This paper shows a modeling technique to forecast probability density curves for the flows that represent the monthly affluence to hydropower reservoirs. Briefly, the factors that require affluence forecast in terms of probabilities, the ranges of existing forecast methods as well as the contradiction between those techniques and the real requirements of decision-making procedures are pointed out. The mentioned contradiction is resolved applying the Fokker-Planck-Kolmogorov equation that describes the time evolution of a stochastic process that can be considered as markovian. We show the numerical scheme for this equation, its initial and boundary conditions, and its application results in the case of Betania's reservoir

Finite elements for partial differential equations: An introductory survey

International Nuclear Information System (INIS)

Succi, S.

1988-03-01

After presentation of the basic ideas behind the theory of the Finite Element Method, the application of the method to three equations of particular interest in Physics and Engineering is discussed in some detail, namely, a one-dimensional Sturm-Liouville problem, a two-dimensional linear Fokker-Planck equation and a two-dimensional nonlinear Navier-Stokes equation. 6 refs, 8 figs

Numerical evaluation of path-integral solutions to Fokker-Planck equations. II. Restricted stochastic processes

International Nuclear Information System (INIS)

Wehner, M.F.

1983-01-01

A path-integral solution is derived for processes described by nonlinear Fokker-Plank equations together with externally imposed boundary conditions. This path-integral solution is written in the form of a path sum for small time steps and contains, in addition to the conventional volume integral, a surface integral which incorporates the boundary conditions. A previously developed numerical method, based on a histogram representation of the probability distribution, is extended to a trapezoidal representation. This improved numerical approach is combined with the present path-integral formalism for restricted processes and is show t give accurate results. 35 refs., 5 figs

Numerical resolution of N-dimensional Fokker-Plank stochastic equations

International Nuclear Information System (INIS)

Garcia-Olivares, A.; Muoz, A.

1992-01-01

This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. the input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetics, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (author) 21 fig. 16 ref

3-D Ray-tracing and 2-D Fokker-Planck Simulations of Radiofrequency Application to Tokamak Plasmas

International Nuclear Information System (INIS)

Cardinali, A.; Paoletti, F.; Bernabei, S.

1999-01-01

A state of the art numerical tool has been developed to simulate the propagation and the absorption of coexisting different types of waves in a tokamak geometry. The code includes a numerical solution of the three-dimensional (R, Z, Φ) toroidal wave equation for the electric field of the different waves in the WKBJ approximation. At each step of integration, the two-dimensional (v parallel, v perpendicular) Fokker-Planck equation is solved in the presence of quasilinear diffusion coefficients. The electron Landau damping of the waves is modeled taking into account the interaction of the wave electric fields with the quasilinearly modified distribution function. Consistently, the code calculates the radial profiles of non-inductively generated current densities, the transmitted power traces and the total power damping curves. Synergistic effects among the different type of waves (e.g., lower hybrid and ion Bernstein waves) are studied through the separation of the contributions of the single wave from the effects due to their coexistence

On self-consistent ray-tracing and Fokker-Planck modeling of the hard X-ray emission during lower-hybrid current driven in Tokamaks

International Nuclear Information System (INIS)

Bizarro, J.P.; Peysson, Y.; Bonoli, P.T.; Carrasco, J.; Dudok de Wit, T.; Fuchs, V.; Hoang, G.T.; Litaudon, X.; Moreau, D.; Pocheau, C.; Shkarofsky, I.P.

1993-04-01

A detailed investigation is presented on the ability of combined ray-tracing and Fokker-Planck calculations to predict the hard x-ray (HXR) emission during lower-hybrid (LH) current drive in tokamaks when toroidally induced-ray-stochasticity is important. A large number of rays is used and the electron distribution function is obtained by self-consistently iterating the appropriate LH power deposition and Fokker-Planck calculations. Most of the experimentally observed features of the HXR emission are correctly predicted. It is found that corrections due to radial diffusion of suprathermal electrons and to radiation scattering by the inner wall can be significant

Analysis of discrete reaction-diffusion equations for autocatalysis and continuum diffusion equations for transport

Energy Technology Data Exchange (ETDEWEB)

Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)

2013-01-01

In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.

Stochastic quantum inflation for a canonical scalar field with linear self-interaction potential

Energy Technology Data Exchange (ETDEWEB)

Panotopoulos, Grigoris [CENTRA, Instituto Superior Tecnico, Universidade de Lisboa, Lisboa (Portugal)

2017-10-15

We apply Starobinsky's formalism of stochastic inflation to the case of a massless minimally coupled scalar field with linear self-interaction potential. We solve the corresponding Fokker-Planck equation exactly, and we obtain analytical expressions for the stochastic expectation values. (orig.)

Fokker-action principle for a system of particles interacting through a linear potential

International Nuclear Information System (INIS)

Rivacoba, A.

1984-01-01

A Fokker-action principle for a system of scalar particles interacting through their time-symmetric relativistic generalization of linear potential is obtained. From this action, motion equations and conservation laws for the total energy and angular momentum of the system, in which field contributions are included, are derived. These equations are exactly applied to the problem suggested by Schild of two particles moving in circular concentric orbits

When is quasi-linear theory exact. [particle acceleration

Science.gov (United States)

Jones, F. C.; Birmingham, T. J.

1975-01-01

We use the cumulant expansion technique of Kubo (1962, 1963) to derive an integrodifferential equation for the average one-particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the equation for this function degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory only for this limited class of fluctuations.

Stationarity-conservation laws for fractional differential equations with variable coefficients

International Nuclear Information System (INIS)

Klimek, Malgorzata

2002-01-01

In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity-conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker-Planck equation in (1+1)- and (d+1)-dimensional space and the fractional Klein-Kramers equation. (author)

Stationarity-conservation laws for fractional differential equations with variable coefficients

Energy Technology Data Exchange (ETDEWEB)

Klimek, Malgorzata [Institute of Mathematics and Computer Science, Technical University of Czestochowa, Czestochowa (Poland)

2002-08-09

In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity-conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker-Planck equation in (1+1)- and (d+1)-dimensional space and the fractional Klein-Kramers equation. (author)

Fokker--Planck/transport analyses of fusion plasmas in contemporary beam-driven tokamaks

International Nuclear Information System (INIS)

Mirin, A.A.; McCoy, M.G.; Killeen, J.; Rensink, M.E.; Shumaker, D.E.; Jassby, D.L.; Post, D.E.

1978-04-01

The properties of deuterium plasmas in experimental tokamaks heated and fueled by intense neutral-beam injection are evaluated with a Fokker-Planck/radial transport code coupled with a Monte Carlo neutrals treatment. Illustrative results are presented for the Poloidal Divertor Experiment at PPPL as a function of beam power and plasma recycling coefficient, R/sub c/. When P/sub beam/ = 8 MW at E/sub b/ = 60 keV, and R/sub c/ = 0.2, then approximately 0.5, [ 2 / 3 ] = 22 keV approximately 6 , and the D-D neutron intensity is 10 16 n/sec

FOKN: a relativistic Fokker-Planck code with large angle scattering and radiation losses

International Nuclear Information System (INIS)

Zimmerman, G.; Scharlemann, T.; Wood, L.; Weaver, T.; Chu, T.; Lee, G.

1976-07-01

FOKN is a computer code which employs a relativistic Fokker-Planck algorithm to evolve the distribution functions of the various mutually interacting components of a multi-species plasma forward in time, with the optional addition of high angle, large energy and momentum transfer interactions between the various charged species of the plasma. As a computational expediency, the latter processes are handled by transfer matrices which are generated separately by another code, RNUX, so that once specific transfer matrices are generated, they can be used over and over by FOKN provided the group structures are compatible

Correlated Levy Noise in Linear Dynamical Systems

International Nuclear Information System (INIS)

Srokowski, T.

2011-01-01

Linear dynamical systems, driven by a non-white noise which has the Levy distribution, are analysed. Noise is modelled by a specific stochastic process which is defined by the Langevin equation with a linear force and the Levy distributed symmetric white noise. Correlation properties of the process are discussed. The Fokker-Planck equation driven by that noise is solved. Distributions have the Levy shape and their width, for a given time, is smaller than for processes in the white noise limit. Applicability of the adiabatic approximation in the case of the linear force is discussed. (author)

Strong plasma shock structures based on the Navier--Stokes equations

International Nuclear Information System (INIS)

Abe, K.

1975-01-01

The structure of a plasma collisional shock wave is examined on the basis of the Navier--Stokes equations and simultaneously on the basis of the Fokker--Planck equation. The resultant structures are compared to check the validity of the Navier--Stokes equations applied to the structures of strong shock waves. The Navier--Stokes equations give quite correct structures for weak shock waves. For the strong shock waves, the detailed structures obtained from the Navier--Stokes equations differ from the results of the Fokker--Planck equation, but the shock thicknesses of the two shock waves are in relatively close agreement

Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment

Science.gov (United States)

Ge, Wenchao; Rodenburg, Brandon; Bhattacharya, M.

2016-08-01

Optically levitated nanoparticles have recently emerged as versatile platforms for investigating macroscopic quantum mechanics and enabling ultrasensitive metrology. In this paper we theoretically consider two damping regimes of an optically levitated nanoparticle cooled by cavityless parametric feedback. Our treatment is based on a generalized Fokker-Planck equation derived from the quantum master equation presented recently and shown to agree very well with experiment [B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, Quantum model of cooling and force sensing with an optically trapped nanoparticle, Optica 3, 318 (2016), 10.1364/OPTICA.3.000318]. For low damping, we find that the resulting Wigner function yields the single-peaked oscillator position distribution and recovers the appropriate energy distribution derived earlier using a classical theory and verified experimentally [J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state, Nat. Nano. 9, 358 (2014), 10.1038/nnano.2014.40]. For high damping, in contrast, we predict a double-peaked position distribution, which we trace to an underlying bistability induced by feedback. Unlike in cavity-based optomechanics, stochastic processes play a major role in determining the bistable behavior. To support our conclusions, we present analytical expressions as well as numerical simulations using the truncated Wigner function approach. Our work opens up the prospect of developing bistability-based devices, characterization of phase-space dynamics, and investigation of the quantum-classical transition using levitated nanoparticles.

Fokker-Planck Modelling of Delayed Loss of Charged Fusion Products in TFTR

International Nuclear Information System (INIS)

Edenstrasser, J.W.; Goloborod'ko, V.Ya.; Reznik, S.N.; Yavorskij, V.A.; Zweben, S.

1998-01-01

The results of a Fokker-Planck simulation of the ripple-induced loss of charged fusion products in the Tokamak Fusion Test Reactor (TFTR) are presented. It is shown that the main features of the measured ''delayed loss'' of partially thermalized fusion products, such as the differences between deuterium-deuterium and deuterium-tritium discharges, the plasma current and major radius dependencies, etc., are in satisfactory agreement with the classical collisional ripple transport mechanism. The inclusion of the inward shift of the vacuum flux surfaces turns out to be necessary for an adequate and consistent explanation of the origin of the partially thermalized fusion product loss to the bottom of TFTR

Quantum Stochastic Trajectories: The Fokker-Planck-Bohm Equation Driven by the Reduced Density Matrix.

Science.gov (United States)

Avanzini, Francesco; Moro, Giorgio J

2018-03-15

The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.

Long-Time Dynamic Response and Stochastic Resonance of Subdiffusive Overdamped Bistable Fractional Fokker-Planck Systems

International Nuclear Information System (INIS)

Yan-Mei, Kang; Yao-Lin, Jiang

2008-01-01

To explore the influence of anomalous diffusion on stochastic resonance (SR) more deeply and effectively, the method of moments is extended to subdiffusive overdamped bistable fractional Fokker-Planck systems for calculating the long-time linear dynamic response. It is found that the method of moments attains high accuracy with the truncation order N = 10, and in normal diffusion such obtained spectral amplification factor (SAF) of the first-order harmonic is also confirmed by stochastic simulation. Observing the SAF of the odd-order harmonics we find some interesting results, i.e. for smaller driving frequency the decrease of sub diffusion exponent inhibits the stochastic resonance (SR), while for larger driving frequency the decrease of sub diffusion exponent enhances the second SR peak, but the first one vanishes and a double SR is induced in the third-order harmonic at the same time. These observations suggest that the anomalous diffusion has important influence on the bistable dynamics

Fokker-Planck description of the scattering of radio frequency waves at the plasma edge

International Nuclear Information System (INIS)

Hizanidis, Kyriakos; Kominis, Yannis; Tsironis, Christos; Ram, Abhay K.

2010-01-01

In magnetic fusion devices, radio frequency (rf) waves in the electron cyclotron (EC) and lower hybrid (LH) range of frequencies are being commonly used to modify the plasma current profile. In ITER, EC waves are expected to stabilize the neoclassical tearing mode (NTM) by providing current in the island region [R. Aymar et al., Nucl. Fusion 41, 1301 (2001)]. The appearance of NTMs severely limits the plasma pressure and leads to the degradation of plasma confinement. LH waves could be used in ITER to modify the current profile closer to the edge of the plasma. These rf waves propagate from the excitation structures to the core of the plasma through an edge region, which is characterized by turbulence--in particular, density fluctuations. These fluctuations, in the form of blobs, can modify the propagation properties of the waves by refraction. In this paper, the effect on rf due to randomly distributed blobs in the edge region is studied. The waves are represented as geometric optics rays and the refractive scattering from a distribution of blobs is formulated as a Fokker-Planck equation. The scattering can have two diffusive effects--one in real space and the other in wave vector space. The scattering can modify the trajectory of rays into the plasma and it can affect the wave vector spectrum. The refraction of EC waves, for example, could make them miss the intended target region where the NTMs occur. The broadening of the wave vector spectrum could broaden the wave generated current profile. The Fokker-Planck formalism for diffusion in real space and wave vector space is used to study the effect of density blobs on EC and LH waves in an ITER type of plasma environment. For EC waves the refractive effects become important since the distance of propagation from the edge to the core in ITER is of the order of a meter. The diffusion in wave vector space is small. For LH waves the refractive effects are insignificant but the diffusion in wave vector space is

Electron acceleration by an obliquely propagating electromagnetic wave in the regime of validity of the Fokker-Planck-Kolmogorov approach

Science.gov (United States)

Hizanidis, Kyriakos; Vlahos, L.; Polymilis, C.

1989-01-01

The relativistic motion of an ensemble of electrons in an intense monochromatic electromagnetic wave propagating obliquely in a uniform external magnetic field is studied. The problem is formulated from the viewpoint of Hamiltonian theory and the Fokker-Planck-Kolmogorov approach analyzed by Hizanidis (1989), leading to a one-dimensional diffusive acceleration along paths of constant zeroth-order generalized Hamiltonian. For values of the wave amplitude and the propagating angle inside the analytically predicted stochastic region, the numerical results suggest that the diffusion probes proceeds in stages. In the first stage, the electrons are accelerated to relatively high energies by sampling the first few overlapping resonances one by one. During that stage, the ensemble-average square deviation of the variable involved scales quadratically with time. During the second stage, they scale linearly with time. For much longer times, deviation from linear scaling slowly sets in.

THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

KAUST Repository

ARNOLD, ANTON; GAMBA, IRENE M.; GUALDANI, MARIA PIA; MISCHLER, STÉ PHANE; MOUHOT, CLEMENT; SPARBER, CHRISTOF

2012-01-01

solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for FokkerPlanck type operators in certain weighted L 2-spaces. In addition we show that the steady state corresponds to a positive density

Path integral solutions of the master equation. [Lagrangian function, Ehrenfest-type theorem, Cauchy method, inverse functions

Energy Technology Data Exchange (ETDEWEB)

Etim, E; Basili, C [Rome Univ. (Italy). Ist. di Matematica

1978-08-21

The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation the authors reproduce the result obtained by integrating over Fourier series coefficients and by other methods.

Vlasov-Fokker-Planck modeling of magnetized plasma

Energy Technology Data Exchange (ETDEWEB)

Thomas, Alexander [Univ. of Michigan, Ann Arbor, MI (United States)

2016-08-01

Understanding the magnetic fields that can develop in high-power-laser interactions with solid-density plasma is important because such fields significantly modify both the magnitude and direction of electron heat fluxes. The dynamics of such fields evidently have consequences for inertial fusion energy applications, as the coupling of the laser beams with the walls or pellet and the development of temperature inhomogeneities are critical to the uniformity of the implosion and potentially the success of, for example, the National Ignition Facility. To study these effects, we used the code Impacta, a two-dimensional, fully implicit, Vlasov-Fokker-Planck code with self-consistent magnetic fields and a hydrodynamic ion model, designed for nanosecond time-scale laser-plasma interactions. Heat-flux effects in Ohm’s law under non-local conditions was investigated; physics that is not well captured by standard numerical models but is nevertheless important in fusion-related scenarios. Under such conditions there are numerous interesting physical effects, such as collisional magnetic instabilities, amplification of magnetic fields, re-emergence of non-locality through magnetic convection, and reconnection of magnetic field lines and redistribution of thermal energy. In this project highlights included the first full-scale kinetic simulations of a magnetized hohlraum and the discovery of a new magnetic reconnection mechanism, as well as a completed PhD thesis and the production of a new code for Inertial Fusion research.

Vlasov-Fokker-Planck modeling of magnetized plasma

International Nuclear Information System (INIS)

Thomas, Alexander

2016-01-01

Understanding the magnetic fields that can develop in high-power-laser interactions with solid-density plasma is important because such fields significantly modify both the magnitude and direction of electron heat fluxes. The dynamics of such fields evidently have consequences for inertial fusion energy applications, as the coupling of the laser beams with the walls or pellet and the development of temperature inhomogeneities are critical to the uniformity of the implosion and potentially the success of, for example, the National Ignition Facility. To study these effects, we used the code Impacta, a two-dimensional, fully implicit, Vlasov-Fokker-Planck code with self-consistent magnetic fields and a hydrodynamic ion model, designed for nanosecond time-scale laser-plasma interactions. Heat-flux effects in Ohm's law under non-local conditions was investigated; physics that is not well captured by standard numerical models but is nevertheless important in fusion-related scenarios. Under such conditions there are numerous interesting physical effects, such as collisional magnetic instabilities, amplification of magnetic fields, re-emergence of non-locality through magnetic convection, and reconnection of magnetic field lines and redistribution of thermal energy. In this project highlights included the first full-scale kinetic simulations of a magnetized hohlraum and the discovery of a new magnetic reconnection mechanism, as well as a completed PhD thesis and the production of a new code for Inertial Fusion research.

Using some results about the Lie evolution of differential operators to obtain the Fokker-Planck equation for non-Hamiltonian dynamical systems of interest

Science.gov (United States)

Bianucci, Marco

2018-05-01

Finding the generalized Fokker-Planck Equation (FPE) for the reduced probability density function of a subpart of a given complex system is a classical issue of statistical mechanics. Zwanzig projection perturbation approach to this issue leads to the trouble of resumming a series of commutators of differential operators that we show to correspond to solving the Lie evolution of first order differential operators along the unperturbed Liouvillian of the dynamical system of interest. In this paper, we develop in a systematic way the procedure to formally solve this problem. In particular, here we show which the basic assumptions are, concerning the dynamical system of interest, necessary for the Lie evolution to be a group on the space of first order differential operators, and we obtain the coefficients of the so-evolved operators. It is thus demonstrated that if the Liouvillian of the system of interest is not a first order differential operator, in general, the FPE structure breaks down and the master equation contains all the power of the partial derivatives, up to infinity. Therefore, this work shed some light on the trouble of the ubiquitous emergence of both thermodynamics from microscopic systems and regular regression laws at macroscopic scales. However these results are very general and can be applied also in other contexts that are non-Hamiltonian as, for example, geophysical fluid dynamics, where important events, like El Niño, can be considered as large time scale phenomena emerging from the observation of few ocean degrees of freedom of a more complex system, including the interaction with the atmosphere.

A Priori Regularity of Parabolic Partial Differential Equations

KAUST Repository

Berkemeier, Francisco

2018-01-01

In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular

Stochastic substitute for coupled rate equations in the modeling of highly ionized transient plasmas

International Nuclear Information System (INIS)

Eliezer, S.; Falquina, R.; Minguez, E.

1994-01-01

Plasmas produced by intense laser pulses incident on solid targets often do not satisfy the conditions for local thermodynamic equilibrium, and so cannot be modeled by transport equations relying on equations of state. A proper description involves an excessively large number of coupled rate equations connecting many quantum states of numerous species having different degrees of ionization. Here we pursue a recent suggestion to model the plasma by a few dominant states perturbed by a stochastic driving force. The driving force is taken to be a Poisson impulse process, giving a Langevin equation which is equivalent to a Fokker-Planck equation for the probability density governing the distribution of electron density. An approximate solution to the Langevin equation permits calculation of the characteristic relaxation rate. An exact stationary solution to the Fokker-Planck equation is given as a function of the strength of the stochastic driving force. This stationary solution is used, along with a Laplace transform, to convert the Fokker-Planck equation to one of Schroedinger type. We consider using the classical Hamiltonian formalism and the WKB method to obtain the time-dependent solution

A Priori Regularity of Parabolic Partial Differential Equations

KAUST Repository

Berkemeier, Francisco

2018-05-13

In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions. First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations. The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.

Similarity solutions of the Fokker–Planck equation with time-dependent coefficients

International Nuclear Information System (INIS)

Lin, W.-T.; Ho, C.-L.

2012-01-01

In this work, we consider the solvability of the Fokker–Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker–Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker–Planck equations are presented, and their properties analyzed. - Highlights: ► Scaling form of the Fokker–Planck equation with time-dependent drift and diffusion coefficients is derived. ► Exact similarity solution of the Fokker–Planck equation is given in closed forms. ► New examples of Fokker–Planck equations exactly solvable by similarity methods are discussed.

Fokker-Planck simulations of knock-on electron runaway avalanche and bursts in tokamaks

International Nuclear Information System (INIS)

Chiu, S.C.; Rosenbluth, M.N.; Harvey, R.W.; Chan, V.S.

1998-01-01

The avalanche of runaway electrons in an ohmic tokamak plasma triggered by knock-on collisions of traces of energetic electrons with the bulk electrons is simulated by the bounce averaged Fokker-Planck code, CQL3D. It is shown that even when the electric field is small for the production of Dreicer runaways, the knock-on collisions can produce significant runaway electrons in a fraction of a second at typical reactor parameters. The energy spectrum of these knock-on runaways has a characteristic temperature. The growth rate and temperature of the runaway distribution are determined and compared with theory. In simulations of pellet injection into high temperature plasmas, it is shown that a burst of Dreicer runaways may also occur depending on the cooling rate due to the pellet injection. Implications of these phenomena on disruption control in reactor plasmas are discussed. (author)

Loading, absorption, and Fokker-Planck calculations for upcoming ICRF experiments on ATF

International Nuclear Information System (INIS)

Shepard, T.D.; Carter, M.D.; Goulding, R.H.; Kwon, M.

1989-01-01

ICRF experiments on ATF at the 100-kW level are planned for the current 1989 operating period. These plans include the 2ω/sub cH/ regime at f/sub RF/ = 28.88 MHz, D(H) at 14.44 MHz, and 4 He( 3 He) and D( 3 He) at 9.63 MHz. ECH target plasmas have n/sub eO/ /approxreverse arrowlt/ 0.15 /times/ 10 20 m/sup /minus/3/ and B = 0.95 T. The density and temperature profiles obtained are broader than those from 1988, owing to recent field error corrections. The values used for target-plasma parameters in the calculations were taken from initial 1989 ATF data. Loading and absorption calculations have been performed using the 3D RF heating code ORION with a helically symmetric equilibrium, and Fokker-Planck calculations were performed using the steady-state code RFTRANS with two velocity dimensions and one spatial dimension. 6 refs., 3 figs

Stochastic differential equation model to Prendiville processes

International Nuclear Information System (INIS)

Granita; Bahar, Arifah

2015-01-01

The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution

Stochastic differential equation model to Prendiville processes

Energy Technology Data Exchange (ETDEWEB)

Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)

2015-10-22

The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.

Addendum: ``The Dynamics of M15: Observations of the Velocity Dispersion Profile and Fokker-Planck Models'' (ApJ, 481, 267 [1997])

Science.gov (United States)

Dull, J. D.; Cohn, H. N.; Lugger, P. M.; Murphy, B. W.; Seitzer, P. O.; Callanan, P. J.; Rutten, R. G. M.; Charles, P. A.

2003-03-01

It has recently come to our attention that there are axis scale errors in three of the figures presented in Dull et al. (1997, hereafter D97). This paper presented Fokker-Planck models for the collapsed-core globular cluster M15 that include a dense, centrally concentrated population of neutron stars and massive white dwarfs. These models do not include a central black hole. Figure 12 of D97, which presents the predicted mass-to-light profile, is of particular interest, since it was used by Gerssen et al. (2002) as an input to their Jeans equation analysis of the Hubble Space Telescope (HST) STIS velocity measurements reported by van der Marel et al. (2002). On the basis of the original, incorrect version of Figure 12, Gerssen et al. (2002) concluded that the D97 models can fit the new data only with the addition of an intermediate-mass black hole. However, this is counter to our previous finding, shown in Figure 6 of D97, that the Fokker-Planck models predict the sort of moderately rising velocity dispersion profile that Gerssen et al. (2002) infer from the new data. Baumgardt et al. (2003) have independently noted this apparent inconsistency. We appreciate the thoughtful cooperation of Roeland van der Marel in resolving this issue. Using our corrected version of Figure 12 (see below), Gerssen et al. (2003) now find that the velocity dispersion profile that they infer from the D97 mass-to-light ratio profile is entirely consistent with the velocity dispersion profile presented in Figure 6 of D97. Gerssen et al. (2003) further find that there is no statistically significant difference between the fit to the van der Marel et al. (2002) velocity measurements provided by the D97 intermediate-phase model and that provided by their model, which supplements this D97 model with a 1.7+2.7-1.7×103Msolar black hole. Thus, the choice between models with and without black holes will require additional model predictions and observational tests. We present corrected versions of

Non-equilibrium effects upon the non-Markovian Caldeira-Leggett quantum master equation

International Nuclear Information System (INIS)

Bolivar, A.O.

2011-01-01

Highlights: → Classical Brownian motion described by a non-Markovian Fokker-Planck equation. → Quantization process. → Quantum Brownian motion described by a non-Markovian Caldeira-Leggett equation. → A non-equilibrium quantum thermal force is predicted. - Abstract: We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker-Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira-Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.

Reduced kinetic equations: An influence functional approach

International Nuclear Information System (INIS)

Wio, H.S.

1985-01-01

The author discusses a scheme for obtaining reduced descriptions of multivariate kinetic equations based on the 'influence functional' method of Feynmann. It is applied to the case of Fokker-Planck equations showing the form that results for the reduced equation. The possibility of Markovian or non-Markovian reduced description is discussed. As a particular example, the reduction of the Kramers equation to the Smoluchwski equation in the limit of high friction is also discussed

Exact statistical analysis of nonlinear dynamic power reactor models by the Fokker--Planck method. Part II. Reactor with on-off control

International Nuclear Information System (INIS)

Debosscher, A.F.; Dutre, W.L.

1979-01-01

The paper deals with the exact stochastic analysis of the low-frequency neutron density fluctuations in an on-off controlled nuclear power reactor without delayed neutrons and perturbed by Gaussian white reactivity noise. The stochastic process, being Markovian, is completely characterized by its first-order probability density function (pdf) and the transition pdf. The first-order pdf is the normalized solution to the time-independent Fokker--Planck equation (FPE). Using this pdf, a general expression for the moments is obtained. The conditions for stochastic stability in probability, in the mean, and in the mean-square are derived. The time-dependent FPE is solved using the Laplace transform technique, which results in four distinct expressions for the transition pdf, according to the relative magnitude of initial and final reactor power with respect to the regulator level. After Laplace inversion, a physical interpretation of the controller's effect on the stochastic process becomes possible. Finally, making use of the obtained pdf's, the spectral density of the reactor power fluctuations is calculated

Development of a Generalized Version of the Poisson-Nernst-Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

DEFF Research Database (Denmark)

Johannesson, Björn

2010-01-01

A numerical scheme for the transient solution of generalized version of the Poisson-Nernst-Planck equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The Poisson......-scale and that it includes the volume fractions of phases in its structure. The background to the Poisson-Nernst-Planck equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with the Gauss’ law used together with consistent constitutive laws. The HMT theory......-Nernst-Planck equations represent a set of diffusion equations for charged species, i.e. dissolved ions. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst-Planck equations describing the diffusion of the ionic species and the Gauss’ law in used are...

Fokker-Planck code for the quasi-linear absorption of electron cyclotron waves in a tokamak plasma

International Nuclear Information System (INIS)

Meyer, R.L.; Giruzzi, G.; Krivenski, V.

1986-01-01

We present the solution of the kinetic equation describing the quasi-linear evolution of the electron momentum distribution function under the influence of the electron cyclotron wave absorption. Coulomb collisions and the dc electric field in a tokamak plasma. The solution of the quasi-linear equation is obtained numerically using a two-dimensional initial value code following an ADI scheme. Most emphasis is given to the full non-linear and self-consistent problem, namely, the wave amplitude is evaluated at any instant and any point in space according to the actual damping. This is necessary since wave damping is a very sensitive function of the slope of the local momentum distribution function because the resonance condition relates the electron momentum to the location of wave energy deposition. (orig.)

Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

International Nuclear Information System (INIS)

Granita; Bahar, A.

2015-01-01

This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found

Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

Energy Technology Data Exchange (ETDEWEB)

Granita, E-mail: granitafc@gmail.com [Dept. Mathematical Education, State Islamic University of Sultan Syarif Kasim Riau, 28293 Indonesia and Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310,Johor (Malaysia); Bahar, A. [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310,Johor Malaysia and UTM Center for Industrial and Applied Mathematics (UTM-CIAM) (Malaysia)

2015-03-09

This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.

A mixed finite element method for nonlinear diffusion equations

KAUST Repository

Burger, Martin; Carrillo, José ; Wolfram, Marie-Therese

2010-01-01

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.

Treatment of constraints in the stochastic quantization method and covariantized Langevin equation

International Nuclear Information System (INIS)

Ikegami, Kenji; Kimura, Tadahiko; Mochizuki, Riuji

1993-01-01

We study the treatment of the constraints in the stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking into account the Ito calculus. Then we obtain an improved Langevin equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to an O(N) non-linear σ model and it is shown that singular terms appearing in the improved Langevin equation cancel out the δ n (0) divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of independent variables, is actually equivalent to the one in the general-coordinate transformation covariant and vielbein-rotation invariant formalism. (orig.)

Full-wave and Fokker Planck analysis of ICRF heating experiments in the Alcator C-Mod tokamak

International Nuclear Information System (INIS)

Bonoli, P.T.; Golovato, S.; Porkolab, M.; Takase, Y.

1996-01-01

The Alcator C-Mod device is a high field, high density, shaped tokamak with parameters a = 0.22 m, R 0 = 0.67 m, B 0 ≤ 9.0 T, κ ≤ 1.8, δ ≤ 0.8, and 1.0 x 10 20 m -3 n e (0) ≤ 1.0 x 10 21 m -3 . Four megawatt of ICRF power is available at 80 MHz. The wide operating range in magnetic field makes several heating schemes possible: (i) Second harmonic heating of hydrogen (f 0 = 2f CH ) at 2.6 T in (D-H); (ii) Fundamental heating of (H) (f 0 = f CH ) at 5.3T in a D-(H) plasma; and (iii) Fundamental heating of ( 3 He) (f 0 = f C 3 He ) at 7.9 T in a D-( 3 He) plasma. The most successful heating regime to date has been (H)-minority heating at 5.3 T. Pellet enhanced performance (PEP) modes have also been achieved in C-Mod in D-(H) at 5.3 T and in D-( 3 He) at 7.9 T, with a combination of intense ICRF heating and Li-pellet injection. A variety of numerical models are used to analyze these heating schemes. A 1-D full-wave code (FELICE) is used to study open-quotes single passclose quotes damping of the ICRF wavefront and damping of mode-converted ion Bernstein waves. A toroidal full-wave code (FISIC) is used to study interference and focussing effects of the ICRF waves as well as damping of the ICRF power upon multiple passes of the ICRF wavefront. A combined bounce averaged Fokker Planck and toroidal full-wave code (FPPRF) is used to study the ion tail formation, orbit losses, and the power partition of the ICRF tail to the background electrons and ions. Full-wave and Fokker Planck analyses confirm the strong single pass absorption of the ICRF power in D-(H) at 5.3 T. Analysis of PEP-mode plasmas in D-( 3 He) indicates improved wave focussing and 3 He-cyclotron absorption of the ICRF waves relative to L-mode. A dramatic increase in the transfer of 3 He tail power to the background deuterium is also found for PEP-mode plasmas

Master equations in the microscopic theory of nuclear collective dynamics

International Nuclear Information System (INIS)

Matsuo, M.; Sakata, F.; Marumori, T.; Zhuo, Y.

1988-07-01

In the first half of this paper, the authors describe briefly a recent theoretical approach where the mechanism of the large-amplitude dissipative collective motions can be investigated on the basis of the microscopic theory of nuclear collective dynamics. Namely, we derive the general coupled master equations which can disclose, in the framework of the TDHF theory, not only non-linear dynamics among the collective and the single-particle modes of motion but also microscopic dynamics responsible for the dissipative processes. In the latter half, the authors investigate, without relying on any statistical hypothesis, one possible microscopic origin which leads us to the transport equation of the Fokker-Planck type so that usefullness of the general framework is demonstrated. (author)

From a stochastic to a macroscopic approach to brownian motion

International Nuclear Information System (INIS)

Bocquet, L.

1998-01-01

In this lecture, we examine the dynamics of suspensions of mesoscopic (Brownian) particles in a molecular fluid, starting from first principles. We introduce the technique of multiple time-scales to derive the Fokker-Planck equation for a single, or for a set of interacting Brownian particles, starting from the Liouville equation for the full system (Brownian particles and discrete bath). The limitations of the Fokker-Planck equation will then be emphasized. In particular, we shall point out that under ''standard'' experimental conditions, the Fokker-Planck description cannot be correct and that non-Markovian effects are expected. A microscopic description in the true experimental limit confirms this breakdown and leads to a ''generalized'' (non-Markovian and non-local in velocity space) Fokker-Planck equation, which describes the thermalization of the Brownian particle. (author)

Design of foam-buffered high gain target with Fokker-Planck implosion simulation for thermal insulation and imprint mitigation

International Nuclear Information System (INIS)

Takeda, T.; Mima, K.; Norimatsu, T.; Nagatomo, H.; Nishiguchi, A.

2003-01-01

It is proposed that a thick foam layer on a plastic capsule of fusion pellet is effective not only for reducing the initial imprint, but also for solving the melting problem of cryogenic deuterium-tritium layer, in a reactor chamber. Investigated are the dependences of gain, thermal insulation for preventing the melting, and imprint mitigation of a foam-buffered target on the foam layer thickness. The imprint mitigation, the Rayleigh-Taylor growth factor and the fusion gain of a foam-buffered target are evaluated by the hydrodynamic implosion code HIMICO [A. Nishiguchi et al., Phys. Fluids B 4, 417 (1992)], which includes a Fokker-Planck transport code. As the result, it is found that high gain can be achieved by the foam-buffered target together with thermal insulation and imprint mitigation

Angular Multigrid Preconditioner for Krylov-Based Solution Techniques Applied to the Sn Equations with Highly Forward-Peaked Scattering

Science.gov (United States)

Turcksin, Bruno; Ragusa, Jean C.; Morel, Jim E.

2012-01-01

It is well known that the diffusion synthetic acceleration (DSA) methods for the Sn equations become ineffective in the Fokker-Planck forward-peaked scattering limit. In response to this deficiency, Morel and Manteuffel (1991) developed an angular multigrid method for the 1-D Sn equations. This method is very effective, costing roughly twice as much as DSA per source iteration, and yielding a maximum spectral radius of approximately 0.6 in the Fokker-Planck limit. Pautz, Adams, and Morel (PAM) (1999) later generalized the angular multigrid to 2-D, but it was found that the method was unstable with sufficiently forward-peaked mappings between the angular grids. The method was stabilized via a filtering technique based on diffusion operators, but this filtering also degraded the effectiveness of the overall scheme. The spectral radius was not bounded away from unity in the Fokker-Planck limit, although the method remained more effective than DSA. The purpose of this article is to recast the multidimensional PAM angular multigrid method without the filtering as an Sn preconditioner and use it in conjunction with the Generalized Minimal RESidual (GMRES) Krylov method. The approach ensures stability and our computational results demonstrate that it is also significantly more efficient than an analogous DSA-preconditioned Krylov method.

Pedestrian Flow in the Mean Field Limit

KAUST Repository

Haji Ali, Abdul Lateef

2012-01-01

-dependent density of two-dimensional pedestrians satisfies a four-dimensional integro-differential Fokker-Planck equation. To approximate the solution of the Fokker-Planck equation we use a time-splitting approach and solve the diffusion part using a Crank

Planck constant as spectral parameter in integrable systems and KZB equations

Science.gov (United States)

Levin, A.; Olshanetsky, M.; Zotov, A.

2014-10-01

We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations with Ñ punctures by deformation of the corresponding quantum gl N rational R-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is τ. At the level of classical mechanics the deformation parameter τ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized R-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in R matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical r-matrices which can be treated as halves of the classical Yang-Baxter equation. At last we discuss the R-matrix valued linear problems which provide gl Ñ CM models and Painlevé equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum R-matrix. When the quantum gl N R-matrix is scalar ( N = 1) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the gl Ñ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.

Application of the Fokker-Planck molecular mixing model to turbulent scalar mixing using moment methods

Science.gov (United States)

Madadi-Kandjani, E.; Fox, R. O.; Passalacqua, A.

2017-06-01

An extended quadrature method of moments using the β kernel density function (β -EQMOM) is used to approximate solutions to the evolution equation for univariate and bivariate composition probability distribution functions (PDFs) of a passive scalar for binary and ternary mixing. The key element of interest is the molecular mixing term, which is described using the Fokker-Planck (FP) molecular mixing model. The direct numerical simulations (DNSs) of Eswaran and Pope ["Direct numerical simulations of the turbulent mixing of a passive scalar," Phys. Fluids 31, 506 (1988)] and the amplitude mapping closure (AMC) of Pope ["Mapping closures for turbulent mixing and reaction," Theor. Comput. Fluid Dyn. 2, 255 (1991)] are taken as reference solutions to establish the accuracy of the FP model in the case of binary mixing. The DNSs of Juneja and Pope ["A DNS study of turbulent mixing of two passive scalars," Phys. Fluids 8, 2161 (1996)] are used to validate the results obtained for ternary mixing. Simulations are performed with both the conditional scalar dissipation rate (CSDR) proposed by Fox [Computational Methods for Turbulent Reacting Flows (Cambridge University Press, 2003)] and the CSDR from AMC, with the scalar dissipation rate provided as input and obtained from the DNS. Using scalar moments up to fourth order, the ability of the FP model to capture the evolution of the shape of the PDF, important in turbulent mixing problems, is demonstrated. Compared to the widely used assumed β -PDF model [S. S. Girimaji, "Assumed β-pdf model for turbulent mixing: Validation and extension to multiple scalar mixing," Combust. Sci. Technol. 78, 177 (1991)], the β -EQMOM solution to the FP model more accurately describes the initial mixing process with a relatively small increase in computational cost.

Quantization of non-Hamiltonian physical systems

International Nuclear Information System (INIS)

Bolivar, A.O.

1998-09-01

We propose a general method of quantization of non-Hamiltonian physical systems. Applying it, for example, to a dissipative system coupled to a thermal reservoir described by the Fokker-Planck equation, we are able to obtain the Caldeira-Leggett master equation, the non-linear Schroedinger-Langevin equation and Caldirola-Kanai equation (with an additional term), as particular cases. (author)

Kramers-Moyal expansion for stochastic differential equations with single and multiple delays: Applications to financial physics and neurophysics

International Nuclear Information System (INIS)

Frank, T.D.

2007-01-01

We present a generalized Kramers-Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker-Planck equation derived earlier in the literature is a special case of the proposed Kramers-Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed

Elementary stochastic cooling

Energy Technology Data Exchange (ETDEWEB)

Tollestrup, A.V.; Dugan, G

1983-12-01

Major headings in this review include: proton sources; antiproton production; antiproton sources and Liouville, the role of the Debuncher; transverse stochastic cooling, time domain; the accumulator; frequency domain; pickups and kickers; Fokker-Planck equation; calculation of constants in the Fokker-Planck equation; and beam feedback. (GHT)

Fokker-Planck simulation of runaway electron generation in disruptions with the hot-tail effect

Energy Technology Data Exchange (ETDEWEB)

Nuga, H., E-mail: nuga@p-grp.nucleng.kyoto-u.ac.jp; Fukuyama, A. [Department of Engineering, Kyoto University, Kyoto 615-8540 (Japan); Yagi, M. [National Institutes for Quantum and Radiological Science and Technology, Aomori 039-3212 (Japan)

2016-06-15

To study runaway electron generation in disruptions, we have extended the three-dimensional (two-dimensional in momentum space; one-dimensional in the radial direction) Fokker-Planck code, which describes the evolution of the relativistic momentum distribution function of electrons and the induced toroidal electric field in a self-consistent manner. A particular focus is placed on the hot-tail effect in two-dimensional momentum space. The effect appears if the drop of the background plasma temperature is sufficiently rapid compared with the electron-electron slowing down time for a few times of the pre-quench thermal velocity. It contributes to not only the enhancement of the primary runaway electron generation but also the broadening of the runaway electron distribution in the pitch angle direction. If the thermal energy loss during the major disruption is assumed to be isotropic, there are hot-tail electrons that have sufficiently large perpendicular momentum, and the runaway electron distribution becomes broader in the pitch angle direction. In addition, the pitch angle scattering also yields the broadening. Since the electric field is reduced due to the burst of runaway electron generation, the time required for accelerating electrons to the runaway region becomes longer. The longer acceleration period makes the pitch-angle scattering more effective.

A Finite-Orbit-Width Fokker-Planck solver for modeling of energetic particle interactions with waves, with application to Helicons in ITER

Directory of Open Access Journals (Sweden)

Petrov Yuri V.

2017-01-01

Full Text Available The bounce-average (BA finite-difference Fokker-Planck (FP code CQL3D [1,2] now includes the essential physics to describe the RF heating of Finite-Orbit-Width (FOW ions in tokamaks. The FP equation is reformulated in terms of Constants-Of-Motion coordinates, which we select to be particle speed, pitch angle, and major radius on the equatorial plane thus obtaining the distribution function directly at this location. Full-orbit, low collisionality neoclassical radial transport emerges from averaging the local friction and diffusion coefficients along guiding center orbits. Similarly, the BA of local quasilinear RF diffusion terms gives rise to additional radial transport. The local RF electric field components needed for the BA operator are usually obtained by a ray-tracing code, such as GENRAY, or in conjunction with full-wave codes. As a new, practical application, the CQL3D-FOW version is used for simulation of alpha-particle heating by high-harmonic waves in ITER. Coupling of high harmonic or helicon fast waves power to electrons is a promising current drive (CD scenario for high beta plasmas. However, the efficiency of current drive can be diminished by parasitic channeling of RF power into fast ions, such as alphas, through finite Larmor-radius effects. We investigate possibilities to reduce the fast ion heating in CD scenarios.

Fractional Poisson-Nernst-Planck Model for Ion Channels I: Basic Formulations and Algorithms.

Science.gov (United States)

Chen, Duan

2017-11-01

In this work, we propose a fractional Poisson-Nernst-Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker-Planck equation. Then, it is generalized to the macroscopic fractional Poisson-Nernst-Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.

Solution of Fokker–Planck equation by finite element and finite ...

Indian Academy of Sciences (India)

The response of a structural system to white noise excitation (delta-correlated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker–Planck and backward Kolmogorov equations. Numerical solution of these equations by finite element and finite ...

Equations of multiparticle dynamics

International Nuclear Information System (INIS)

Chao, A.W.

1987-01-01

The description of the motion of charged-particle beams in an accelerator proceeds in steps of increasing complexity. The first step is to consider a single-particle picture in which the beam is represented as a collection on non-interacting test particles moving in a prescribed external electromagnetic field. Knowing the external field, it is then possible to calculate the beam motion to a high accuracy. The real beam consists of a large number of particles, typically 10 11 per beam bunch. It is sometimes inconvenient, or even impossible, to treat the real beam behavior using the single particle approach. One way to approach this problem is to supplement the single particle by another qualitatively different picture. The commonly used tools in accelerator physics for this purpose are the Vlasov and the Fokker-Planck equations. These equations assume smooth beam distributions and are therefore strictly valid in the limit of infinite number of micro-particles, each carrying an infinitesimal charge. The hope is that by studying the two extremes -- the single particle picture and the picture of smooth beam distributions -- we will be able to describe the behavior of our 10 11 -particle system. As mentioned, the most notable use of the smooth distribution picture is the study of collective beam instabilities. However, the purpose of this lecture is not to address this more advanced subject. Rather, it has the limited goal to familiarize the reader with the analytical tools, namely the Vlasov and the Fokker-Planck equations, as a preparation for dealing with the more advanced problems at later times. We will first derive these equations and then illustrate their applications by several examples which allow exact solutions

Brownian motion of classical spins: Anomalous dissipation and generalized Langevin equation

Science.gov (United States)

Bandyopadhyay, Malay; Jayannavar, A. M.

2017-10-01

In this work, we derive the Langevin equation (LE) of a classical spin interacting with a heat bath through momentum variables, starting from the fully dynamical Hamiltonian description. The derived LE with anomalous dissipation is analyzed in detail. The obtained LE is non-Markovian with multiplicative noise terms. The concomitant dissipative terms obey the fluctuation-dissipation theorem. The Markovian limit correctly produces the Kubo and Hashitsume equation. The perturbative treatment of our equations produces the Landau-Lifshitz equation and the Seshadri-Lindenberg equation. Then we derive the Fokker-Planck equation corresponding to LE and the concept of equilibrium probability distribution is analyzed.

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

International Nuclear Information System (INIS)

Guo, Ran; Du, Jiulin

2015-01-01

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

Energy Technology Data Exchange (ETDEWEB)

Guo, Ran; Du, Jiulin, E-mail: jiulindu@aliyun.com

2015-08-15

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution.

Unsteady analytical solutions to the Poisson–Nernst–Planck equations

International Nuclear Information System (INIS)

Schönke, Johannes

2012-01-01

It is shown that the Poisson–Nernst–Planck equations for a single ion species can be formulated as one equation in terms of the electric field. This previously not analyzed equation shows similarities to the vector Burgers equation and is identical with it in the one dimensional case. Several unsteady exact solutions for one and multidimensional cases are presented. Besides new mathematical insights which these first known unsteady solutions give, they can serve as test cases in computer simulations to analyze numerical algorithms and to verify code. (paper)

Pulse-coupled mixed-mode oscillators: Cluster states and extreme noise sensitivity

Science.gov (United States)

Karamchandani, Avinash J.; Graham, James N.; Riecke, Hermann

2018-04-01

Motivated by rhythms in the olfactory system of the brain, we investigate the synchronization of all-to-all pulse-coupled neuronal oscillators exhibiting various types of mixed-mode oscillations (MMOs) composed of sub-threshold oscillations (STOs) and action potentials ("spikes"). We focus particularly on the impact of the delay in the interaction. In the weak-coupling regime, we reduce the system to a Kuramoto-type equation with non-sinusoidal phase coupling and the associated Fokker-Planck equation. Its linear stability analysis identifies the appearance of various cluster states. Their type depends sensitively on the delay and the width of the pulses. Interestingly, long delays do not imply slow population rhythms, and the number of emerging clusters only loosely depends on the number of STOs. Direct simulations of the oscillator equations reveal that for quantitative agreement of the weak-coupling theory the coupling strength and the noise have to be extremely small. Even moderate noise leads to significant skipping of STO cycles, which can enhance the diffusion coefficient in the Fokker-Planck equation by two orders of magnitude. Introducing an effective diffusion coefficient extends the range of agreement significantly. Numerical simulations of the Fokker-Planck equation reveal bistability and solutions with oscillatory order parameters that result from nonlinear mode interactions. These are confirmed in simulations of the full spiking model.

Non deterministic methods for charged particle transport

International Nuclear Information System (INIS)

Besnard, D.C.; Buresi, E.; Hermeline, F.; Wagon, F.

1985-04-01

The coupling of Monte-Carlo methods for solving Fokker Planck equation with ICF inertial confinement fusion codes requires them to be economical and to preserve gross conservation properties. Besides, the presence in FPE Fokker-Planck equation of diffusion terms due to collisions between test particles and the background plasma challenges standard M.C. (Monte-Carlo) techniques if this phenomenon is dominant. We address these problems through the use of a fixed mesh in phase space which allows us to handle highly variable sources, avoiding any Russian Roulette for lowering the size of the sample. Also on this mesh are solved diffusion equations obtained from a splitting of FPE. Any non linear diffusion terms of FPE can be handled in this manner. Another method, also presented here is to use a direct particle method for solving the full FPE

Flow Equation Approach to the Statistics of Nonlinear Dynamical Systems

Science.gov (United States)

Marston, J. B.; Hastings, M. B.

2005-03-01

The probability distribution function of non-linear dynamical systems is governed by a linear framework that resembles quantum many-body theory, in which stochastic forcing and/or averaging over initial conditions play the role of non-zero . Besides the well-known Fokker-Planck approach, there is a related Hopf functional methodootnotetextUriel Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, 1995) chapter 9.5.; in both formalisms, zero modes of linear operators describe the stationary non-equilibrium statistics. To access the statistics, we investigate the method of continuous unitary transformationsootnotetextS. D. Glazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993); Phys. Rev. D 49, 4214 (1994). (also known as the flow equation approachootnotetextF. Wegner, Ann. Phys. 3, 77 (1994).), suitably generalized to the diagonalization of non-Hermitian matrices. Comparison to the more traditional cumulant expansion method is illustrated with low-dimensional attractors. The treatment of high-dimensional dynamical systems is also discussed.

Mean-field games with logistic population dynamics

KAUST Repository

Gomes, Diogo A.; De Lima Ribeiro, Ricardo

2013-01-01

In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE.

Mean-field games with logistic population dynamics

KAUST Repository

Gomes, Diogo A.

2013-12-01

In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE.

An efficient explicit numerical scheme for diffusion-type equations with a highly inhomogeneous and highly anisotropic diffusion tensor

International Nuclear Information System (INIS)

Larroche, O.

2007-01-01

A locally split-step explicit (LSSE) algorithm was developed for efficiently solving a multi-dimensional advection-diffusion type equation involving a highly inhomogeneous and highly anisotropic diffusion tensor, which makes the problem very ill-conditioned for standard implicit methods involving the iterative solution of large linear systems. The need for such an optimized algorithm arises, in particular, in the frame of thermonuclear fusion applications, for the purpose of simulating fast charged-particle slowing-down with an ion Fokker-Planck code. The LSSE algorithm is presented in this paper along with the results of a model slowing-down problem to which it has been applied

Computer models for kinetic equations of magnetically confined plasmas

International Nuclear Information System (INIS)

Killeen, J.; Kerbel, G.D.; McCoy, M.G.; Mirin, A.A.; Horowitz, E.J.; Shumaker, D.E.

1987-01-01

This paper presents four working computer models developed by the computational physics group of the National Magnetic Fusion Energy Computer Center. All of the models employ a kinetic description of plasma species. Three of the models are collisional, i.e., they include the solution of the Fokker-Planck equation in velocity space. The fourth model is collisionless and treats the plasma ions by a fully three-dimensional particle-in-cell method

Two-dimensional spectroscopy for harmonic vibrational modes with nonlinear system-bath interactions. I. Gaussian-white case

NARCIS (Netherlands)

Steffen, T; Tanimura, Y

The quantum Fokker-Planck equation is derived for a system nonlinearly coupled to a harmonic oscillator bath. The system-bath interaction is assumed to be linear in the bath coordinates but quadratic in the system coordinate. The relaxation induced dynamics of a harmonic system are investigated by

Prediction of the behavior of structural materials under irradiation through modeling of the microstructure. Progress report, April 1, 1978-August 30, 1979

International Nuclear Information System (INIS)

Wolfer, W.G.

1979-08-01

The development of the radiation-induced microstructure occurs in several stages: formation of small defect clusters, formation of dislocation loops, nucleation and growth of voids, and regeneration of the dislocation network. With the exception of the latter, these processes can be modeled with rate equations of similar form. However, instead of using one rate equation for each defect cluster of a given size, the discrete formulation is transformed in a continuous one leading to Fokker-Planck equations. It is shown by comparing the steady-state nucleation rates of both formulations that the Fokker-Planck equation derived is the correct continuous description. A path-integral solution for the Fokker-Planck equation was derived to provide the basis for a numerical solution procedure, capable of dealing with the vastly different time scales involved in cluster formation, nucleation, and growth

A continuous stochastic model for non-equilibrium dense gases

Science.gov (United States)

Sadr, M.; Gorji, M. H.

2017-12-01

While accurate simulations of dense gas flows far from the equilibrium can be achieved by direct simulation adapted to the Enskog equation, the significant computational demand required for collisions appears as a major constraint. In order to cope with that, an efficient yet accurate solution algorithm based on the Fokker-Planck approximation of the Enskog equation is devised in this paper; the approximation is very much associated with the Fokker-Planck model derived from the Boltzmann equation by Jenny et al. ["A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion," J. Comput. Phys. 229, 1077-1098 (2010)] and Gorji et al. ["Fokker-Planck model for computational studies of monatomic rarefied gas flows," J. Fluid Mech. 680, 574-601 (2011)]. The idea behind these Fokker-Planck descriptions is to project the dynamics of discrete collisions implied by the molecular encounters into a set of continuous Markovian processes subject to the drift and diffusion. Thereby, the evolution of particles representing the governing stochastic process becomes independent from each other and thus very efficient numerical schemes can be constructed. By close inspection of the Enskog operator, it is observed that the dense gas effects contribute further to the advection of molecular quantities. That motivates a modelling approach where the dense gas corrections can be cast in the extra advection of particles. Therefore, the corresponding Fokker-Planck approximation is derived such that the evolution in the physical space accounts for the dense effects present in the pressure, stress tensor, and heat fluxes. Hence the consistency between the devised Fokker-Planck approximation and the Enskog operator is shown for the velocity moments up to the heat fluxes. For validation studies, a homogeneous gas inside a box besides Fourier, Couette, and lid-driven cavity flow setups is considered. The results based on the Fokker-Planck model are

Time-space limitations of Nernst-Planck equations

International Nuclear Information System (INIS)

Pellicer, J.; Aguilera, V.M.; Mafe, S.

1988-01-01

The nature and applicability of Nernst-Planck and Poisson equations are considered, concerning the problem of electrolyte transport in non-homogeneous solutions. Some approximations related to the model of transport are discussed, specially those referring to the electrodynamical aspects. Thus, the connection between the classical electrostatics approximations and the time-space limitations of the model is shown. A detailed analysis leads to conclude that some of the aspects of the charge separation process have not been completely understood. (Author)

Introduction to fractional and pseudo-differential equations with singular symbols

CERN Document Server

Umarov, Sabir

2015-01-01

The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations.

From BBGKY hierarchy to non-Markovian evolution equations

International Nuclear Information System (INIS)

Gerasimenko, V.I.; Shtyk, V.O.; Zagorodny, A.G.

2009-01-01

The problem of description of the evolution of the microscopic phase density and its generalizations is discussed. With this purpose, the sequence of marginal microscopic phase densities is introduced, and the appropriate BBGKY hierarchy for these microscopic distributions and their average values is formulated. The microscopic derivation of the generalized evolution equation for the average value of the microscopic phase density is given, and the non-Markovian generalization of the Fokker-Planck collision integral is proposed

Potential in stochastic differential equations: novel construction

International Nuclear Information System (INIS)

Ao, P

2004-01-01

There is a whole range of emergent phenomena in a complex network such as robustness, adaptiveness, multiple-equilibrium, hysteresis, oscillation and feedback. Those non-equilibrium behaviours can often be described by a set of stochastic differential equations. One persistent important question is the existence of a potential function. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. We present an explicit construction procedure to obtain the potential and relevant quantities. In the procedure no reference to the Fokker-Planck equation is needed. The availability of the potential suggests that powerful statistical mechanics tools can be used in nonequilibrium situations. (letter to the editor)

Steady state solution of the Poisson-Nernst-Planck equations

International Nuclear Information System (INIS)

Golovnev, A.; Trimper, S.

2010-01-01

The exact steady state solution of the Poisson-Nernst-Planck equations (PNP) is given in terms of Jacobi elliptic functions. A more tractable approximate solution is derived which can be used to compare the results with experimental observations in binary electrolytes. The breakdown of the PNP for high concentration and high applied voltage is discussed.

Application of the Poisson-Nernst-Planck equations to the migration test

DEFF Research Database (Denmark)

Krabbenhøft, Kristian; Krabbenhøft, Jørgen

2008-01-01

The Poisson-Nernst-Planck (PNP) equations are applied to model the migration test. A detailed analysis of the equations is presented and the effects of a number of common, simplifying assumptions are quantified. In addition, closed-form solutions for the effective chloride diffusivity based...... on the full PNP equations are derived, a number of experiments are analyzed in detail, and a new, truly accelerated migration test is proposed. Finally, we present a finite element procedure for numerical solution of the PNP equations....

The effect of an alternating electric field on a totally ionised plasma; Action d'un champ electrique alternatif sur un plasma totalement ionise

Energy Technology Data Exchange (ETDEWEB)

Baglin, H; Brin, A; Ozias, Y; Salmon, J [Commissariat a l' Energie Atomique, Saclay (France).Centre d' Etudes Nucleaires; Delcroix, J L [Ecole Normale Superieure, 75 - Paris (France)

1960-07-01

The equation giving the distribution function of the electrons in a steady-state, for a fully ionized plasma in an a.c. field, are provided from the Fokker-Planck equation. The electric conductivity is complex and depends on the frequency. (author) [French] L'equation qui donne la fonction de distribution des electrons dans un etat stationnaire pour un plasma totalement ionise dans un champ electrique alternatif est fournie par l'equation de Fokker-Planck. La conductibilite electrique est complexe et depend de la frequence. (auteur)

Oligarchy as a phase transition: The effect of wealth-attained advantage in a Fokker-Planck description of asset exchange

Science.gov (United States)

Boghosian, Bruce M.; Devitt-Lee, Adrian; Johnson, Merek; Li, Jie; Marcq, Jeremy A.; Wang, Hongyan

2017-06-01

The ;Yard-Sale Model; of asset exchange is known to result in complete inequality-all of the wealth in the hands of a single agent. It is also known that, when this model is modified by introducing a simple model of redistribution based on the Ornstein-Uhlenbeck process, it admits a steady state exhibiting some features similar to the celebrated Pareto Law of wealth distribution. In the present work, we analyze the form of this steady-state distribution in much greater detail, using a combination of analytic and numerical techniques. We find that, while Pareto's Law is approximately valid for low redistribution, it gives way to something more similar to Gibrat's Law when redistribution is higher. Additionally, we prove in this work that, while this Pareto or Gibrat behavior may persist over many orders of magnitude, it ultimately gives way to gaussian decay at extremely large wealth. Also in this work, we introduce a bias in favor of the wealthier agent-what we call Wealth-Attained Advantage (WAA)-and show that this leads to the phenomenon of ;wealth condensation; when the bias exceeds a certain critical value. In the wealth-condensed state, a finite fraction of the total wealth of the population ;condenses; to the wealthiest agent. We examine this phenomenon in some detail, and derive the corresponding modification to the Fokker-Planck equation. We observe a second-order phase transition to a state of coexistence between an oligarch and a distribution of non-oligarchs. Finally, by studying the asymptotic behavior of the distribution in some detail, we show that the onset of wealth condensation has an abrupt reciprocal effect on the character of the non-oligarchical part of the distribution. Specifically, we show that the above-mentioned gaussian decay at extremely large wealth is valid both above and below criticality, but degenerates to exponential decay precisely at criticality.

Langevin equation method for the rotational Brownian motion and orientational relaxation in liquids: II. Symmetrical top molecules

CERN Document Server

Coffey, W T; Titov, S V

2003-01-01

A theory of orientational relaxation for the inertial rotational Brownian motion of a symmetric top molecule is developed using the Langevin equation rather than the Fokker-Planck equation. The infinite hierarchy of differential-recurrence relations for the orientational correlation functions for the relaxation behaviour is derived by averaging the corresponding Euler-Langevin equations. The solution of this hierarchy is obtained using matrix continued fractions allowing the calculation of the correlation times and the spectra of the orientational correlation functions for typical values of the model parameters.

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions.

Science.gov (United States)

Lu, Benzhuo; Holst, Michael J; McCammon, J Andrew; Zhou, Y C

2010-09-20

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.

Green's functions and trace formulas for generalized Sturm-Liouville problems related by Darboux transformations

International Nuclear Information System (INIS)

Schulze-Halberg, Axel

2010-01-01

We study Green's functions of the generalized Sturm-Liouville problems that are related to each other by Darboux -equivalently, supersymmetrical - transformations. We establish an explicit relation between the corresponding Green's functions and derive a simple formula for their trace. The class of equations considered here includes the conventional Schroedinger equation and generalizations, such as for position-dependent mass and with linearly energy-dependent potential, as well as the stationary Fokker-Planck equation.

Stochastic Calculus and Differential Equations for Physics and Finance

Science.gov (United States)

McCauley, Joseph L.

2013-02-01

1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman-Kolmogorov; 8. Non Markov Ito processes; 9. Black-Scholes, martingales, and Feynman-Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.

Study of the interaction operator between two groups of particles in a completely ionised plasma. Development of the distribution functions in a series of orthogonal polynomials (1963); Etude de l'operateur d'interaction entre deux groupes de particules dans un plasma completement ionise. Developpement des fonctions de distribution en series de polynomes orthogonaux (1963)

Energy Technology Data Exchange (ETDEWEB)

Fain, A [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires

1963-05-15

A completely ionised medium is considered in which the variations of the distribution functions for each species of particles is governed by the Fokker-Planck equation. The interaction operator is of a form given by Rosenbluth - Macdonald - Judd. The distribution functions are expanded into orthogonal polynomial series in the velocity space. In a first stage these functions are first split up into spherical harmonic series (or, in an equivalent form, into series of scalar products of irreducible cartesian tensors), with coefficients which are a function of the velocity modulus as well as space and time coordinates. In the second stage these coefficients are expanded into series of orthogonal functions of the velocity modulus; the 1 order harmonic is represented by the product of a Maxwell distribution and of a SONINE polynomial series, having an index of 1 + 1 / 2, which have as variable the reduced energy of the particles (in terms of a basic temperature), with coefficients which then only depend on the space and time coordinates. In the first part the relationship is established between the expansion coefficients and the moments of the distribution function, as well as the hydrodynamic values. In the second part the expansion using spherical harmonics is applied to the Fokker-Planck equation. The general expression for the second member is given as well as the particular expressions corresponding to the cases where the operator is linearized. In the third part the complete expansion in orthogonal polynomial series is applied to the Fokker-Planck equation. The expression of the generating functions is given for all the harmonics in the case of the linearized operator, as well as the transport equations for the first four harmonics. (author) [French] On considere un milieu completement ionise ou l'evolution des fonctions de distribution pour chaque espece de particules est regie par l'equation de FOKKER-PLANCK. L'operateur d'interaction se met sous la forme

Response of sliding structures to seismic excitation: bibliographical study

International Nuclear Information System (INIS)

Sarh, K.; Duval, C.

1992-11-01

Calculation of the seismic response of structures on sliding supports involves the dual problem of ''non-linear'' and ''random'' dynamic behaviour. After a review of the non-linearities common in dynamics, slipping is compared with a hysteresis phenomenon. Simple examples are then used to present the Fokker-Planck equation and the equivalent linearization method. Finally, the methods for modification of the excitation spectrum intended for the engineering calculations are recalled. (authors). 21 figs., 23 refs

On the Nernst-Planck equation.

Science.gov (United States)

Maex, Reinoud

2017-01-01

This review first discusses Nernst's and Planck's early papers on electro-diffusion, the brief priority conflict that followed, and the role these papers played in shaping the emerging concept of membrane excitability. The second part discusses in greater detail the constraints of the Nernst-Planck theory, and shows more recent examples of its applicability for neuronal modelling.

The Markov process admits a consistent steady-state thermodynamic formalism

Science.gov (United States)

Peng, Liangrong; Zhu, Yi; Hong, Liu

2018-01-01

The search for a unified formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for the master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that (1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; (2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; (3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that general Markov processes admit a unified and self-consistent non-equilibrium steady-state thermodynamic formalism, regardless of underlying detailed models.

External non-white noise and nonequilibrium phase transitions

International Nuclear Information System (INIS)

Sancho, J.M.; San Miguel, M.

1980-01-01

Langevin equations with external non-white noise are considered. A Fokker Planck equation valid in general in first order of the correlation time tau of the noise is derived. In some cases its validity can be extended to any value of tau. The effect of a finite tau in the nonequilibrium phase transitions induced by the noise is analyzed, by means of such Fokker Planck equation, in general, for the Verhulst equation under two different kind of fluctuations, and for a genetic model. It is shown that new transitions can appear and that the threshold value of the parameter can be changed. (orig.)

Derivation of stochastic differential equations for scrape-off layer plasma fluctuations from experimentally measured statistics

Energy Technology Data Exchange (ETDEWEB)

Mekkaoui, Abdessamad [IEK-4 Forschungszentrum Juelich 52428 (Germany)

2013-07-01

A method to derive stochastic differential equations for intermittent plasma density dynamics in magnetic fusion edge plasma is presented. It uses a measured first four moments (mean, variance, Skewness and Kurtosis) and the correlation time of turbulence to write a Pearson equation for the probability distribution function of fluctuations. The Fokker-Planck equation is then used to derive a Langevin equation for the plasma density fluctuations. A theoretical expectations are used as a constraints to fix the nonlinearity structure of the stochastic differential equation. In particular when the quadratically nonlinear dynamics is assumed, then it is shown that the plasma density is driven by a multiplicative Wiener process and evolves on the turbulence correlation time scale, while the linear growth is quadratically damped by the fluctuation level. Strong criteria for statistical discrimination of experimental time series are proposed as an alternative to the Kurtosis-Skewness scaling. This scaling is broadly used in contemporary literature to characterize edge turbulence, but it is inappropriate because a large family of distributions could share this scaling. Strong criteria allow us to focus on the relevant candidate distribution and approach a nonlinear structure of edge turbulence model.

Independence of the effective dielectric constant of an electrolytic solution on the ionic distribution in the linear Poisson-Nernst-Planck model.

Science.gov (United States)

Alexe-Ionescu, A L; Barbero, G; Lelidis, I

2014-08-28

We consider the influence of the spatial dependence of the ions distribution on the effective dielectric constant of an electrolytic solution. We show that in the linear version of the Poisson-Nernst-Planck model, the effective dielectric constant of the solution has to be considered independent of any ionic distribution induced by the external field. This result follows from the fact that, in the linear approximation of the Poisson-Nernst-Planck model, the redistribution of the ions in the solvent due to the external field gives rise to a variation of the dielectric constant that is of the first order in the effective potential, and therefore it has to be neglected in the Poisson's equation that relates the actual electric potential across the electrolytic cell to the bulk density of ions. The analysis is performed in the case where the electrodes are perfectly blocking and the adsorption at the electrodes is negligible, and in the absence of any ion dissociation-recombination effect.

Pedestrian Flow in the Mean Field Limit

KAUST Repository

Haji Ali, Abdul Lateef

2012-11-01

We study the mean-field limit of a particle-based system modeling the behavior of many indistinguishable pedestrians as their number increases. The base model is a modified version of Helbing\\'s social force model. In the mean-field limit, the time-dependent density of two-dimensional pedestrians satisfies a four-dimensional integro-differential Fokker-Planck equation. To approximate the solution of the Fokker-Planck equation we use a time-splitting approach and solve the diffusion part using a Crank-Nicholson method. The advection part is solved using a Lax-Wendroff-Leveque method or an upwind Backward Euler method depending on the advection speed. Moreover, we use multilevel Monte Carlo to estimate observables from the particle-based system. We discuss these numerical methods, and present numerical results showing the convergence of observables that were calculated using the particle-based model as the number of pedestrians increases to those calculated using the probability density function satisfying the Fokker-Planck equation.

A kind of iteration algorithm for fast wave heating

International Nuclear Information System (INIS)

Zhu Xueguang; Kuang Guangli; Zhao Yanping; Li Youyi; Xie Jikang

1998-03-01

The standard normal distribution for particles in Tokamak geometry is usually assumed in fast wave heating. In fact, due to the quasi-linear diffusion effect, the parallel and vertical temperature of resonant particles is not equal, so, this will bring some error. For this case, the Fokker-Planck equation is introduced, and iteration algorithm is adopted to solve the problem well

Moment equation approach to neoclassical transport theory

International Nuclear Information System (INIS)

Hirshman, S.P.

1978-01-01

The neoclassical cross-field fluxes for a toroidally confined, axisymmetric plasma are calculated in terms of the thermodynamic forces from the fluid continuity and momentum balance equations. This macroscopic formulation of neoclassical transport theory unifies the numerous complex expressions for the transport coefficients, previously obtained by solving the Fokker--Planck equation, and elucidates their physical basis. In the large aspect ratio limit, the continuous transition in the scaling of the diffusion coefficient throughout various collisionality regimes is shown to depend on the ratio of parallel viscosity coefficients of the plasma species. Comparison of the present results with the kinetic theory expressions for the neoclassical fluxes determines the parallel viscosity coefficients for a multispecies plasma in the long-mean-free-path regime

Analytical solution of the Poisson-Nernst-Planck equations for an electrochemical system close to electroneutrality

International Nuclear Information System (INIS)

Pabst, M.

2014-01-01

Single charge densities and the potential are used to describe models of electrochemical systems. These quantities can be calculated by solving a system of time dependent nonlinear coupled partial differential equations, the Poisson-Nernst-Planck equations. Assuming small deviations from the electroneutral equilibrium, the linearized and decoupled equations are solved for a radial symmetric geometry, which represents the interface between a cell and a sensor device. The densities and the potential are expressed by Fourier-Bessels series. The system considered has a ratio between the Debye-length and its geometric dimension on the order of 10 −4 so the Fourier-Bessel series can be approximated by elementary functions. The time development of the system is characterized by two time constants, τ c and τ g . The constant τ c describes the approach to the stationary state of the total charge and the potential. τ c is several orders of magnitude smaller than the geometry-dependent constant τ g , which is on the order of 10 ms characterizing the transition to the stationary state of the single ion densities

Stochastic dynamics of an inflationary model and initial distribution of universes

International Nuclear Information System (INIS)

Nambu, Yasusada.

1989-01-01

We investigate the stationary solution of the modified Fokker-Planck equation which governs the global dynamics of the inflation. Contrary to the original FP equation which is for a Hubble horizon size region, we found that the normalizable stationary solution can exist for modified Fokker-Planck equation which is for many Hubble horizon size regions. For a chaotic inflationary model with the potential λψ 2n , we get initial distribution of classical universes using this solution, and discussed the physical meaning of it. Especially for n = 2, this distribution obeys power-law and classical universes which created from the Planck energy region make the fractal structure. Other cases n ≠ 2, creation of large classical universes are strongly suppressed. (author)

Isostable reduction with applications to time-dependent partial differential equations.

Science.gov (United States)

Wilson, Dan; Moehlis, Jeff

2016-07-01

Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.

Some models of spin coherence and decoherence in storage rings

International Nuclear Information System (INIS)

Heinemann, K.

1997-09-01

I present some simple exactly solvable models of spin diffusion caused by synchrotron radiation noise in storage rings. I am able to use standard stochastic differential equation and Fokker-Planck methods and I thereby introduce, and exploit, the polarization density. This quantity obeys a linear evolution equation of the Bloch type, which is, like the Fokker-Planck equation, universal in the sense that it is independent of the state of the system. I also briefly consider Bloch equations for other local polarization quantities derived from the polarization density. One of the models chosen is of relevance for some existing and proposed low energy electron (positron) storage rings which need polarization. I present numerical results for a ring with parameters typical of HERA and show that, where applicable, the results of my approach are in satisfactory agreement with calculations using SLIM. These calculations provide a numerical check of a basic tenet of the conventional method of calculating depolarization using the n-vector-axis. I also investigate the equilibrium behaviour of the spin ensemble when there is no synchrotron radiation. Finally, I summarize other results which I have obtained using the polarization density and which will be published separately. (orig.)

Stochastic theory of molecular collisions. II. Application to atom--vibrotor collisions

International Nuclear Information System (INIS)

Augustin, S.D.; Rabitz, H.

1977-01-01

In this work stochastic theory is applied to the treatment of atom--vibrotor collisions. This is an extension of a previous paper which described molecular collisions by a Pauli master equation or a Fokker--Planck equation. In this framework an energy conserving classical path model is explored, and methods for solving the equations numerically are discussed. The coefficients of the Fokker--Planck equation are shown to be expressible as simple functions of the interaction potential. Estimates of the computational labor are also discussed. Finally as a followup on the initial work, numerical solutions of the master equation for the collinear vibrational excitation problem of Secrest and Johnson are presented in an Appendix

Time dependence of entropy flux and entropy production for a dynamical system driven by noises with coloured cross-correlation

Institute of Scientific and Technical Information of China (English)

Xie Wen-Xian; Xu Wei; Cai Li

2007-01-01

This paper shows the Fokker-Planck equation of a dynamical system driven by coloured cross-correlated white noises in the absence and presence of a small external force. Based on the Fokker-Planck equation and the definition of Shannon's information entropy, the time dependence of entropy flux and entropy production can be calculated. The present results can be used to explain the extremal behaviour of time dependence of entropy flux and entropy production in view of the dissipative parameter γ of the system, coloured cross-correlation time τ and coloured cross-correlation strength λ.

Time dependent mean-field games

KAUST Repository

Gomes, Diogo A.

2014-01-06

We consider time dependent mean-field games (MFG) with a local power-like dependence on the measure and Hamiltonians satisfying both sub and superquadratic growth conditions. We establish existence of smooth solutions under a certain set of conditions depending both on the growth of the Hamiltonian as well as on the dimension. In the subquadratic case this is done by combining a Gagliardo-Nirenberg type of argument with a new class of polynomial estimates for solutions of the Fokker-Planck equation in terms of LrLp- norms of DpH. These techniques do not apply to the superquadratic case. In this setting we recur to a delicate argument that combines the non-linear adjoint method with polynomial estimates for solutions of the Fokker-Planck equation in terms of L1L1-norms of DpH. Concerning the subquadratic case, we substantially improve and extend the results previously obtained. Furthermore, to the best of our knowledge, the superquadratic case has not been addressed in the literature yet. In fact, it is likely that our estimates may also add to the current understanding of Hamilton-Jacobi equations with superquadratic Hamiltonians.

Quantum diffusion

International Nuclear Information System (INIS)

Habib, S.

1994-01-01

We consider a simple quantum system subjected to a classical random force. Under certain conditions it is shown that the noise-averaged Wigner function of the system follows an integro-differential stochastic Liouville equation. In the simple case of polynomial noise-couplings this equation reduces to a generalized Fokker-Planck form. With nonlinear noise injection new ''quantum diffusion'' terms rise that have no counterpart in the classical case. Two special examples that are not of a Fokker-Planck form are discussed: the first with a localized noise source and the other with a spatially modulated noise source

Poisson-Nernst-Planck equations with steric effects - non-convexity and multiple stationary solutions

Science.gov (United States)

Gavish, Nir

2018-04-01

We study the existence and stability of stationary solutions of Poisson-Nernst-Planck equations with steric effects (PNP-steric equations) with two counter-charged species. We show that within a range of parameters, steric effects give rise to multiple solutions of the corresponding stationary equation that are smooth. The PNP-steric equation, however, is found to be ill-posed at the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP-Cahn-Hilliard model, show that it is well-posed and that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.

Stochastic diffusion of dust grains by the interplanetary magnetic field

International Nuclear Information System (INIS)

Hassan, M.H.A.; Wallis, M.K.

1983-10-01

The effects of the sectored Interplanetary Magnetic Field on charged dust grains orbiting around the sun under radiation pressure and Poynting-Robertson drag forces are examined for initially circular and non-inclined orbits. The distribution function of the charged grains satisfies a Fokker-Planck equation in which the sectored field is taken as a source of stochastic impulses. By adopting the integrals of the impulse-free motion as variable parameters, the Fokker-Planck equation can be properly treated as a diffusion equation. Analytic solutions of the resulting diffusion equation show that dust grains injected near the ecliptic plane are scattered strongly to high helio-latitudes. The scattering is more pronounced for small grains injected at large distances from the Sun. (author)

Fractional Bhatnagar-Gross-Krook kinetic equation

Science.gov (United States)

Goychuk, Igor

2017-11-01

The linear Boltzmann equation (LBE) approach is generalized to describe fractional superdiffusive transport of the Lévy walk type in external force fields. The time distribution between scattering events is assumed to have a finite mean value and infinite variance. It is completely characterized by the two scattering rates, one fractional and a normal one, which defines also the mean scattering rate. We formulate a general fractional LBE approach and exemplify it with a particularly simple case of the Bohm and Gross scattering integral leading to a fractional generalization of the Bhatnagar, Gross and Krook (BGK) kinetic equation. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given fixed temperature. We show that the retardation effects are indispensable even in the limit of infinite mean scattering rate and argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers-Fokker-Planck (KFP) equation by Barkai and Silbey and its generalization by Friedrich et al. based on the picture of divergent mean time between scattering events. The case of divergent mean time is also discussed at length and compared with the earlier results obtained within the fractional KFP. Also a phenomenological fractional BGK equation without retardation effects is proposed in the limit of infinite scattering rates. It cannot be, however, rigorously derived from a scattering model, being rather clever postulated. It this respect, this retardationless equation is similar to the fractional KFP by Barkai and Silbey. However, it corresponds to the opposite, much more physical limit and, therefore, also presents a viable alternative.

Nonlinear drift-diffusion model of gating in K and nACh ion channels

Energy Technology Data Exchange (ETDEWEB)

Vaccaro, S.R. [Department of Physics, University of Adelaide, Adelaide, South Australia 5005 (Australia)], E-mail: svaccaro@physics.adelaide.edu.au

2007-09-03

The configuration of a sensor regulates the transition between the closed and open states of both voltage and ligand gated channels. The closed state dwell-time distribution f{sub c}(t) derived from a Fokker-Planck equation with a nonlinear diffusion coefficient is in good agreement with experimental data and can account for the power law approximation to f{sub c}(t) for a delayed rectifier K channel and a nicotinic acetylcholine (nACh) ion channel. The solution of a master equation which approximates the Fokker-Planck equation provides a better description of the small time behaviour of the dwell-time distribution and can account for the empirical rate-amplitude correlation for these ion channels.

New and precise construction of the local interstellar electron spectrum from the radio background and an application to the solar modulation of cosmic rays showing an incompatability of the electron and nuclei modulation using the spherically symmetric Fokker-Planck equation

International Nuclear Information System (INIS)

Rockstroh, J.M.

1977-01-01

Cosmic-ray electrons generate the observed radio-frequency background. Previous attempts in the literature to reconcile quantitatively the measured radio-frequency intensity with the intensity deduced from the electron spectrum measured at earth have culminated in the problem that to get the respective emissivities to agree, an unacceptably high interstellar B field must be chosen. In the light of new experimental data on the emissivity as deduced from H II region studies and on the functional dependence of the diffusion coefficient with solar radius and particle rigidity, the assumptions under which the electron emissivity comparison has been made have been reexamined closely. The paradox between predicted and measured emissivity was resolved by ascribing to the magnetic fields of the galaxy a distribution of magnetic field strengths. From modified synchrotron formulas, the interstellar electron spectrum has been constructed from the radio frequency emission data with greatly improved precision. The interstellar electron spectrum has been determined independently of the solar modulation and provides, therefore, an estimate of the absolute depth of the electron modulation. Then the measured electron, proton, and helium-nuclei fluxes were systematically compared to the predictions of the spherically symmetric Fokker-Planck equation using the electron modulation as a base. A previously unnoticed non-tracking of the modulation parameters was observed during the recent recovery that did not occur during the 1965 to 1969 period. Although the argument could be presented just as well by attributing the anomaly to the nuclei, the discussion here arbitrarily tailored it to the electrons, and this new phenomenon was named, the modulation reluctance of the cosmic-ray electrons

Macroscopic plasma properties and stability theory

International Nuclear Information System (INIS)

Sakanaka, P.H.

1981-01-01

1. Two-fluid equations: (a) Boltzmann equation: complete set of equations; collision models - Vlasov, BGK, Fokker-Planck-Landau, Boltzmann. (b) Moments of the Boltzmann equation: problem of closure. (c) Two-fluid equations. 2. One-fluid equation: (a) One-fluid variables. (b) One-fluid equations: quasi-neutrality. (c) Resistive MHD equations. (d) Ideal MHD equations: one-adiabatic approximation; double-adiabatic approximation - CGL. 3. MHD stability problem - energy principle: (a) Linearized ideal MHD equations: force-operator equation. (b) Boundary conditions. (c) Self-adjointness of force operator. (d) The energy principle. 4. Stability problems: application of the energy principle; stability of sharp-boundary plasmas. 5. Thermodynamic approach for stability of plasmas: Newcomb and Rosenbluth's stability criteria. (author)

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

Science.gov (United States)

Chaudhry, Jehanzeb Hameed; Comer, Jeffrey; Aksimentiev, Aleksei; Olson, Luke N.

2013-01-01

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current. PMID:24363784

Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

KAUST Repository

Erban, Radek

2009-01-01

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. © 2009 Society for Industrial and Applied Mathematics.

Time dependent mean-field games

KAUST Repository

Gomes, Diogo A.; Pimentel, Edgard; Sá nchez-Morgado, Hé ctor

2014-01-01

In this setting we recur to a delicate argument that combines the non-linear adjoint method with polynomial estimates for solutions of the Fokker-Planck equation in terms of L1L1-norms of DpH. Concerning the subquadratic case, we substantially improve and extend the results previously obtained. Furthermore, to the best of our knowledge, the superquadratic case has not been addressed in the literature yet. In fact, it is likely that our estimates may also add to the current understanding of Hamilton-Jacobi equations with superquadratic Hamiltonians.

Is the Langevin phase equation an efficient model for oscillating neurons?

Science.gov (United States)

Ota, Keisuke; Tsunoda, Takamasa; Omori, Toshiaki; Watanabe, Shigeo; Miyakawa, Hiroyoshi; Okada, Masato; Aonishi, Toru

2009-12-01

The Langevin phase model is an important canonical model for capturing coherent oscillations of neural populations. However, little attention has been given to verifying its applicability. In this paper, we demonstrate that the Langevin phase equation is an efficient model for neural oscillators by using the machine learning method in two steps: (a) Learning of the Langevin phase model. We estimated the parameters of the Langevin phase equation, i.e., a phase response curve and the intensity of white noise from physiological data measured in the hippocampal CA1 pyramidal neurons. (b) Test of the estimated model. We verified whether a Fokker-Planck equation derived from the Langevin phase equation with the estimated parameters could capture the stochastic oscillatory behavior of the same neurons disturbed by periodic perturbations. The estimated model could predict the neural behavior, so we can say that the Langevin phase equation is an efficient model for oscillating neurons.

Is the Langevin phase equation an efficient model for oscillating neurons?

International Nuclear Information System (INIS)

Ota, Keisuke; Tsunoda, Takamasa; Aonishi, Toru; Omori, Toshiaki; Okada, Masato; Watanabe, Shigeo; Miyakawa, Hiroyoshi

2009-01-01

The Langevin phase model is an important canonical model for capturing coherent oscillations of neural populations. However, little attention has been given to verifying its applicability. In this paper, we demonstrate that the Langevin phase equation is an efficient model for neural oscillators by using the machine learning method in two steps: (a) Learning of the Langevin phase model. We estimated the parameters of the Langevin phase equation, i.e., a phase response curve and the intensity of white noise from physiological data measured in the hippocampal CA1 pyramidal neurons. (b) Test of the estimated model. We verified whether a Fokker-Planck equation derived from the Langevin phase equation with the estimated parameters could capture the stochastic oscillatory behavior of the same neurons disturbed by periodic perturbations. The estimated model could predict the neural behavior, so we can say that the Langevin phase equation is an efficient model for oscillating neurons.

Statistical properties of single-mode emission in free-electron lasers

International Nuclear Information System (INIS)

Bertolotti, M.; Luks, A.; Perina, J.; Perinova, V.; Sibilia, C.

1984-01-01

The authors of this paper discuss the statistical properties of radiation produced in the free electron laser, in the case of singlemode emission when the system is used as an amplifier, with very small gain. The coherent states technique and the q-c number correspondence is employed, starting from the master-equation and obtaining the generalized Fokker-Planck equation for the anti-normal quasidistribution function. Solutions of Fokker-Planck equation provide the photocounting distribution and its factorial moments. No losses are included. It is shown that, in the short-time approximation, the radiation field exhibits antibunching, and that the photocounting distributions, when some suitable conditions on the field intensities are fulfilled, in the stationary regime shows a two-peak behavior, evidencing the existence of bistable states

Linear integral equations and soliton systems

International Nuclear Information System (INIS)

Quispel, G.R.W.

1983-01-01

A study is presented of classical integrable dynamical systems in one temporal and one spatial dimension. The direct linearizations are given of several nonlinear partial differential equations, for example the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the sine-Gordon equation, the nonlinear Schroedinger equation, and the equation of motion for the isotropic Heisenberg spin chain; the author also discusses several relations between these equations. The Baecklund transformations of these partial differential equations are treated on the basis of a singular transformation of the measure (or equivalently of the plane-wave factor) occurring in the corresponding linear integral equations, and the Baecklund transformations are used to derive the direct linearization of a chain of so-called modified partial differential equations. Finally it is shown that the singular linear integral equations lead in a natural way to the direct linearizations of various nonlinear difference-difference equations. (Auth.)

A Planck Vacuum Cosmology

Directory of Open Access Journals (Sweden)

Daywitt W. C.

2009-04-01

Full Text Available Both the big-bang and the quasi-steady-state cosmologies originate in some type of Planck state. This paper presents a new cosmological theory based on the Planck- vacuum negative-energy state, a state consisting of a degenerate collection of negative- energy Planck particles. A heuristic look at the Einstein field equation provides a con- vincing argument that such a vacuum state could provide a theoretical explanation for the visible universe.

Quantum linear Boltzmann equation

International Nuclear Information System (INIS)

Vacchini, Bassano; Hornberger, Klaus

2009-01-01

We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of translation-covariance and on the relation to the classical linear Boltzmann equation. After analyzing its general symmetry properties and the associated relaxation dynamics, we discuss a quantum Monte Carlo method for its numerical solution. We then review important limiting forms of the quantum linear Boltzmann equation, such as the case of quantum Brownian motion and pure collisional decoherence, as well as the application to matter wave optics. Finally, we point to the incorporation of quantum degeneracies and self-interactions in the gas by relating the equation to the dynamic structure factor of the ambient medium, and we provide an extension of the equation to include internal degrees of freedom.

Inherent noise can facilitate coherence in collective swarm motion

KAUST Repository

Yates, C. A.; Erban, R.; Escudero, C.; Couzin, I. D.; Buhl, J.; Kevrekidis, I. G.; Maini, P. K.; Sumpter, D. J. T.

2009-01-01

Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the self-propelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.

Inherent noise can facilitate coherence in collective swarm motion

KAUST Repository

Yates, C. A.

2009-03-31

Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the self-propelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.

Subdiffusive master equation with space-dependent anomalous exponent and structural instability

Science.gov (United States)

Fedotov, Sergei; Falconer, Steven

2012-03-01

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent μ in a bounded domain [0,L] are not structurally stable with respect to the nonhomogeneous variations of parameter μ. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of μ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a “Black Swan,” the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.

Coupled energy-drift and force-balance equations for high-field hot-carrier transport

International Nuclear Information System (INIS)

Huang, Danhong; Alsing, P.M.; Apostolova, T.; Cardimona, D.A.

2005-01-01

Coupled energy-drift and force-balance equations that contain a frictional force for the center-of-mass motion of electrons are derived for hot-electron transport under a strong dc electric field. The frictional force is found to be related to the net rate of phonon emission, which takes away the momentum of a phonon from an electron during each phonon-emission event. The net rate of phonon emission is determined by the Boltzmann scattering equation, which depends on the distribution of electrons interacting with phonons. The work done by the frictional force is included into the energy-drift equation for the electron-relative scattering motion and is found to increase the thermal energy of the electrons. The importance of the hot-electron effect in the energy-drift term under a strong dc field is demonstrated in reducing the field-dependent drift velocity and mobility. The Doppler shift in the energy conservation of scattering electrons interacting with impurities and phonons is found to lead to an anisotropic distribution of electrons in the momentum space along the field direction. The importance of this anisotropic distribution is demonstrated through a comparison with the isotropic energy-balance equation, from which we find that defining a state-independent electron temperature becomes impossible. To the leading order, the energy-drift equation is linearized with a distribution function by expanding it into a Fokker-Planck-type equation, along with the expansions of both the force-balance equation and the Boltzmann scattering equation for hot phonons

Particle Dynamics under Quasi-linear Interaction with Electromagnetic Waves

Energy Technology Data Exchange (ETDEWEB)

Castejon, F.; Eguilior, S.

2003-07-01

Langevin equations for quasi-linear wave particle interaction are obtained taking advantage of the unique vocal equivalence between Fokker-Plank equation and the former ones. Langevin equations are solved numerically and, hence, the evolution of a single particle embedded in an electromagnetic field in momentum space is obtained. The equations are relativistic and valid for any wave. It is also shown that the stochastic part of the equations is negligible in comparison with the deterministic term, except for the momentum to the resonance condition for the main parallel refractive index. (Author) 24 refs.

Particle Dynamics under Quasi-linear Interaction with Electromagnetic Waves

International Nuclear Information System (INIS)

Castejon, F.; Eguilior, S.

2003-01-01

Langevin equations for quasi-linear wave particle interaction are obtained taking advantage of the unique vocal equivalence between Fokker-Plank equation and the former ones. Langevin equations are solved numerically and, hence, the evolution of a single particle embedded in an electromagnetic field in momentum space is obtained. The equations are relativistic and valid for any wave. It is also shown that the stochastic part of the equations is negligible in comparison with the deterministic term, except for the momentum to the resonance condition for the main parallel refractive index. (Author) 24 refs

Nucleation rate of critical droplets on an elastic string in a φ6 potential

International Nuclear Information System (INIS)

Kerr, W.C.; Graham, A.J.

2004-01-01

We obtain the nucleation rate of critical droplets for an elastic string moving in a φ 6 local potential and subject to noise and damping forces. The critical droplet is a bound soliton-antisoliton pair that carries a section of the string out of the metastable central minimum into one of the stable side minima. The frequencies of small oscillations about the critical droplet are obtained from a Heun equation. We solve the Fokker-Planck equation for the phase-space probability density by projecting it onto the eigenfunction basis obtained from the Heun equation. We employ Farkas' 'flux-overpopulation' method to obtain boundary conditions for solving the Fokker-Planck equation; these restrict the validity of our solution to the moderate to heavy damping regime. We present results for the rate as a function of temperature, well depth, and damping

Slowing down of test particles in a plasma (1961); Ralentissement de particules test dans un plasma (1961)

Energy Technology Data Exchange (ETDEWEB)

Belayche, P; Chavy, P; Dupoux, M; Salmon, J [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires

1961-07-01

Numerical solution of the Fokker-Planck equation applied to the slowing down of tritons in a deuterium plasma. After the equations and the boundary conditions have been written, some attention is paid to the numerical tricks used to run the problem on a high speed electronic computer. The numerical results thus obtained are then analyzed and as far as possible, mathematically explained. (authors) [French] Resolution numerique de l'equation de Fokker-Planck appliquee au ralentissement de tritons dans un plasma de deuterium. Apres avoir rappele les equations, les conditions aux limites, l'accent est mis sur les artifices numeriques utilises pour traiter le probleme sur une calculatrice a grande vitesse. Les resultats numeriques obtenus sont ensuite analyses et si possible expliques mathematiquement. En particulier ils peuvent se rattacher a ceux obtenus par application directe de la formule de Spitzer. (auteurs)

Non-Markovian noise

International Nuclear Information System (INIS)

Fulinski, A.

1994-01-01

The properties of non-Markovian noises with exponentially correlated memory are discussed. Considered are dichotomic noise, white shot noise, Gaussian white noise, and Gaussian colored noise. The stationary correlation functions of the non-Markovian versions of these noises are given by linear combinations of two or three exponential functions (colored noises) or of the δ function and exponential function (white noises). The non-Markovian white noises are well defined only when the kernel of the non-Markovian master equation contains a nonzero admixture of a Markovian term. Approximate equations governing the probability densities for processes driven by such non-Markovian noises are derived, including non-Markovian versions of the Fokker-Planck equation and the telegrapher's equation. As an example, it is shown how the non-Markovian nature changes the behavior of the driven linear process

Adaptive Finite Element Method Assisted by Stochastic Simulation of Chemical Systems

KAUST Repository

Cotter, Simon L.; Vejchodský , Tomá š; Erban, Radek

2013-01-01

Stochastic models of chemical systems are often analyzed by solving the corresponding Fokker-Planck equation, which is a drift-diffusion partial differential equation for the probability distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with nonnegligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the stationary probability density. Numerical examples demonstrate that the presented method is competitive with existing a posteriori methods. © 2013 Society for Industrial and Applied Mathematics.

Linear superposition solutions to nonlinear wave equations

International Nuclear Information System (INIS)

Liu Yu

2012-01-01

The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n, n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed

Linear and quasi-linear equations of parabolic type

CERN Document Server

Ladyženskaja, O A; Ural′ceva, N N; Uralceva, N N

1968-01-01

Equations of parabolic type are encountered in many areas of mathematics and mathematical physics, and those encountered most frequently are linear and quasi-linear parabolic equations of the second order. In this volume, boundary value problems for such equations are studied from two points of view: solvability, unique or otherwise, and the effect of smoothness properties of the functions entering the initial and boundary conditions on the smoothness of the solutions.

The exact fundamental solution for the Benes tracking problem

Science.gov (United States)

Balaji, Bhashyam

2009-05-01

The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

Modeling for cardiac excitation propagation based on the Nernst-Planck equation and homogenization.

Science.gov (United States)

Okada, Jun-ichi; Sugiura, Seiryo; Hisada, Toshiaki

2013-06-01

The bidomain model is a commonly used mathematical model of the electrical properties of the cardiac muscle that takes into account the anisotropy of both the intracellular and extracellular spaces. However, the equations contain self-contradiction such that the update of ion concentrations does not consider intracellular or extracellular ion movements due to the gradient of electric potential and the membrane charge as capacitive currents in spite of the fact that those currents are taken into account in forming Kirchhoff's first law. To overcome this problem, we start with the Nernst-Planck equation, the ionic conservation law, and the electroneutrality condition at the cellular level, and by introducing a homogenization method and assuming uniformity of variables at the microscopic scale, we derive rational bidomain equations at the macroscopic level.

Poisson-Boltzmann-Nernst-Planck model

International Nuclear Information System (INIS)

Zheng Qiong; Wei Guowei

2011-01-01

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external

Neoclassical transport including collisional nonlinearity.

Science.gov (United States)

Candy, J; Belli, E A

2011-06-10

In the standard δf theory of neoclassical transport, the zeroth-order (Maxwellian) solution is obtained analytically via the solution of a nonlinear equation. The first-order correction δf is subsequently computed as the solution of a linear, inhomogeneous equation that includes the linearized Fokker-Planck collision operator. This equation admits analytic solutions only in extreme asymptotic limits (banana, plateau, Pfirsch-Schlüter), and so must be solved numerically for realistic plasma parameters. Recently, numerical codes have appeared which attempt to compute the total distribution f more accurately than in the standard ordering by retaining some nonlinear terms related to finite-orbit width, while simultaneously reusing some form of the linearized collision operator. In this work we show that higher-order corrections to the distribution function may be unphysical if collisional nonlinearities are ignored.

Dynamical renormalization group approach to transport in ultrarelativistic plasmas: The electrical conductivity in high temperature QED

International Nuclear Information System (INIS)

Boyanovsky, Daniel; Vega, Hector J. de; Wang Shangyung

2003-01-01

The dc electrical conductivity of an ultrarelativistic QED plasma is studied in real time by implementing the dynamical renormalization group. The conductivity is obtained from the real-time dependence of a dissipative kernel closely related to the retarded photon polarization. Pinch singularities in the imaginary part of the polarization are manifest as secular terms that grow in time in the perturbative expansion of this kernel. The leading secular terms are studied explicitly and it is shown that they are insensitive to the anomalous damping of hard fermions as a result of a cancellation between self-energy and vertex corrections. The resummation of the secular terms via the dynamical renormalization group leads directly to a renormalization group equation in real time, which is the Boltzmann equation for the (gauge invariant) fermion distribution function. A direct correspondence between the perturbative expansion and the linearized Boltzmann equation is established, allowing a direct identification of the self-energy and vertex contributions to the collision term. We obtain a Fokker-Planck equation in momentum space that describes the dynamics of the departure from equilibrium to leading logarithmic order in the coupling. This equation determines that the transport time scale is given by t tr =24 π/e 4 T ln(1/e). The solution of the Fokker-Planck equation approaches asymptotically the steady-state solution as ∼e -t/(4.038...t tr ) . The steady-state solution leads to the conductivity σ=15.698 T/e 2 ln(1/e) to leading logarithmic order. We discuss the contributions beyond leading logarithms as well as beyond the Boltzmann equation. The dynamical renormalization group provides a link between linear response in quantum field theory and kinetic theory

On the Hughes model and numerical aspects

KAUST Repository

Gomes, Diogo A.; Machado Velho, Roberto

2017-01-01

We study a crowd model proposed by R. Hughes in [11] and we describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an eikonal equation with Dirichlet or Neumann data. First, we establish a priori

Some Numerical Aspects on Crowd Motion - The Hughes Model

KAUST Repository

Gomes, Diogo A.; Machado Velho, Roberto

2016-01-01

Here, we study a crowd model proposed by R. Hughes in [5] and we describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. First, we establish a priori

The effect of an alternating electric field on a totally ionised plasma

International Nuclear Information System (INIS)

Baglin, H.; Brin, A.; Ozias, Y.; Salmon, J.

1960-01-01

The equation giving the distribution function of the electrons in a steady-state, for a fully ionized plasma in an a.c. field, are provided from the Fokker-Planck equation. The electric conductivity is complex and depends on the frequency. (author) [fr

Linear determining equations for differential constraints

International Nuclear Information System (INIS)

Kaptsov, O V

1998-01-01

A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach equations of an ideal incompressible fluid and non-linear heat equations are discussed

Transport equation and shock waves

International Nuclear Information System (INIS)

Besnard, D.

1981-04-01

A multi-group method is derived from a one dimensional transport equation for the slowing down and spatial transport of energetic positive ions in a plasma. This method is used to calculate the behaviour of energetic charged particles in non homogeneous and non stationary plasma, and the effect of energy deposition of the particles on the heating of the plasma. In that purpose, an equation for the density of fast ions is obtained from the Fokker-Planck equation, and a closure condition for the second moment of this equation is deduced from phenomenological considerations. This method leads to a numerical method, simple and very efficient, which doesn't require much computer storage. Two types of numerical results are obtained. First, results on the slowing down of 3.5 MeV alpha particles in a 50 keV plasma plublished by Corman and al and Moses are compared with the results obtained with both our method and a Monte Carlo type method. Good agreement was obtained, even for energy deposition on the ions of the plasma. Secondly, we have calculated propagation of alpha particles heating a cold plasma. These results are in very good agreement with those given by an accurate Monte Carlo method, for both the thermal velocity, and the energy deposition in the plasma

Computational-physics program of the National MFE Computer Center

International Nuclear Information System (INIS)

Mirin, A.A.

1982-02-01

The computational physics group is ivolved in several areas of fusion research. One main area is the application of multidimensional Fokker-Planck, transport and combined Fokker-Planck/transport codes to both toroidal and mirror devices. Another major area is the investigation of linear and nonlinear resistive magnetohydrodynamics in two and three dimensions, with applications to all types of fusion devices. The MHD work is often coupled with the task of numerically generating equilibria which model experimental devices. In addition to these computational physics studies, investigations of more efficient numerical algorithms are being carried out

Construction and accuracy of partial differential equation approximations to the chemical master equation.

Science.gov (United States)

Grima, Ramon

2011-11-01

The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

Nernst-Planck modeling of multicomponent ion transport in a Nafion membrane at high current density

NARCIS (Netherlands)

Moshtari Khah, S.; Oppers, N.A.W.; de Groot, M.T.; Keurentjes, J.T.F.; Schouten, J.C.; van der Schaaf, J.

A mathematical model of multicomponent ion transport through a cation-exchange membrane is developed based on the Nernst–Planck equation. A correlation for the non-linear potential gradient is derived from current density relation with fluxes. The boundary conditions are determined with the Donnan

Determination of the macroscopic chloride diffusivity in cementitious by porous materials coupling periodic homogenization of Nernst-Planck equation with experimental protocol

Directory of Open Access Journals (Sweden)

Olivier Millet

2008-03-01

Full Text Available In this paper, we propose a macroscopic migration model for cementitious porous media obtained from periodic homogenization technique. The dimensional analysis of Nernst-Planck equation leads to dimensionless numbers characterizing the problem. According to the order of magnitude of the dimensionless numbers, the homogenization of Nernst-Planck equation leads at the leading order to a macroscopic model where several rates can be coupled or not. For a large applied electrical field accelerating the transfer of ionic species, we obtain a macroscopic model only involving migration. A simple experimental procedure of measurement of the homogenized chlorides diffusivity is then proposed for cement-based materials.

Poisson–Boltzmann–Nernst–Planck model

Science.gov (United States)

Zheng, Qiong; Wei, Guo-Wei

2011-01-01

The Poisson–Nernst–Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst–Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst–Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst–Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson–Boltzmann and Nernst–Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations

Poisson-Boltzmann-Nernst-Planck model.

Science.gov (United States)

Zheng, Qiong; Wei, Guo-Wei

2011-05-21

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external

Analysis of unexpected exits using the Fokker - Planck equation

NARCIS (Netherlands)

Herwaarden, van O.A.

1996-01-01

In this thesis exit problems are considered for stochastic dynamical systems with small random fluctuations. We study exit from a domain in the state space through a boundary, or a specified part of the boundary, that is unattainable in the underlying deterministic system. We analyze

Channeling experiments at planar diamond and silicon single crystals with electrons from the Mainz Microtron MAMI

Science.gov (United States)

Backe, H.; Lauth, W.; Tran Thi, T. N.

2018-04-01

Line structures were observed for (110) planar channeling of electrons in a diamond single crystal even at a beam energy of 180 MeV . This observation motivated us to initiate dechanneling length measurements as function of the beam energy since the occupation of quantum states in the channeling potential is expected to enhance the dechanneling length. High energy loss signals, generated as a result of emission of a bremsstrahlung photon with about half the beam energy at channeling of 450 and 855 MeV electrons, were measured as function of the crystal thickness. The analysis required additional assumptions which were extracted from the numerical solution of the Fokker-Planck equation. Preliminary results for diamond are presented. In addition, we reanalyzed dechanneling length measurements at silicon single crystals performed previously at the Mainz Microtron MAMI at beam energies between 195 and 855 MeV from which we conclude that the quality of our experimental data set is not sufficient to derive definite conclusions on the dechanneling length. Our experimental results are below the predictions of the Fokker-Planck equation and somewhat above the results of simulation calculations of A. V. Korol and A. V. Solov'yov et al. on the basis of the MBN Explorer simulation package. We somehow conservatively conclude that the prediction of the asymptotic dechanneling length on the basis of the Fokker-Planck equation represents an upper limit.

From quantum to semiclassical kinetic equations: Nuclear matter estimates

International Nuclear Information System (INIS)

Galetti, D.; Mizrahi, S.S.; Nemes, M.C.; Toledo Piza, A.F.R. de

1985-01-01

Starting from the exact microscopic time evolution of the quantum one body density associated with a many fermion system semiclassical approximations are derived to it. In the limit where small momentum transfer two body collisions are dominant we get a Fokker-Planck equation and work out friction and diffusion tensors explicitly for nuclear matter. If arbitrary momentum transfers are considered a Boltzmann equation is derived and used to calculate the viscosity coefficient of nuclear matter. A derivation is given of the collision term used by Landau to describe the damping of zero sound waves at low temperature in Plasmas. Memory effects are essential for this. The damping of zero sound waves in nuclear matter is also calculated and the value so obtained associated with the bulk value of the damping of giant resonances in finite nuclei. The bulk value is estimated to be quite small indicating the importance of the nuclear surface for the damping. (Author) [pt

Derivation of a reduced kinetic equation using Lie-transform techniques

International Nuclear Information System (INIS)

Brizard, A.

1991-01-01

The asymptotic elimination of fast time scales from a general kinetic equation, of the form: ∂ t f+z·∂ x f = C[f], facilitates the study of the long time behavior of its solution f(z,t). Here z describe the single-particle Hamiltonian dynamics and the operator C, which may possess nonlinear functional dependence on f, describes processes (such as discrete-particle effects, resonant wave-particle effects, or effects due to external sources) which cause changes in f as it is convectively transported along a Hamiltonian phase-space trajectory. When a fast time scale is associated with z through the dependence on a fast angle θ (whose frequency θ = Ω satisfies ε ≡ 1/Ωτ much-lt 1, where τ is a slow time scale of interest), a near-identity phase-space transformation T ε :z→Z (carried out with Lie-transform techniques) yields reduced Hamiltonian dynamical equations Z ε which are θ-independent. The corresponding transformed kinetic equation is derived. Averaging this equation over the fast angle θ yields a kinetic equation for left-angle F right-angle, the θ-averaged part of F. In general, the θ-dependence of C ε couples the kinetic equations for left-angle F right-angle and F, the θ-dependent part of F. One solves for the Fourier coefficient F l (associated with e ilθ ) as a functional of left-angle F right-angle. One obtains a reduced kinetic equation for left-angle F right-angle: d R left-angle F right-angle/dt = C R [left-angle F right-angle]. General expressions for C R are given, as well as expressions for the guiding-center and oscillation-center phase-space transformations of a linear Fokker-Planck operator. A discussion of the relationship with Mynick's work is presented

Decoupling of the nernst-planck and poisson equations. Application to a membrane system at overlimiting currents.

Science.gov (United States)

Urtenov, Mahamet A-Kh; Kirillova, Evgeniya V; Seidova, Natalia M; Nikonenko, Victor V

2007-12-27

This paper deals with one-dimensional stationary Nernst-Planck and Poisson (NPP) equations describing ion electrodiffusion in multicomponent solution/electrode or ion-conductive membrane systems. A general method for resolving ordinary and singularly perturbed problems with these equations is developed. This method is based on the decoupling of NPP equations that results in deduction of an equation containing only the terms with different powers of the electrical field and its derivatives. Then, the solution of this equation, analytical in several cases or numerical, is substituted into the Nernst-Planck equations for calculating the concentration profile for each ion present in the system. Different ionic species are grouped in valency classes that allows one to reduce the dimension of the original set of equations and leads to a relatively easy treatment of multi-ion systems. When applying the method developed, the main attention is paid to ion transfer at limiting and overlimiting currents, where a significant deviation from local electroneutrality occurs. The boundary conditions and different approximations are examined: the local electroneutrality (LEN) assumption and the original assumption of quasi-uniform distribution of the space charge density (QCD). The relations between the ion fluxes at limiting and overlimiting currents are discussed. In particular, attention is paid to the "exaltation" of counterion transfer toward an ion-exchange membrane by co-ion flux leaking through the membrane or generated at the membrane/solution interface. The structure of the multi-ion concentration field in a depleted diffusion boundary layer (DBL) near an ion-exchange membrane at overlimiting currents is analyzed. The presence of salt ions and hydrogen and hydroxyl ions generated in the course of the water "splitting" reaction is considered. The thickness of the DBL and its different zones, as functions of applied current density, are found by fitting experimental current

A stochastic Friedman Universe with dissipation

International Nuclear Information System (INIS)

Gruszczak, J.

1985-01-01

A probabilistic measure is constructed for the radiation-filled Friedman Universe with bulk viscosity and the equation of state perturbed by a ''white noise''. The corresponding Fokker-Planck equation is solved. In the stochastic evolution singularities turn out to be irrelevant. 3 refs., 1 fig. (author)

Effect of Ponderomotive Terms on Heat Flux in Laser-Produced Plasmas

Science.gov (United States)

Li, G.

2005-10-01

A laser electromagnetic field introduces ponderomotive termsootnotetextV. N. Goncharov and G. Li, Phys. Plasmas 11, 5680 (2004). in the heat flux in a plasma. To account for the nonlocal effects in the ponderomotive terms, first, the kinetic equation coupled with the Maxwell equations is numerically solved for the isotropic part of the electron distribution function. Such an equation includes self-consistent electromagnetic fields and laser absorption through the inverse bremsstrahlung. Then, the anisotropic part is found by solving a simplified Fokker--Planck equation. Using the distribution function, the electric current and heat flux are obtained and substituted into the hydrocode LILAC to simulate ICF implosions. The simulation results are compared against the existing nonlocal electron conduction modelsootnotetextG. P. Schurtz, P. D. Nicola"i, and M. Busquet, Phys. Plasmas 9, 4238 (2000). and Fokker--Planck simulations. This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-92SF19460.

Current-drive theory II: the lower-hybrid wave

International Nuclear Information System (INIS)

Fisch, N.J.

1986-01-01

The theory of current-drive seeks to predict the efficiency with which an external power source can produce current in a plasma torus. The theory, which is now well supported by experimental data, becomes especially simple in the important limit of lower-hybrid or electron-cyclotron waves interacting with superthermal electrons. The solution of an equation adjoint to the linearized Fokker-Planck equation gives both the steady-state and ramp-up current-drive efficiencies. Other phenomena, such as rf-induced runaway rates, rf-induced radiation, etc., may be calculated by this method, and analytical solutions have been obtained in several limiting cases. 12 refs

Isomorphism of Intransitive Linear Lie Equations

Directory of Open Access Journals (Sweden)

Jose Miguel Martins Veloso

2009-11-01

Full Text Available We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.

Steady-State Electrodiffusion from the Nernst-Planck Equation Coupled to Local Equilibrium Monte Carlo Simulations.

Science.gov (United States)

Boda, Dezső; Gillespie, Dirk

2012-03-13

We propose a procedure to compute the steady-state transport of charged particles based on the Nernst-Planck (NP) equation of electrodiffusion. To close the NP equation and to establish a relation between the concentration and electrochemical potential profiles, we introduce the Local Equilibrium Monte Carlo (LEMC) method. In this method, Grand Canonical Monte Carlo simulations are performed using the electrochemical potential specified for the distinct volume elements. An iteration procedure that self-consistently solves the NP and flux continuity equations with LEMC is shown to converge quickly. This NP+LEMC technique can be used in systems with diffusion of charged or uncharged particles in complex three-dimensional geometries, including systems with low concentrations and small applied voltages that are difficult for other particle simulation techniques.

Parabolic equations in biology growth, reaction, movement and diffusion

CERN Document Server

Perthame, Benoît

2015-01-01

This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.

The plasma transport equations derived by multiple time-scale expansions and turbulent transport. I. General theory

International Nuclear Information System (INIS)

Edenstrasser, J.W.

1995-01-01

A multiple time-scale derivative expansion scheme is applied to the dimensionless Fokker--Planck equation and to Maxwell's equations, where the parameter range of a typical fusion plasma was assumed. Within kinetic theory, the four time scales considered are those of Larmor gyration, particle transit, collisions, and classical transport. The corresponding magnetohydrodynamic (MHD) time scales are those of ion Larmor gyration, Alfven, MHD collision, and resistive diffusion. The solution of the zeroth-order equations results in the force-free equilibria and ideal Ohm's law. The solution of the first-order equations leads under the assumption of a weak collisional plasma to the ideal MHD equations. On the MHD-collision time scale, not only the full set of the MHD transport equations is obtained, but also turbulent terms, where the related transport quantities are one order in the expansion parameter larger than those of classical transport. Finally, at the resistive diffusion time scale the known transport equations are arrived at including, however, also turbulent contributions. copyright 1995 American Institute of Physics

Coding of Information in Limit Cycle Oscillators

Science.gov (United States)

Schleimer, Jan-Hendrik; Stemmler, Martin

2009-12-01

Starting from a general description of noisy limit cycle oscillators, we derive from the Fokker-Planck equations the linear response of the instantaneous oscillator frequency to a time-varying external force. We consider the time series of zero crossings of the oscillator’s phase and compute the mutual information between it and the driving force. A direct link is established between the phase response curve summarizing the oscillator dynamics and the ability of a limit cycle oscillator, such as a heart cell or neuron, to encode information in the timing of peaks in the oscillation.

Students’ difficulties in solving linear equation problems

Science.gov (United States)

Wati, S.; Fitriana, L.; Mardiyana

2018-03-01

A linear equation is an algebra material that exists in junior high school to university. It is a very important material for students in order to learn more advanced mathematics topics. Therefore, linear equation material is essential to be mastered. However, the result of 2016 national examination in Indonesia showed that students’ achievement in solving linear equation problem was low. This fact became a background to investigate students’ difficulties in solving linear equation problems. This study used qualitative descriptive method. An individual written test on linear equation tasks was administered, followed by interviews. Twenty-one sample students of grade VIII of SMPIT Insan Kamil Karanganyar did the written test, and 6 of them were interviewed afterward. The result showed that students with high mathematics achievement donot have difficulties, students with medium mathematics achievement have factual difficulties, and students with low mathematics achievement have factual, conceptual, operational, and principle difficulties. Based on the result there is a need of meaningfulness teaching strategy to help students to overcome difficulties in solving linear equation problems.

Radio-frequency wave enhanced runaway production rate

International Nuclear Information System (INIS)

Chan, V.S.; McClain, F.W.

1983-01-01

Enhancement of runaway electron production (over that of an Ohmic discharge) can be achieved by the addition of radio-frequency waves. This effect is studied analytically and numerically using a two-dimensional Fokker--Planck quasilinear equation

Computing with linear equations and matrices

International Nuclear Information System (INIS)

Churchhouse, R.F.

1983-01-01

Systems of linear equations and matrices arise in many disciplines. The equations may accurately represent conditions satisfied by a system or, more likely, provide an approximation to a more complex system of non-linear or differential equations. The system may involve a few or many thousand unknowns and each individual equation may involve few or many of them. Over the past 50 years a vast literature on methods for solving systems of linear equations and the associated problems of finding the inverse or eigenvalues of a matrix has been produced. These lectures cover those methods which have been found to be most useful for dealing with such types of problem. References are given where appropriate and attention is drawn to the possibility of improved methods for use on vector and parallel processors. (orig.)

Correct Linearization of Einstein's Equations

Directory of Open Access Journals (Sweden)

Rabounski D.

2006-06-01

Full Text Available Regularly Einstein's equations can be reduced to a wave form (linearly dependent from the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel's symbols. As shown here, the origin of the problem is that one uses the general covariant theory of measurement. Here the wave form of Einstein's equations is obtained in the terms of Zelmanov's chronometric invariants (physically observable projections on the observer's time line and spatial section. The obtained equations depend on solely the second derivatives even if gravitation, the space rotation and Christoffel's symbols. The correct linearization proves: the Einstein equations are completely compatible with weak waves of the metric.

Diffusive limits for linear transport equations

International Nuclear Information System (INIS)

Pomraning, G.C.

1992-01-01

The authors show that the Hibert and Chapman-Enskog asymptotic treatments that reduce the nonlinear Boltzmann equation to the Euler and Navier-Stokes fluid equations have analogs in linear transport theory. In this linear setting, these fluid limits are described by diffusion equations, involving familiar and less familiar diffusion coefficients. Because of the linearity extant, one can carry out explicitly the initial and boundary layer analyses required to obtain asymptotically consistent initial and boundary conditions for the diffusion equations. In particular, the effects of boundary curvature and boundary condition variation along the surface can be included in the boundary layer analysis. A brief review of heuristic (nonasymptotic) diffusion description derivations is also included in our discussion

Spectral theories for linear differential equations

International Nuclear Information System (INIS)

Sell, G.R.

1976-01-01

The use of spectral analysis in the study of linear differential equations with constant coefficients is not only a fundamental technique but also leads to far-reaching consequences in describing the qualitative behaviour of the solutions. The spectral analysis, via the Jordan canonical form, will not only lead to a representation theorem for a basis of solutions, but will also give a rather precise statement of the (exponential) growth rates of various solutions. Various attempts have been made to extend this analysis to linear differential equations with time-varying coefficients. The most complete such extensions is the Floquet theory for equations with periodic coefficients. For time-varying linear differential equations with aperiodic coefficients several authors have attempted to ''extend'' the Foquet theory. The precise meaning of such an extension is itself a problem, and we present here several attempts in this direction that are related to the general problem of extending the spectral analysis of equations with constant coefficients. The main purpose of this paper is to introduce some problems of current research. The primary problem we shall examine occurs in the context of linear differential equations with almost periodic coefficients. We call it ''the Floquet problem''. (author)

Effects of Drift-Shell Splitting by Chorus Waves on Radiation Belt Electrons

Science.gov (United States)

Chan, A. A.; Zheng, L.; O'Brien, T. P., III; Tu, W.; Cunningham, G.; Elkington, S. R.; Albert, J.

2015-12-01

Drift shell splitting in the radiation belts breaks all three adiabatic invariants of charged particle motion via pitch angle scattering, and produces new diffusion terms that fully populate the diffusion tensor in the Fokker-Planck equation. Based on the stochastic differential equation method, the Radbelt Electron Model (REM) simulation code allows us to solve such a fully three-dimensional Fokker-Planck equation, and to elucidate the sources and transport mechanisms behind the phase space density variations. REM has been used to perform simulations with an empirical initial phase space density followed by a seed electron injection, with a Tsyganenko 1989 magnetic field model, and with chorus wave and ULF wave diffusion models. Our simulation results show that adding drift shell splitting changes the phase space location of the source to smaller L shells, which typically reduces local electron energization (compared to neglecting drift-shell splitting effects). Simulation results with and without drift-shell splitting effects are compared with Van Allen Probe measurements.

Two-dimensional and relativistic effects in lower-hybrid current drive

International Nuclear Information System (INIS)

Hewett, D.; Hizanidis, K.; Krapchev, V.; Bers, A.

1983-06-01

We present new numerical and analytic solutions of the two-dimensional Fokker-Planck equation supplemented by a parallel quasilinear diffusion term. The results show a large enhancement of the perpendicular temperature of both the electrons resonant with the applied RF fields and the more energetic electrons in the tail. Both the RF-generated current and power dissipated are substantially increased by the perpendicular energy broadening in the resonant region. In the presence of a small DC electric field the RF current generated is very much enhanced, much more than in a simple additive fashion. In addition, we present a relativistic formulation of the two-dimensional Fokker-Planck quasilinear equation. From conservation equations, based upon this formulation, we derive the characteristics of RF current drive with energetic electrons. These show how the RF-driven current and its figure of merit (I/P/sub d/) increase with the energy of the current-carrying electrons, and that their perpendicular, random momentum must also increase

Lie algebras and linear differential equations.

Science.gov (United States)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

Transient finite element analysis of electric double layer using Nernst-Planck-Poisson equations with a modified Stern layer.

Science.gov (United States)

Lim, Jongil; Whitcomb, John; Boyd, James; Varghese, Julian

2007-01-01

A finite element implementation of the transient nonlinear Nernst-Planck-Poisson (NPP) and Nernst-Planck-Poisson-modified Stern (NPPMS) models is presented. The NPPMS model uses multipoint constraints to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The Poisson-Boltzmann equation is used to provide a limited check of the transient models for low surface potential and dilute bulk solutions. The effects of the surface potential and bulk molarity on the electric potential and ion concentrations as functions of space and time are studied. The ability of the models to predict realistic energy storage capacity is investigated. The predicted energy is much more sensitive to surface potential than to bulk solution molarity.

Diffusion-controlled reactions modeling in Geant4-DNA

Czech Academy of Sciences Publication Activity Database

Karamitros, M.; Luan, S.; Bernal, M. A.; Allison, J.; Baldacchino, G.; Davídková, Marie; Francis, Z.; Friedland, W.; Ivanchenko, A.; Ivanchenko, V.; Mantero, A.; Nieminen, P.; Santin, G.; Tran, H. N.; Stepan, V.; Incerti, S.

2014-01-01

Roč. 274, OCT (2014), s. 841-882 ISSN 0021-9991 Institutional support: RVO:61389005 Keywords : chemical kinetics simulation * radiation chemistry * Fokker-Planck equation * Smoluchowski diffusion equation * Brownian bridge * dynamical time steps * k-d tree * radiolysis * radiobiology * Geant4-DNA * Brownian dynamics Subject RIV: BO - Biophysics Impact factor: 2.434, year: 2014

Transport theory of deep-inelastic collisions between heavy nuclei

International Nuclear Information System (INIS)

Ayik, S.; Noerenberg, W.; Schuermann, B.

1975-01-01

In collisions between heavy nuclei, the major part of the total cross-section is due to deep-inelastic processes. These processes have been studied within a quantum-statistical approach leading to transport equations of the Fokker-Planck type (generalized diffusion equation). Transport coefficients have been studied within a model. (orig./WL) [de

The validity of quantum-classical multi-channel diffusion equations describing interlevel transitions in the condensed phase. The adiabatic representation

CERN Document Server

Basilevsky, M V

2002-01-01

We develop an approach for derivation of quantum-classical relaxation equations for a two-channel problem. The treatment is based on the adiabatic channel wavefunctions and the system-bath coupling is modelled as a bilinear interaction in momentum representation. In the quantum-classical limit we obtain Liouville equations with the relaxation operator containing diffusion terms diagonal in Liouvillian space and the off-diagonal part which is responsible for thermal interlevel transitions. The high-frequency interlevel quantum beats are fully taken into account in this relaxation term. In the framework of the present formulation and as a consequence of the momentum-dependent interaction the Smoluchovsky diffusion limit can be reached without invoking Fokker-Planck equations as an intermediate step. The inherent property of equations so obtained is that the partial rates of interlevel transitions obey the principle of detailed balance. This result could not be gained in earlier treatments of the two-level diffu...

Linear q-nonuniform difference equations

International Nuclear Information System (INIS)

Bangerezako, Gaspard

2010-01-01

We introduce basic concepts of q-nonuniform differentiation and integration and study linear q-nonuniform difference equations and systems, as well as their application in q-nonuniform difference linear control systems. (author)

First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations

Energy Technology Data Exchange (ETDEWEB)

Schmuck, Markus [Imperial College, London (United Kingdom). Depts. of Chemical Engineering and Mathematics

2012-04-15

We study the well-accepted Poisson-Nernst-Planck equations modeling transport of charged particles. By formal multiscale expansions we rederive the porous media formulation obtained by two-scale convergence in [42, 43]. The main result is the derivation of the error which occurs after replacing a highly heterogeneous solid-electrolyte composite by a homogeneous one. The derived estimates show that the approximation errors for both, the ion densities quantified in L{sup 2}-norm and the electric potential measured in H{sup 1}-norm, are of order O(s{sup 1/2}). (orig.)

From Brownian Dynamics to Markov Chain: An Ion Channel Example

KAUST Repository

Chen, Wan; Erban, Radek; Chapman, S. Jonathan

2014-01-01

is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal

Stochastic Stokes' Drift, Homogenized Functional Inequalities, and Large Time Behavior of Brownian Ratchets

KAUST Repository

Blanchet, Adrien; Dolbeault, Jean; Kowalczyk, MichaŁ

2009-01-01

-similar variables attached to t he center of mass to a stationary solution of a Fokker-Planck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front

An x-space analysis of evolution equations: Soffer's inequality and the non-forward evolution

International Nuclear Information System (INIS)

Cafarella, Alessandro; Coriano, Claudio; Guzzi, Marco

2003-01-01

We analyze the use of algorithms based in x-space for the solution of renormalization group equations of DGLAP-type and test their consistency by studying bounds among partons distributions - in our specific case Soffer's inequality and the perturbative behaviour of the nucleon tensor charge - to next-to-leading order in QCD. A discussion of the perturbative resummation implicit in these expansions using Mellin moments is included. We also comment on the (kinetic) proof of positivity of the evolution of h1, using a kinetic analogy and illustrate the extension of the algorithm to the evolution of generalized parton distributions. We prove positivity of the non-forward evolution in a special case and illustrate a Fokker-Planck approximation to it. (author)

Fokker-type dynamics with three-body correlations

International Nuclear Information System (INIS)

Salas, A.; Sanchez-Ron, J.M.

1981-01-01

Dynamical systems of N point particles without internal degrees of freedom are studied. Their equations of motion are derived from a Fokker-type variational principle with n-body correlations (n = 2,3,...,N), with special emphasis on the case n = 3. The distinction between n-body correlation and n-body effective force is analyzed in detail, with the help of an example. Maximal sets of independent three-body Poincare-invariant scalars are given. An example of three-body correlation formally similar to the usual two-body long-range scalar correlation is given and discussed. (author)

Dissipation and fluctuations within a timedependent microscopic description of deep inelastic collisions

International Nuclear Information System (INIS)

Hofmann, H.

1978-01-01

In this talk I want to give a brief description of the linear response approach to deep inelastic collisions. In this model the relative motion of the two fragments as well as all slow 'shape-degrees' of the composite system are treated within the same physical picture. Among these 'shape-degrees' the most important ones will be those which represent: mass asymmetry, collective vibrations, and, possibly, rotations. The goal of the approach is an equation of motion which accounts for dissipative effects as well as for statistical fluctuations of all these macroscopic degrees and their conjugate momenta. It will be a Fokker Planck equation for the probability distribution in the classical phase space. (orig.) [de

Schwarz maps of algebraic linear ordinary differential equations

Science.gov (United States)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods

KAUST Repository

Machado Velho, Roberto

2017-09-10

In this dissertation, we present two research projects, namely finite-state mean-field games and the Hughes model for the motion of crowds. In the first part, we describe finite-state mean-field games and some applications to socio-economic sciences. Examples include paradigm shifts in the scientific community and the consumer choice behavior in a free market. The corresponding finite-state mean-field game models are hyperbolic systems of partial differential equations, for which we propose and validate a new numerical method. Next, we consider the dual formulation to two-state mean-field games, and we discuss numerical methods for these problems. We then depict different computational experiments, exhibiting a variety of behaviors, including shock formation, lack of invertibility, and monotonicity loss. We conclude the first part of this dissertation with an investigation of the shock structure for two-state problems. In the second part, we consider a model for the movement of crowds proposed by R. Hughes in [56] and describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. We first establish a priori estimates for the solutions. Next, we consider radial solutions, and we identify a shock formation mechanism. Subsequently, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. We also propose a new numerical method for the solution of Fokker-Planck equations and then to systems of PDEs composed by a Fokker-Planck equation and a potential type equation. Finally, we illustrate the use of the numerical method both to the Hughes model and mean-field games. We also depict cases such as the evacuation of a room and the movement of persons around Kaaba (Saudi Arabia).

Noise-induced multistability in chemical systems: Discrete versus continuum modeling

Czech Academy of Sciences Publication Activity Database

Duncan, A.; Liao, S.; Vejchodský, Tomáš; Erban, R.; Grima, R.

2015-01-01

Roč. 91, č. 4 (2015), s. 042111 ISSN 1539-3755 EU Projects: European Commission(XE) 328008 - STOCHDETBIOMODEL Institutional support: RVO:67985840 Keywords : chemical master equation * chemical Fokker-Planck equation * multimodality Subject RIV: BA - General Mathematics Impact factor: 2.288, year: 2014 http://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.042111

Noise-induced multistability in chemical systems: Discrete versus continuum modeling

Czech Academy of Sciences Publication Activity Database

Duncan, A.; Liao, S.; Vejchodský, Tomáš; Erban, R.; Grima, R.

2015-01-01

Roč. 91, č. 4 (2015), s. 042111 ISSN 1539-3755 EU Projects: European Commission(XE) 328008 - STOCHDETBIOMODEL Institutional support: RVO:67985840 Keywords : chemical master equation * chemical Fokker-Planck equation * multimodality Subject RIV: BA - General Mathematics Impact factor: 2.288, year: 2014 http://journals. aps .org/pre/abstract/10.1103/PhysRevE.91.042111

Response statistics of rotating shaft with non-linear elastic restoring forces by path integration

Science.gov (United States)

Gaidai, Oleg; Naess, Arvid; Dimentberg, Michael

2017-07-01

Extreme statistics of random vibrations is studied for a Jeffcott rotor under uniaxial white noise excitation. Restoring force is modelled as elastic non-linear; comparison is done with linearized restoring force to see the force non-linearity effect on the response statistics. While for the linear model analytical solutions and stability conditions are available, it is not generally the case for non-linear system except for some special cases. The statistics of non-linear case is studied by applying path integration (PI) method, which is based on the Markov property of the coupled dynamic system. The Jeffcott rotor response statistics can be obtained by solving the Fokker-Planck (FP) equation of the 4D dynamic system. An efficient implementation of PI algorithm is applied, namely fast Fourier transform (FFT) is used to simulate dynamic system additive noise. The latter allows significantly reduce computational time, compared to the classical PI. Excitation is modelled as Gaussian white noise, however any kind distributed white noise can be implemented with the same PI technique. Also multidirectional Markov noise can be modelled with PI in the same way as unidirectional. PI is accelerated by using Monte Carlo (MC) estimated joint probability density function (PDF) as initial input. Symmetry of dynamic system was utilized to afford higher mesh resolution. Both internal (rotating) and external damping are included in mechanical model of the rotor. The main advantage of using PI rather than MC is that PI offers high accuracy in the probability distribution tail. The latter is of critical importance for e.g. extreme value statistics, system reliability, and first passage probability.

Estimations hypoelliptiques globales et compacit\\'e de la r\\'esolvante Estimations hypoelliptiques globales et compacit\\'e de la r\\'esolvante pour des op\\'erateurs de Fokker-Planck ou des laplaciens de Witten

CERN Document Server

Helffer, B

2004-01-01

Ces derni\\`eres ann\\'ees, les estimations hypoelliptiques ont connu une nouvelle jeunesse en liaison avec des questions provenant de la th\\'eorie cin\\'etique des gaz. Dans cette direction de nombreux auteurs ont en effet eu besoin de d\\'emontrer des estimations maximales pour en d\\'eduire la compacit\\'e de l'op\\'erateur de Fokker-Planck et avoir des estimations sur la r\\'esolvante permettant d'aborder la question du retour \\`a l'\\'equilibre. Dans un article tr\\`es r\\'ecent, F.~H\\'erau et F.~Nier (inspir\\'es par des calculs explicites du livre de Risken) ont mis en \\'evidence les liens \\'etroits entre ces questions et des questions analogues pour un laplacien de Witten. L'\\'etude de ces liens est poursuivie et syst\\'ematis\\'ee dans un livre en pr\\'eparation \\'ecrit en collaboration avec F.~Nier dont nous allons pr\\'esenter quelques aspects ici en pr\\'esentant parfois un \\'eclairage diff\\'erent sur un probl\\`eme qui laisse encore beaucoup de conjectures non r\\'esolues.

Basic linear partial differential equations

CERN Document Server

Treves, Francois

1975-01-01

Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches.

Science.gov (United States)

Xu, Zhenli; Ma, Manman; Liu, Pei

2014-07-01

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Non-local quasi-linear parabolic equations

International Nuclear Information System (INIS)

Amann, H

2005-01-01

This is a survey of the most common approaches to quasi-linear parabolic evolution equations, a discussion of their advantages and drawbacks, and a presentation of an entirely new approach based on maximal L p regularity. The general results here apply, above all, to parabolic initial-boundary value problems that are non-local in time. This is illustrated by indicating their relevance for quasi-linear parabolic equations with memory and, in particular, for time-regularized versions of the Perona-Malik equation of image processing

The Cauchy problem for non-linear Klein-Gordon equations

International Nuclear Information System (INIS)

Simon, J.C.H.; Taflin, E.

1993-01-01

We consider in R n+1 , n≥2, the non-linear Klein-Gordon equation. We prove for such an equation that there is neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincare covariant then the non-linear representation of the Poincare-Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincare group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincare group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra. (orig.)

Multiple solutions of steady-state Poisson–Nernst–Planck equations with steric effects

International Nuclear Information System (INIS)

Lin, Tai-Chia; Eisenberg, Bob

2015-01-01

Experiments measuring currents through single protein channels show unstable currents. Channels switch between ‘open’ or ‘closed’ states in a spontaneous stochastic process called gating. Currents are either (nearly) zero or at a definite level, characteristic of each type of protein, independent of time, once the channel is open. The steady state Poisson–Nernst–Planck equations with steric effects (PNP-steric equations) describe steady current through the open channel quite well, in a wide variety of conditions. Here we study the existence of multiple solutions of steady state PNP-steric equations to see if they themselves, without modification or augmentation, can describe two levels of current. We prove that there are two steady state solutions of PNP-steric equations for (a) three types of ion species (two types of cations and one type of anion) with a positive constant permanent charge, and (b) four types of ion species (two types of cations and their counter-ions) with a constant permanent charge but no sign condition. The excess currents (due to steric effects) associated with these two steady state solutions are derived and expressed as two distinct formulas. Our results indicate that PNP-steric equations may become a useful model to study spontaneous gating of ion channels. Spontaneous gating is thought to involve small structural changes in the channel protein that perhaps produce large changes in the profiles of free energy that determine ion flow. Gating is known to be modulated by external structures. Both can be included in future extensions of our present analysis. (paper)

Mobility and volatility: What is behind the rising income inequality in the United States

Science.gov (United States)

Wu, Huixuan; Li, Yao

2018-02-01

Inequality of family incomes in the United States has increased significantly in the past four decades. This is largely interpreted as a result of unequal mobility, e.g., the rich can get richer at a faster pace than the rest of the population. However, using nationally representative data and the Fokker-Planck equation, our study shows that income mobility in the United States has remained stable. Instead, we find another factor - income volatility, which measures the instability of incomes - has increased considerably and caused the surge of income inequality. In addition, the rising volatility is associated with the plummeting of income-growth opportunity, creating the feeling that the American Dream is in decline. Volatility has often been overlooked in previous studies on inequality, partially because mobility and volatility are usually studied separately. By contrast, the Fokker-Planck equation takes both mobility and volatility into consideration, making it a more comprehensive model.

Hamiltonian structures of some non-linear evolution equations

International Nuclear Information System (INIS)

Tu, G.Z.

1983-06-01

The Hamiltonian structure of the O(2,1) non-linear sigma model, generalized AKNS equations, are discussed. By reducing the O(2,1) non-linear sigma model to its Hamiltonian form some new conservation laws are derived. A new hierarchy of non-linear evolution equations is proposed and shown to be generalized Hamiltonian equations with an infinite number of conservation laws. (author)

Liouville's theorem and phase-space cooling

International Nuclear Information System (INIS)

Mills, R.L.; Sessler, A.M.

1993-01-01

A discussion is presented of Liouville's theorem and its consequences for conservative dynamical systems. A formal proof of Liouville's theorem is given. The Boltzmann equation is derived, and the collisionless Boltzmann equation is shown to be rigorously true for a continuous medium. The Fokker-Planck equation is derived. Discussion is given as to when the various equations are applicable and, in particular, under what circumstances phase space cooling may occur

Solving polynomial differential equations by transforming them to linear functional-differential equations

OpenAIRE

Nahay, John Michael

2008-01-01

We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generali...

Amplitude growth due to random, correlated kicks

International Nuclear Information System (INIS)

Michelotti, L.; Mills, F.

1989-03-01

Historically, stochastic processes, such as gas scattering or stochastic cooling, have been treated by the Fokker-Planck equation. In this approach, usually considered for one dimension only, the equation can be considered as a continuity equation for a variable which would be a constant of the motion in the absence of the stochastic process, for example, the action variable, I = ε/2π for betatron oscillations, where ε is the area of the Courant-Snyder ellipse, or energy in the case of unbunched beams, or the action variable for phase oscillations in case the beam is bunched. A flux, /Phi/, including diffusive terms can be defined, usually to second order. /Phi/ = M 1 F(I) + M 2 ∂F/∂I + /hor ellipsis/. M 1 and M 2 are the expectation values of δI and (δI) 2 due to the individual stochastic kicks over some period of time, long enough that the variance of these quantities is sufficiently small. Then the Fokker-Planck equation is just ∂F/∂I + ∂/Phi//∂I = 0. In many cases those where the beam distribution has already achieved its final shape, it is sufficient to find the rate of increase of by taking simple averages over the Fokker-Planck equation. At the time this work was begun, there was good knowledge of the second moment for general stochastic processes due to stochastic cooling theory, but the form of the first moment was known only for extremely wideband processes. The purposes of this note are to derive an expression relating the expected single particle amplitude growth to the noise autocorrelation function and to obtain, thereby, the form of M 1 for narrow band processes. 4 refs

A finite difference method for numerical solution of the Nernst-Planck equations when convective flux and electric current are involved

International Nuclear Information System (INIS)

Aguilera, V.M.; Garrido, J.; Mafe, S.; Pellicer, J.

1985-01-01

An algorithm for the solution of Nernst-Planck equations with simultaneous convective flux and electric current has been developed without using Poisson's equation. The numerical simulation which has been developed reproduces the behaviour of the system employing their experimental variables as parameters of the algorithm. However, other procedures are only capable of dealing with some of the experimental conditions described here. The agreement between the theoretically predicted values and the experimentally obtained is quite reasonable. (author)

The Dynamics of M15: Observations of the Velocity Dispersion Profile and Fokker-Planck Models

Science.gov (United States)

Dull, J. D.; Cohn, H. N.; Lugger, P. M.; Murphy, B. W.; Seitzer, P. O.; Callanan, P. J.; Rutten, R. G. M.; Charles, P. A.

1997-05-01

We report a new measurement of the velocity dispersion profile within 1' (3 pc) of the center of the globular cluster M15 (NGC 7078), using long-slit spectra from the 4.2 m William Herschel Telescope at La Palma Observatory. We obtained spatially resolved spectra for a total of 23 slit positions during two observing runs. During each run, a set of parallel slit positions was used to map out the central region of the cluster; the position angle used during the second run was orthogonal to that used for the first. The spectra are centered in wavelength near the Ca II infrared triplet at 8650 Å, with a spectral range of about 450 Å. We determined radial velocities by cross-correlation techniques for 131 cluster members. A total of 32 stars were observed more than once. Internal and external comparisons indicate a velocity accuracy of about 4 km s-1. The velocity dispersion profile rises from about σ = 7.2 +/- 1.4 km s-1 near 1' from the center of the cluster to σ = 13.9 +/- 1.8 km s-1 at 20". Inside of 20", the dispersion remains approximately constant at about 10.2 +/- 1.4 km s-1 with no evidence for a sharp rise near the center. This last result stands in contrast with that of Peterson, Seitzer, & Cudworth who found a central velocity dispersion of 25 +/- 7 km s-1, based on a line-broadening measurement. Our velocity dispersion profile is in good agreement with those determined in the recent studies of Gebhardt et al. and Dubath & Meylan. We have developed a new set of Fokker-Planck models and have fitted these to the surface brightness and velocity dispersion profiles of M15. We also use the two measured millisecond pulsar accelerations as constraints. The best-fitting model has a mass function slope of x = 0.9 (where 1.35 is the slope of the Salpeter mass function) and a total mass of 4.9 × 105 M⊙. This model contains approximately 104 neutron stars (3% of the total mass), the majority of which lie within 6" (0.2 pc) of the cluster center. Since the

Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback.

Science.gov (United States)

Gaudreault, Mathieu; Drolet, François; Viñals, Jorge

2010-11-01

Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average , where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.

Monte Carlo particle simulation and finite-element techniques for tandem mirror transport

International Nuclear Information System (INIS)

Rognlien, T.D.; Cohen, B.I.; Matsuda, Y.; Stewart, J.J. Jr.

1987-01-01

A description is given of numerical methods used in the study of axial transport in tandem mirrors owing to Coulomb collisions and rf diffusion. The methods are Monte Carlo particle simulations and direct solution to the Fokker-Planck equations by finite-element expansion. (author)

Two-dimensional spectroscopy for harmonic vibrational modes with nonlinear system-bath interactions. II. Gaussian-Markovian case

NARCIS (Netherlands)

Tanimura, Y; Steffen, T

2000-01-01

The relaxation processes in a quantum system nonlinearly coupled to a harmonic Gaussian-Markovian heat bath are investigated by the quantum Fokker-Planck equation in the hierarchy form. This model describes frequency fluctuations in the quantum system with an arbitrary correlation time and thus

Non-equipotential magnetic surfaces and mode-transition in tokamaks

International Nuclear Information System (INIS)

Li Xingzhong

1988-01-01

The solution of the Fokker-Planck equation is used to describe a phase transition in velocity space. This transition is related to the mode-transition in tokamaks. After the transition the electrostatic potential on a magnetic surface cannot be considered as a constant. (orig.)

Monte Carlo particle simulation and finite-element techniques for tandem mirror transport

International Nuclear Information System (INIS)

Rognlien, T.D.; Cohen, B.I.; Matsuda, Y.; Stewart, J.J. Jr.

1985-12-01

A description is given of numerical methods used in the study of axial transport in tandem mirrors owing to Coulomb collisions and rf diffusion. The methods are Monte Carlo particle simulations and direct solution to the Fokker-Planck equations by finite-element expansion. 11 refs

Linear differential equations to solve nonlinear mechanical problems: A novel approach

OpenAIRE

Nair, C. Radhakrishnan

2004-01-01

Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential equation is known. Using the known solution of the non-linear differential equation, linear differential equations are set up. The solutions of these linear differential equations are found using standard techniques. Then the solutions of the linear differe...

Linear measure functional differential equations with infinite delay

OpenAIRE

Monteiro, G. (Giselle Antunes); Slavík, A.

2014-01-01

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

Science.gov (United States)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

A kinetic theory of diffusion in general relativity with cosmological scalar field

International Nuclear Information System (INIS)

Calogero, Simone

2011-01-01

A new model to describe the dynamics of particles undergoing diffusion in general relativity is proposed. The evolution of the particle system is described by a Fokker-Planck equation without friction on the tangent bundle of spacetime. It is shown that the energy-momentum tensor for this matter model is not divergence-free, which makes it inconsistent to couple the Fokker-Planck equation to the Einstein equations. This problem can be solved by postulating the existence of additional matter fields in spacetime or by modifying the Einstein equations. The case of a cosmological scalar field term added to the left hand side of the Einstein equations is studied in some details. For the simplest cosmological model, namely the flat Robertson-Walker spacetime, it is shown that, depending on the initial value of the cosmological scalar field, which can be identified with the present observed value of the cosmological constant, either unlimited expansion or the formation of a singularity in finite time will occur in the future. Future collapse into a singularity also takes place for a suitable small but positive present value of the cosmological constant, in contrast to the standard diffusion-free scenario

Saturation and linear transport equation

International Nuclear Information System (INIS)

Kutak, K.

2009-03-01

We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)

Examination of various postulates of irreversibility

Energy Technology Data Exchange (ETDEWEB)

Salmon, J [Conservatoire National des Arts et Metiers (CNAM), 75 - Paris (France)

1977-01-01

Firstly, it is shown that it is necessary to break the reversible character of the B.B.G.K.Y. system of equations by means of a postulate of irreversibility to obtain a kinetic equation compatible with the second principle of thermodynamics. Next, three postulates of irreversibility are examined: that of molecular chaos, that of linear relaxation and, finally, that of superposition. Then the corresponding kinetic equations and the expressions for the viscosity coefficient to which they lead are determined. Comparison with experiment is made each time. Lastly, an attempt to obtain an irreversible kinetic equation without introducing a postulate of irreversibility in the B.B.G.K.Y. system is realized. This consists in adding a complementary irreversible term to the fundamental equation of the dynamics of a particle. The suggested term is of quantum origin and leads to a kinetic equation of the Fokker-Planck type.

Risk-sensitive mean-field games

KAUST Repository

Tembine, Hamidou

2014-04-01

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics. © 1963-2012 IEEE.

Risk-sensitive mean-field games

KAUST Repository

Tembine, Hamidou; Zhu, Quanyan; Başar, Tamer

2014-01-01

In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics. © 1963-2012 IEEE.

Linearized pseudo-Einstein equations on the Heisenberg group

Science.gov (United States)

Barletta, Elisabetta; Dragomir, Sorin; Jacobowitz, Howard

2017-02-01

We study the pseudo-Einstein equation R11bar = 0 on the Heisenberg group H1 = C × R. We consider first order perturbations θɛ =θ0 + ɛ θ and linearize the pseudo-Einstein equation about θ0 (the canonical Tanaka-Webster flat contact form on H1 thought of as a strictly pseudoconvex CR manifold). If θ =e2uθ0 the linearized pseudo-Einstein equation is Δb u - 4 | Lu|2 = 0 where Δb is the sublaplacian of (H1 ,θ0) and L bar is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain Ω ⊂H1 by applying subelliptic theory i.e. existence and regularity results for weak subelliptic harmonic maps. We determine a solution u to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that u(x) → - ∞ as | x | → + ∞.

Stochastic quantization for the axial model

International Nuclear Information System (INIS)

Farina, C.; Montani, H.; Albuquerque, L.C.

1991-01-01

We use bosonization ideas to solve the axial model in the stochastic quantization framework. We obtain the fermion propagator of the theory decoupling directly the Langevin equation, instead of the Fokker-Planck equation. In the Appendix we calculate explicitly the anomalous divergence of the axial-vector current by using a regularization that does not break the Markovian character of the stochastic process

Statistical approach to bistable behaviour of a nonlinear system in a stationary field

International Nuclear Information System (INIS)

Luks, A.; Perina, J.; Perinova, V.; Bertolotti, M.; Sibilia, C.

1984-01-01

The quantum statistical properties of an elastic scattering process are investigated comprising crossed light beams which are in interaction with a particle (electron) beam treated as ''two-step'' system. Using the master equation and the generalized Fokker-Planck equation techniques, the integrated intensities are characterized by their probability distributions and it is demonstrated that single modes exhibit two-peak bistable behaviour. (author)

Quasi-steady State Reduction of Molecular Motor-Based Models of Directed Intermittent Search

KAUST Repository

Newby, Jay M.; Bressloff, Paul C.

2010-01-01

at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker-Planck (FP) equation, which has

From Brownian Dynamics to Markov Chain: An Ion Channel Example

KAUST Repository

Chen, Wan

2014-02-27

A discrete rate theory for multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model, one can determine the Markovian transition rates. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed. © 2014 Society for Industrial and Applied Mathematics.

Electron dynamics with radiation and nonlinear wigglers

International Nuclear Information System (INIS)

Jowett, J.M.

1986-06-01

The physics of electron motion in storage rings is described by supplementing the Hamiltonian equations of motion with fluctuating radiation reaction forces to describe the effects of synchrotron radiation. This leads to a description of radiation damping and quantum diffusion in single-particle phase-space by means of Fokker-Planck equations. For practical purposes, most storage rings remain in the regime of linear damping and diffusion; this is discussed in some detail with examples, concentrating on longitudinal phase space. However special devices such as nonlinear wigglers may permit the new generation of very large rings to go beyond this into regimes of nonlinear damping. It is shown how a special combined-function wiggler can be used to modify the energy distribution and current profile of electron bunches

Matter power spectrum in hidden neutrino interacting dark matter models: a closer look at the collision term

Energy Technology Data Exchange (ETDEWEB)

Binder, Tobias; Covi, Laura [Institute for Theoretical Physics, Georg-August University Göttingen,Friedrich-Hund-Platz 1, Göttingen, D-37077 (Germany); Kamada, Ayuki [Department of Physics and Astronomy, University of California,Riverside, California 92521 (United States); Murayama, Hitoshi [Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo Institutes for Advanced Study, University of Tokyo,Kashiwa 277-8583 (Japan); Department of Physics, University of California, Berkeley,Berkeley, California 94720 (United States); Theoretical Physics Group, Lawrence Berkeley National Laboratory,Berkeley, California 94720 (United States); Takahashi, Tomo [Department of Physics, Saga University,Saga 840-8502 (Japan); Yoshida, Naoki [Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo Institutes for Advanced Study, University of Tokyo,Kashiwa 277-8583 (Japan); Department of Physics, University of Tokyo,Tokyo 113-0033 (Japan); CREST, Japan Science and Technology Agency,4-1-8 Honcho, Kawaguchi, Saitama, 332-0012 (Japan)

2016-11-21

Dark Matter (DM) models providing possible alternative solutions to the small-scale crisis of the standard cosmology are nowadays of growing interest. We consider DM interacting with light hidden fermions via well-motivated fundamental operators showing the resultant matter power spectrum is suppressed on subgalactic scales within a plausible parameter region. Our basic description of the evolution of cosmological perturbations relies on a fully consistent first principles derivation of a perturbed Fokker-Planck type equation, generalizing existing literature. The cosmological perturbation of the Fokker-Planck equation is presented for the first time in two different gauges, where the results transform into each other according to the rules of gauge transformation. Furthermore, our focus lies on a derivation of a broadly applicable and easily computable collision term showing important phenomenological differences to other existing approximations. As one of the main results and concerning the small-scale crisis, we show the equal importance of vector and scalar boson mediated interactions between the DM and the light fermions.

Brownian quasi-particles and quantum quasi-particles

International Nuclear Information System (INIS)

Fronteau, J.

1987-01-01

The concept of quasi-particles is used in Statistical Mechanics as well as in Quantum Mechanics, to associate differentiable trajectories to the equations of evolution, trajectories on which a maximum of informations is concentrated concerning the phenomena studied. Two cases are treated numerically, that of the Fokker-Planck equation with an x - x 3 field, and that of the Schroedinger equation with null potential, in a situation of interference [fr

Linear causal modeling with structural equations

CERN Document Server

Mulaik, Stanley A

2009-01-01

Emphasizing causation as a functional relationship between variables that describe objects, Linear Causal Modeling with Structural Equations integrates a general philosophical theory of causation with structural equation modeling (SEM) that concerns the special case of linear causal relations. In addition to describing how the functional relation concept may be generalized to treat probabilistic causation, the book reviews historical treatments of causation and explores recent developments in experimental psychology on studies of the perception of causation. It looks at how to perceive causal

Planck driven by vision, broken by war

CERN Document Server

Brown, Brandon R

2015-01-01

Planck's Law, an equation used by physicists to determine the radiation leaking from any object in the universe, was described by Albert Einstein as "the basis of all twentieth-century physics." Max Planck is credited with being the father of quantum theory, and his work laid the foundation for our modern understanding of matter and energetic processes. But Planck's story is not well known, especially in the United States. A German physicist working during the first half of the twentieth century, his library, personal journals, notebooks, and letters were all destroyed with his home in World War II. What remains, other than his contributions to science, are handwritten letters in German shorthand, and tributes from other scientists of the time, including his close friend Albert Einstein. In Planck: Driven by Vision, Broken by War, Brandon R. Brown interweaves the voices and writings of Planck, his family, and his contemporaries-with many passages appearing in English for the first time-to create a portrait of...

Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation

Directory of Open Access Journals (Sweden)

Hamidreza Rezazadeh

2014-05-01

Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

Systems of Inhomogeneous Linear Equations

Science.gov (United States)

Scherer, Philipp O. J.

Many problems in physics and especially computational physics involve systems of linear equations which arise e.g. from linearization of a general nonlinear problem or from discretization of differential equations. If the dimension of the system is not too large standard methods like Gaussian elimination or QR decomposition are sufficient. Systems with a tridiagonal matrix are important for cubic spline interpolation and numerical second derivatives. They can be solved very efficiently with a specialized Gaussian elimination method. Practical applications often involve very large dimensions and require iterative methods. Convergence of Jacobi and Gauss-Seidel methods is slow and can be improved by relaxation or over-relaxation. An alternative for large systems is the method of conjugate gradients.

Diffusion phenomenon for linear dissipative wave equations

KAUST Repository

Said-Houari, Belkacem

2012-01-01

In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.

Ergodicity of the Stochastic Nosé-Hoover Heat Bath

Science.gov (United States)

Wei Chung Lo,; Baowen Li,

2010-07-01

We numerically study the ergodicity of the stochastic Nosé-Hoover heat bath whose formalism is based on the Markovian approximation for the Nosé-Hoover equation [J. Phys. Soc. Jpn. 77 (2008) 103001]. The approximation leads to a Langevin-like equation driven by a fluctuating dissipative force and multiplicative Gaussian white noise. The steady state solution of the associated Fokker-Planck equation is the canonical distribution. We investigate the dynamics of this method for the case of (i) free particle, (ii) nonlinear oscillators and (iii) lattice chains. We derive the Fokker-Planck equation for the free particle and present approximate analytical solution for the stationary distribution in the context of the Markovian approximation. Numerical simulation results for nonlinear oscillators show that this method results in a Gaussian distribution for the particles velocity. We also employ the method as heat baths to study nonequilibrium heat flow in one-dimensional Fermi-Pasta-Ulam (FPU-β) and Frenkel-Kontorova (FK) lattices. The establishment of well-defined temperature profiles are observed only when the lattice size is large. Our results provide numerical justification for such Markovian approximation for classical single- and many-body systems.

A new linearized equation for servo valve in hydraulic control systems

International Nuclear Information System (INIS)

Kim, Tae Hyung; Lee, Ill Yeong

2002-01-01

In the procedure of the hydraulic control system analysis, a linearized approximate equation described by the first order term of Taylor's series has been widely used. Such a linearized equation is effective just near the operating point. And, as of now, there are no general standards on how to determine the operating point of a servo valve in the process of applying the linearized equation. So, in this study, a new linearized equation for valve characteristics is proposed as a modified form of the existing linearized equation. And, a method for selecting an optimal operating point is proposed for the new linearized equation. The effectiveness of the new linearized equation is confirmed through numerical simulations and experiments for a model hydraulic control system

Tail anisotropy instability during plasma current rise by lower-hybrid waves in a tokamak

International Nuclear Information System (INIS)

Yamagiwa, Mitsuru.

1986-01-01

Tail anisotropy instability during lower-hybrid current rise is investigated. Tail formation by lower-hybrid waves is studied by using a Fokker-Planck equation combined with the return field and the rf associated terms. Quasi-linear relaxation of the electron tail distribution under the influence of the plasma waves excited due to the instability is examined. It is found that the instability condition is related to the strength of the parallel diffusion by lower-hybrid waves and the ratio of the electron cyclotron frequency to the electron plasma frequency. The time scale between the instability spikes and the suppression of the instability by electron cyclotron heating are also discussed. (author)

Aggressive time step selection for the time asymptotic velocity diffusion problem

International Nuclear Information System (INIS)

Hewett, D.W.; Krapchev, V.B.; Hizanidis, K.; Bers, A.

1984-12-01

An aggressive time step selector for an ADI algorithm is preseneted that is applied to the linearized 2-D Fokker-Planck equation including an externally imposed quasilinear diffusion term. This method provides a reduction in CPU requirements by factors of two or three compared to standard ADI. More important, the robustness of the procedure greatly reduces the work load of the user. The procedure selects a nearly optimal Δt with a minimum of intervention by the user thus relieving the need to supervise the algorithm. In effect, the algorithm does its own supervision by discarding time steps made with Δt too large

General solutions of second-order linear difference equations of Euler type

Directory of Open Access Journals (Sweden)

Akane Hongyo

2017-01-01

Full Text Available The purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation \\(y^{\\prime\\prime}+(\\lambda/t^2y=0\\ or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations.

Simulations of Electron Transport in Laser Hot Spots

International Nuclear Information System (INIS)

Brunner, S.; Valeo, E.

2001-01-01

Simulations of electron transport are carried out by solving the Fokker-Planck equation in the diffusive approximation. The system of a single laser hot spot, with open boundary conditions, is systematically studied by performing a scan over a wide range of the two relevant parameters: (1) Ratio of the stopping length over the width of the hot spot. (2) Relative importance of the heating through inverse Bremsstrahlung compared to the thermalization through self-collisions. As for uniform illumination [J.P. Matte et al., Plasma Phys. Controlled Fusion 30 (1988) 1665], the bulk of the velocity distribution functions (VDFs) present a super-Gaussian dependence. However, as a result of spatial transport, the tails are observed to be well represented by a Maxwellian. A similar dependence of the distributions is also found for multiple hot spot systems. For its relevance with respect to stimulated Raman scattering, the linear Landau damping of the electron plasma wave is estimated for such VD Fs. Finally, the nonlinear Fokker-Planck simulations of the single laser hot spot system are also compared to the results obtained with the linear non-local hydrodynamic approach [A.V. Brantov et al., Phys. Plasmas 5 (1998) 2742], thus providing a quantitative limit to the latter method: The hydrodynamic approach presents more than 10% inaccuracy in the presence of temperature variations of the order delta T/T greater than or equal to 1%, and similar levels of deformation of the Gaussian shape of the Maxwellian background

PHYSICS OF ELECTRON CYCLOTRON CURRENT DRIVE ON DIII-D

International Nuclear Information System (INIS)

PETTY, C.C.; PRATER, R.; LUCE, T.C.; ELLIS, R.A.; HARVEY, R.W.; KINSEY, J.E.; LAO, L.L.; LOHR, J.; MAKOWSKI, M.A.

2002-01-01

OAK A271 PHYSICS OF ELECTRON CYCLOTRON CURRENT DRIVE ON DIII-D. Recent experiments on the DIII-D tokamak have focused on determining the effect of trapped particles on the electron cyclotron current drive (ECCD) efficiency. The measured ECCD efficiency increases as the deposition location is moved towards the inboard midplane or towards smaller minor radius for both co and counter injection. The measured ECCD efficiency also increases with increasing electron density and/or temperature. The experimental ECCD is compared to both the linear theory (Toray-GA) as well as a quasilinear Fokker-Planck model (CQL3D). The experimental ECCD is found to be in better agreement with the more complete Fokker-Planck calculation, especially for cases of high rf power density and/or loop voltage

Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

OpenAIRE

Nakamura, Gen; Vashisth, Manmohan

2017-01-01

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\\geq 3$. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\\sqrt{\\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotr...

Stochastic quantization and mean field approximation

International Nuclear Information System (INIS)

Jengo, R.; Parga, N.

1983-09-01

In the context of the stochastic quantization we propose factorized approximate solutions for the Fokker-Planck equation for the XY and Zsub(N) spin systems in D dimensions. The resulting differential equation for a factor can be solved and it is found to give in the limit of t→infinity the mean field or, in the more general case, the Bethe-Peierls approximation. (author)

Slowing down of test particles in a plasma (1961)

International Nuclear Information System (INIS)

Belayche, P.; Chavy, P.; Dupoux, M.; Salmon, J.

1961-01-01

Numerical solution of the Fokker-Planck equation applied to the slowing down of tritons in a deuterium plasma. After the equations and the boundary conditions have been written, some attention is paid to the numerical tricks used to run the problem on a high speed electronic computer. The numerical results thus obtained are then analyzed and as far as possible, mathematically explained. (authors) [fr

The numerical solution of linear multi-term fractional differential equations: systems of equations

Science.gov (United States)

Edwards, John T.; Ford, Neville J.; Simpson, A. Charles

2002-11-01

In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.

Stochastic evolutions and hadronization of highly excited hadronic matter

International Nuclear Information System (INIS)

Carruthers, P.

1984-01-01

Stochastic ingredients of high energy hadronic collisions are analyzed, with emphasis on multiplicity distributions. The conceptual simplicity of the k-cell negative binomial distribution is related to the evolution of probability distributions via the Fokker-Planck and related equations. The connection to underlying field theory ideas is sketched. 17 references

Passing to the limit in a Wasserstein gradient flow : from diffusion to reaction

NARCIS (Netherlands)

Arnrich, S.; Mielke, A.; Peletier, M.A.; Savaré, G.; Veneroni, M.

2012-01-01

We study a singular-limit problem arising in the modelling of chemical reactions. At finite e > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1 / e and in the limit e --> 0, the solution concentrates

Infinite sets of conservation laws for linear and non-linear field equations

International Nuclear Information System (INIS)

Niederle, J.

1984-01-01

The work was motivated by a desire to understand group theoretically the existence of an infinite set of conservation laws for non-interacting fields and to carry over these conservation laws to the case of interacting fields. The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of its space-time symmetry group was established. It is shown that in the case of the Korteweg-de Vries (KdV) equation to each symmetry of the corresponding linear equation delta sub(o)uxxx=u sub() determined by an element of the enveloping algebra of the space translation algebra, there corresponds a symmetry of the full KdV equation

Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations.

Science.gov (United States)

Kilic, Mustafa Sabri; Bazant, Martin Z; Ajdari, Armand

2007-02-01

In situations involving large potentials or surface charges, the Poisson-Boltzman (PB) equation has shortcomings because it neglects ion-ion interactions and steric effects. This has been widely recognized by the electrochemistry community, leading to the development of various alternative models resulting in different sets "modified PB equations," which have had at least qualitative success in predicting equilibrium ion distributions. On the other hand, the literature is scarce in terms of descriptions of concentration dynamics in these regimes. Here, adapting strategies developed to modify the PB equation, we propose a simple modification of the widely used Poisson-Nernst-Planck (PNP) equations for ionic transport, which at least qualitatively accounts for steric effects. We analyze numerical solutions of these modified PNP equations on the model problem of the charging of a simple electrolyte cell, and compare the outcome to that of the standard PNP equations. Finally, we repeat the asymptotic analysis of Bazant, Thornton, and Ajdari [Phys. Rev. E 70, 021506 (2004)] for this new system of equations to further document the interest and limits of validity of the simpler equivalent electrical circuit models introduced in Part I [Kilic, Bazant, and Ajdari, Phys. Rev. E 75, 021502 (2007)] for such problems.

On the Solution of the Continuity Equation for Precipitating Electrons in Solar Flares

Science.gov (United States)

Emslie, A. Gordon; Holman, Gordon D.; Litvinenko, Yuri E.

2014-01-01

Electrons accelerated in solar flares are injected into the surrounding plasma, where they are subjected to the influence of collisional (Coulomb) energy losses. Their evolution is modeled by a partial differential equation describing continuity of electron number. In a recent paper, Dobranskis & Zharkova claim to have found an "updated exact analytical solution" to this continuity equation. Their solution contains an additional term that drives an exponential decrease in electron density with depth, leading them to assert that the well-known solution derived by Brown, Syrovatskii & Shmeleva, and many others is invalid. We show that the solution of Dobranskis & Zharkova results from a fundamental error in the application of the method of characteristics and is hence incorrect. Further, their comparison of the "new" analytical solution with numerical solutions of the Fokker-Planck equation fails to lend support to their result.We conclude that Dobranskis & Zharkova's solution of the universally accepted and well-established continuity equation is incorrect, and that their criticism of the correct solution is unfounded. We also demonstrate the formal equivalence of the approaches of Syrovatskii & Shmeleva and Brown, with particular reference to the evolution of the electron flux and number density (both differential in energy) in a collisional thick target. We strongly urge use of these long-established, correct solutions in future works.

Observational constraints on variable equation of state parameters of dark matter and dark energy after Planck

Directory of Open Access Journals (Sweden)

Suresh Kumar

2014-10-01

Full Text Available In this paper, we study a cosmological model in general relativity within the framework of spatially flat Friedmann–Robertson–Walker space–time filled with ordinary matter (baryonic, radiation, dark matter and dark energy, where the latter two components are described by Chevallier–Polarski–Linder equation of state parameters. We utilize the observational data sets from SNLS3, BAO and Planck + WMAP9 + WiggleZ measurements of matter power spectrum to constrain the model parameters. We find that the current observational data offer tight constraints on the equation of state parameter of dark matter. We consider the perturbations and study the behavior of dark matter by observing its effects on CMB and matter power spectra. We find that the current observational data favor the cold dark matter scenario with the cosmological constant type dark energy at the present epoch.

Observational constraints on variable equation of state parameters of dark matter and dark energy after Planck

International Nuclear Information System (INIS)

Kumar, Suresh; Xu, Lixin

2014-01-01

In this paper, we study a cosmological model in general relativity within the framework of spatially flat Friedmann–Robertson–Walker space–time filled with ordinary matter (baryonic), radiation, dark matter and dark energy, where the latter two components are described by Chevallier–Polarski–Linder equation of state parameters. We utilize the observational data sets from SNLS3, BAO and Planck + WMAP9 + WiggleZ measurements of matter power spectrum to constrain the model parameters. We find that the current observational data offer tight constraints on the equation of state parameter of dark matter. We consider the perturbations and study the behavior of dark matter by observing its effects on CMB and matter power spectra. We find that the current observational data favor the cold dark matter scenario with the cosmological constant type dark energy at the present epoch

Invariant imbedding equations for linear scattering problems

International Nuclear Information System (INIS)

Apresyan, L.

1988-01-01

A general form of the invariant imbedding equations is investigated for the linear problem of scattering by a bounded scattering volume. The conditions for the derivability of such equations are described. It is noted that the possibility of the explicit representation of these equations for a sphere and for a layer involves the separation of variables in the unperturbed wave equation

Microscopic model for the non-linear fluctuating hydrodynamic of 4 He superfluid helium deduced by maximum entropy method

International Nuclear Information System (INIS)

Alvarez R, J.T.

1998-01-01

This thesis presents a microscopic model for the non-linear fluctuating hydrodynamic of superfluid helium ( 4 He), model developed by means of the Maximum Entropy Method (Maxent). In the chapter 1, it is demonstrated the necessity to developing a microscopic model for the fluctuating hydrodynamic of the superfluid helium, starting from to show a brief overview of the theories and experiments developed in order to explain the behavior of the superfluid helium. On the other hand, it is presented the Morozov heuristic method for the construction of the non-linear hydrodynamic fluctuating of simple fluid. Method that will be generalized for the construction of the non-linear fluctuating hydrodynamic of the superfluid helium. Besides, it is presented a brief summary of the content of the thesis. In the chapter 2, it is reproduced the construction of a Generalized Fokker-Planck equation, (GFP), for a distribution function associated with the coarse grained variables. Function defined with aid of a nonequilibrium statistical operator ρhut FP that is evaluated as Wigneris function through ρ CG obtained by Maxent. Later this equation of GFP is reduced to a non-linear local FP equation from considering a slow and Markov process in the coarse grained variables. In this equation appears a matrix D mn defined with a nonequilibrium coarse grained statistical operator ρhut CG , matrix elements are used in the construction of the non-linear fluctuating hydrodynamics equations of the superfluid helium. In the chapter 3, the Lagrange multipliers are evaluated for to determine ρhut CG by means of the local equilibrium statistical operator ρhut l -tilde with the hypothesis that the system presents small fluctuations. Also are determined the currents associated with the coarse grained variables and furthermore are evaluated the matrix elements D mn but with aid of a quasi equilibrium statistical operator ρhut qe instead of the local equilibrium operator ρhut l -tilde. Matrix

Problems of phenomenological simulation of the Dst variation

International Nuclear Information System (INIS)

Gul'el'mi, A.V.

1988-01-01

Stochastic generalization of RBM model, describing the D st -variation is suggested. The corresponding Fokker-Planck equation contains a new phenomenological parameter enabling to obtain the interval estimation of D st forecast. The structure of sources and sinks forming the D st -variation is considered from the viewpoint of critical phenomenon theory

Time-dependent solutions for stochastic systems with delays: Perturbation theory and applications to financial physics

International Nuclear Information System (INIS)

Frank, T.D.

2006-01-01

First-order approximations of time-dependent solutions are determined for stochastic systems perturbed by time-delayed feedback forces. To this end, the theory of delay Fokker-Planck equations is applied in combination with Bayes' theorem. Applications to a time-delayed Ornstein-Uhlenbeck process and the geometric Brownian walk of financial physics are discussed

Ion energy spectrum just after the application of current pulse for turbulent heating in the TRIAM-1 tokamak

Energy Technology Data Exchange (ETDEWEB)

Nakamura, K; Nakamura, Y; Hiraki, N; Itoh, S [Kyushu Univ., Fukuoka (Japan). Research Inst. for Applied Mechanics

1981-07-01

Temporal evolution and spatial profile of ion energy spectrum just after the application of current pulse for turbulent heating are investigated experimentally in TRIAM-1 and numerically with a Fokker-Planck equation. Two-component ion energy spectrum formed by turbulent heating relaxes to single one within tau sub(i) (ion collision time).

Ion energy spectrum just after the application of current pulse for turbulent heating in the TRIAM-1 tokamak

International Nuclear Information System (INIS)

Nakamura, Kazuo; Nakamura, Yukio; Hiraki, Naoji; Itoh, Satoshi

1981-01-01

Temporal evolution and spatial profile of ion energy spectrum just after the application of current pulse for turbulent heating are investigated experimentally in TRIAM-1 and numerically with a Fokker-Planck equation. Two-component ion energy spectrum formed by turbulent heating relaxes to single one within tau sub(i) (ion collision time). (author)

Linear Einstein equations and Kerr-Schild maps

International Nuclear Information System (INIS)

Gergely, Laszlo A

2002-01-01

We prove that given a solution of the Einstein equations g ab for the matter field T ab , an autoparallel null vector field l a and a solution (l a l c , T ac ) of the linearized Einstein equation on the given background, the Kerr-Schild metric g ac + λl a l c (λ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor T ac + λT ac + λ 2 l (a T c)b l b . The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed spacetime due to Guerses and Guersey

Analytical exact solution of the non-linear Schroedinger equation

International Nuclear Information System (INIS)

Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da

2011-01-01

Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)

Macroscopic Modeling of a One-Dimensional Electrochemical Cell using the Poisson-Nernst-Planck Equations

Science.gov (United States)

Yan, David

This thesis presents the one-dimensional equations, numerical method and simulations of a model to characterize the dynamical operation of an electrochemical cell. This model extends the current state-of-the art in that it accounts, in a primitive way, for the physics of the electrolyte/electrode interface and incorporates diffuse-charge dynamics, temperature coupling, surface coverage, and polarization phenomena. The one-dimensional equations account for a system with one or two mobile ions of opposite charge, and the electrode reaction we consider (when one is needed) is a one-electron electrodeposition reaction. Though the modeled system is far from representing a realistic electrochemical device, our results show a range of dynamics and behaviors which have not been observed previously, and explore the numerical challenges required when adding more complexity to a model. Furthermore, the basic transport equations (which are developed in three spatial dimensions) can in future accomodate the inclusion of additional physics, and coupling to more complex boundary conditions that incorporate two-dimensional surface phenomena and multi-rate reactions. In the model, the Poisson-Nernst-Planck equations are used to model diffusion and electromigration in an electrolyte, and the generalized Frumkin-Butler-Volmer equation is used to model reaction kinetics at electrodes. An energy balance equation is derived and coupled to the diffusion-migration equation. The model also includes dielectric polarization effects by introducing different values of the dielectric permittivity in different regions of the bulk, as well as accounting for surface coverage effects due to adsorption, and finite size "crowding", or steric effects. Advection effects are not modeled but could in future be incorporated. In order to solve the coupled PDE's, we use a variable step size second order scheme in time and finite differencing in space. Numerical tests are performed on a simplified system and

Non-Markovian Effects on the Brownian Motion of a Free Particle

OpenAIRE

Bolivar, A. O.

2010-01-01

Non-Markovian effects upon the Brownian movement of a free particle in the presence as well as in the absence of inertial force are investigated within the framework of Fokker-Planck equations (Rayleigh and Smoluchowski equations). More specifically, it is predicted that non-Markovian features can enhance the values of the mean square displacement and momentum, thereby assuring the mathematical property of differentiability of the these physically observable quantities.

Non-linear wave equations:Mathematical techniques

International Nuclear Information System (INIS)

1978-01-01

An account of certain well-established mathematical methods, which prove useful to deal with non-linear partial differential equations is presented. Within the strict framework of Functional Analysis, it describes Semigroup Techniques in Banach Spaces as well as variational approaches towards critical points. Detailed proofs are given of the existence of local and global solutions of the Cauchy problem and of the stability of stationary solutions. The formal approach based upon invariance under Lie transformations deserves attention due to its wide range of applicability, even if the explicit solutions thus obtained do not allow for a deep analysis of the equations. A compre ensive introduction to the inverse scattering approach and to the solution concept for certain non-linear equations of physical interest are also presented. A detailed discussion is made about certain convergence and stability problems which arise in importance need not be emphasized. (author) [es

Stochastic theory of polarized light in nonlinear birefringent media: An application to optical rotation

Science.gov (United States)

Tsuchida, Satoshi; Kuratsuji, Hiroshi

2018-05-01

A stochastic theory is developed for the light transmitting the optical media exhibiting linear and nonlinear birefringence. The starting point is the two-component nonlinear Schrödinger equation (NLSE). On the basis of the ansatz of “soliton” solution for the NLSE, the evolution equation for the Stokes parameters is derived, which turns out to be the Langevin equation by taking account of randomness and dissipation inherent in the birefringent media. The Langevin equation is converted to the Fokker-Planck (FP) equation for the probability distribution by employing the technique of functional integral on the assumption of the Gaussian white noise for the random fluctuation. The specific application is considered for the optical rotation, which is described by the ellipticity (third component of the Stokes parameters) alone: (i) The asymptotic analysis is given for the functional integral, which leads to the transition rate on the Poincaré sphere. (ii) The FP equation is analyzed in the strong coupling approximation, by which the diffusive behavior is obtained for the linear and nonlinear birefringence. These would provide with a basis of statistical analysis for the polarization phenomena in nonlinear birefringent media.

Sensitivity theory for general non-linear algebraic equations with constraints

International Nuclear Information System (INIS)

Oblow, E.M.

1977-04-01

Sensitivity theory has been developed to a high state of sophistication for applications involving solutions of the linear Boltzmann equation or approximations to it. The success of this theory in the field of radiation transport has prompted study of possible extensions of the method to more general systems of non-linear equations. Initial work in the U.S. and in Europe on the reactor fuel cycle shows that the sensitivity methodology works equally well for those non-linear problems studied to date. The general non-linear theory for algebraic equations is summarized and applied to a class of problems whose solutions are characterized by constrained extrema. Such equations form the basis of much work on energy systems modelling and the econometrics of power production and distribution. It is valuable to have a sensitivity theory available for these problem areas since it is difficult to repeatedly solve complex non-linear equations to find out the effects of alternative input assumptions or the uncertainties associated with predictions of system behavior. The sensitivity theory for a linear system of algebraic equations with constraints which can be solved using linear programming techniques is discussed. The role of the constraints in simplifying the problem so that sensitivity methodology can be applied is highlighted. The general non-linear method is summarized and applied to a non-linear programming problem in particular. Conclusions are drawn in about the applicability of the method for practical problems

Heavy quarks thermalization in heavy-ion ultrarelativistic collisions: elastic or radiative?

International Nuclear Information System (INIS)

Gossiaux, Pol Bernard; Guiho, Vincent; Aichelin, Joerg

2006-01-01

We present a dynamical model of heavy quark evolution in the quark-gluon plasma (QGP) based on the Fokker-Planck equation. We then apply this model to the case of ultrarelativistic nucleus-nucleus collisions performed at RHIC in order to investigate which experimental observables might help to discriminate the fundamental process leading to thermalization

Consistent dynamical and statistical description of fission and comparison

Energy Technology Data Exchange (ETDEWEB)

Shunuan, Wang [Chinese Nuclear Data Center, Beijing, BJ (China)

1996-06-01

The research survey of consistent dynamical and statistical description of fission is briefly introduced. The channel theory of fission with diffusive dynamics based on Bohr channel theory of fission and Fokker-Planck equation and Kramers-modified Bohr-Wheeler expression according to Strutinsky method given by P.Frobrich et al. are compared and analyzed. (2 figs.).

Dissipative nucleus-nucleus collisions: study of memory effects

International Nuclear Information System (INIS)

Agarwal, K.C.; Yadav, H.L.

2002-01-01

Dissipative collisions between two heavy nuclei are described in terms of a macroscopic dynamical model within the framework of a multi-dimensional Fokker-Planck equation. The reaction 86 Kr(8.18 MeV/u) + 166 Er has been used as a prototype to study and demonstrate the memory effects for dissipation and diffusion processes

Dynamical symmetries of semi-linear Schrodinger and diffusion equations

International Nuclear Information System (INIS)

Stoimenov, Stoimen; Henkel, Malte

2005-01-01

Conditional and Lie symmetries of semi-linear 1D Schrodinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrodinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf 3 ) C . We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of (conf 3 ) C are classified and the complete list of conditionally invariant semi-linear Schrodinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed

Half-trek criterion for generic identifiability of linear structural equation models

NARCIS (Netherlands)

Foygel, R.; Draisma, J.; Drton, M.

2012-01-01

A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations, and bidirected edges indicate possible correlations

Half-trek criterion for generic identifiability of linear structural equation models

NARCIS (Netherlands)

Foygel, R.; Draisma, J.; Drton, M.

2011-01-01

A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations, and bidirected edges indicate possible correlations

NUMERICAL RESEARCH ON THE THREE-DIMENSIONAL FIBER ORIENTATION DISTRIBUTION IN PLANAR SUSPENSION FLOWS

Directory of Open Access Journals (Sweden)

Qihua Zhang

Full Text Available Abstract To describe flow-induced fiber orientation, the Fokker-Planck equation is widely applied in the processing of composites and fiber suspensions. The analytical solution only exists when the Péclet number is infinite. So developing a numerical method covering a full range of Péclet number is of great significance. To accurately solve the Fokker-Planck equation, a numerical scheme based on the finite volume method is developed. Using spherical symmetry, the boundary is discretized and formulated into a cyclic tridiagonal matrix which is further solved by the CTDMA algorithm. To examine its validity, benchmark tests over a wide range of Péclet number are performed in a simple shear flow. For Pe=∞, the results agree well with the analytical solutions. For the other Pe numbers, the results are compared to results available in the literature. The tests show that this algorithm is accurate, stable, and globally conservative. Furthermore, this algorithm can be extended and used to predict the three-dimensional orientation distribution of complex suspension flows.

ZORNOC: a 1 1/2-D tokamak data analysis code for studying noncircular high beta plasmas

International Nuclear Information System (INIS)

Zurro, B.; Wieland, R.M.; Murakami, M.; Swain, D.W.

1980-03-01

A new tokamak data analysis code, ZORNOC, was developed to study noncircular, high beta plasmas in the Impurity Study Experiment (ISX-B). These plasmas exhibit significant flux surface shifts and elongation in both ohmically heated and beam-heated discharges. The MHD equilibrium flux surface geometry is determined by solving the Grad-Shafranov equation based on: (1) the shape of the outermost flux surface, deduced from the magnetic loop probes; (2) a pressure profile, deduced by means of Thomson scattering data (electrons), charge exchange data (ions), and a Fokker-Planck model (fast ions); and (3) a safety factor profile, determined from the experimental data using a simple model (Z/sub eff/ = const) that is self-consistently altered while the plasma equilibrium is iterated. For beam-heated discharches the beam deposition profile is determined by means of a Monte Carlo scheme and the slowing down of the fast ions by means of an analytical solution of the Fokker-Planck equation. The code also carries out an electron power balance and calculates various confinement parameters. The code is described and examples of its operation are given

Introduction to linear systems of differential equations

CERN Document Server

Adrianova, L Ya

1995-01-01

The theory of linear systems of differential equations is one of the cornerstones of the whole theory of differential equations. At its root is the concept of the Lyapunov characteristic exponent. In this book, Adrianova presents introductory material and further detailed discussions of Lyapunov exponents. She also discusses the structure of the space of solutions of linear systems. Classes of linear systems examined are from the narrowest to widest: 1)�autonomous, 2)�periodic, 3)�reducible to autonomous, 4)�nearly reducible to autonomous, 5)�regular. In addition, Adrianova considers the following: stability of linear systems and the influence of perturbations of the coefficients on the stability the criteria of uniform stability and of uniform asymptotic stability in terms of properties of the solutions several estimates of the growth rate of solutions of a linear system in terms of its coefficients How perturbations of the coefficients change all the elements of the spectrum of the system is defin...

GLOBAL LINEARIZATION OF DIFFERENTIAL EQUATIONS WITH SPECIAL STRUCTURES

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

This paper introduces the global linearization of the differential equations with special structures.The function in the differential equation is unbounded.We prove that the differential equation with unbounded function can be topologically linearlized if it has a special structure.

Geometric Insight into Scalar Combination of Linear Equations

Indian Academy of Sciences (India)

... Journals; Resonance – Journal of Science Education; Volume 14; Issue 11. Geometric Insight into Scalar Combination of Linear Equations. Ranjit Konkar. Classroom Volume 14 Issue 11 November 2009 pp 1092-1097 ... Keywords. Linear algebra; linear dependence; linear combination; family of lines; family of planes.

Second RPA dynamics at finite temperature: time-evolutions of dynamical operators

International Nuclear Information System (INIS)

Jang, S.

1989-01-01

Time-evolutions of dynamical operators, in particular the generalized density matrix comprising both diagonal and off-diagonal elements, are investigated within the framework of second RPA dynamics at finite temperature. The calculation of the density matrix previously carried out through the appliance of the second RPA master equation by retaining only the slowly oscillating coupling terms is extended to include in the interaction Hamiltonian both the rapidly and slowly oscillating coupling terms. The extended second RPA master equation, thereby formulated without making use of the so-called resonant approximation, is analytically solved and a closed expression for the generalized density matrix is extracted. We provide illustrative examples of the generalized density matrix for various specific initial conditions. We turn particularly our attention to the Poisson distribution type of initial condition for which we deduce specifically a particular form of the density matrix from the solution of the Fokker-Planck equation for the coherent state representation. The relation of the Fokker-Planck equation to the second RPA master equation and its properties are briefly discussed. The oversight incurred in the time-evolution of operators by the resonant approximation is elucidated. The first and second moments of collective coordinates are also computed in relation to the expectation value of various dynamical operators involved in the extended master equation

What happens to linear properties as we move from the Klein-Gordon equation to the sine-Gordon equation

International Nuclear Information System (INIS)

Kovalyov, Mikhail

2010-01-01

In this article the sets of solutions of the sine-Gordon equation and its linearization the Klein-Gordon equation are discussed and compared. It is shown that the set of solutions of the sine-Gordon equation possesses a richer structure which partly disappears during linearization. Just like the solutions of the Klein-Gordon equation satisfy the linear superposition principle, the solutions of the sine-Gordon equation satisfy a nonlinear superposition principle.

Emmy Noether and Linear Evolution Equations

Directory of Open Access Journals (Sweden)

P. G. L. Leach

2013-01-01

Full Text Available Noether’s Theorem relates the Action Integral of a Lagrangian with symmetries which leave it invariant and the first integrals consequent upon the variational principle and the existence of the symmetries. These each have an equivalent in the Schrödinger Equation corresponding to the Lagrangian and by extension to linear evolution equations in general. The implications of these connections are investigated.

Simplified Linear Equation Solvers users manual

Energy Technology Data Exchange (ETDEWEB)

Gropp, W. [Argonne National Lab., IL (United States); Smith, B. [California Univ., Los Angeles, CA (United States)

1993-02-01

The solution of large sparse systems of linear equations is at the heart of many algorithms in scientific computing. The SLES package is a set of easy-to-use yet powerful and extensible routines for solving large sparse linear systems. The design of the package allows new techniques to be used in existing applications without any source code changes in the applications.

A linearizing transformation for the Korteweg-de Vries equation; generalizations to higher-dimensional nonlinear partial differential equations

NARCIS (Netherlands)

Dorren, H.J.S.

1998-01-01

It is shown that the Korteweg–de Vries (KdV) equation can be transformed into an ordinary linear partial differential equation in the wave number domain. Explicit solutions of the KdV equation can be obtained by subsequently solving this linear differential equation and by applying a cascade of

Construction of a Roe linearization for the ideal MHD equations

International Nuclear Information System (INIS)

Cargo, P.; Gallice, G.; Raviart, P.A.

1996-01-01

In [3], Munz has constructed a Roe linearization for the equations of gas dynamics in Lagrangian coordinates. We extend this construction to the case of the ideal magnetohydrodynamics equations again in Lagrangian coordinates. As a consequence we obtain a Roe linearization for the MHD equations in Eulerian coordinates. (author)

Noise-induced modulation of the relaxation kinetics around a non-equilibrium steady state of non-linear chemical reaction networks.

Science.gov (United States)

Ramaswamy, Rajesh; Sbalzarini, Ivo F; González-Segredo, Nélido

2011-01-28

Stochastic effects from correlated noise non-trivially modulate the kinetics of non-linear chemical reaction networks. This is especially important in systems where reactions are confined to small volumes and reactants are delivered in bursts. We characterise how the two noise sources confinement and burst modulate the relaxation kinetics of a non-linear reaction network around a non-equilibrium steady state. We find that the lifetimes of species change with burst input and confinement. Confinement increases the lifetimes of all species that are involved in any non-linear reaction as a reactant. Burst monotonically increases or decreases lifetimes. Competition between burst-induced and confinement-induced modulation may hence lead to a non-monotonic modulation. We quantify lifetime as the integral of the time autocorrelation function (ACF) of concentration fluctuations around a non-equilibrium steady state of the reaction network. Furthermore, we look at the first and second derivatives of the ACF, each of which is affected in opposite ways by burst and confinement. This allows discriminating between these two noise sources. We analytically derive the ACF from the linear Fokker-Planck approximation of the chemical master equation in order to establish a baseline for the burst-induced modulation at low confinement. Effects of higher confinement are then studied using a partial-propensity stochastic simulation algorithm. The results presented here may help understand the mechanisms that deviate stochastic kinetics from its deterministic counterpart. In addition, they may be instrumental when using fluorescence-lifetime imaging microscopy (FLIM) or fluorescence-correlation spectroscopy (FCS) to measure confinement and burst in systems with known reaction rates, or, alternatively, to correct for the effects of confinement and burst when experimentally measuring reaction rates.

Variational linear algebraic equations method

International Nuclear Information System (INIS)

Moiseiwitsch, B.L.

1982-01-01

A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)

Time evolution of the fission-decay width under the influence of dissipation

International Nuclear Information System (INIS)

Jurado, B.; Schmidt, K.H.; Benlliure, J.

2002-12-01

Different analytical approximations to the time-dependent fission-decay width used to extract the influence of dissipation on the fission process are critically examined. Calculations with a new, highly realistic analytical approximation to the exact solution of the Fokker-Planck equation sheds doubts on previous conclusions on the dissipation strength made on the basis of less realistic approximations. (orig.)

Modeling Populations of Thermostatic Loads with Switching Rate Actuation

DEFF Research Database (Denmark)

Totu, Luminita Cristiana; Wisniewski, Rafal; Leth, John-Josef

2015-01-01

We model thermostatic devices using a stochastic hybrid description, and introduce an external actuation mechanism that creates random switch events in the discrete dynamics. We then conjecture the form of the Fokker-Planck equation and successfully verify it numerically using Monte Carlo...... simulations. The actuation mechanism and subsequent modeling result are relevant for power system operation....

Currents driven by electron cyclotron waves

International Nuclear Information System (INIS)

Karney, C.F.F.; Fisch, N.J.

1981-07-01

Certain aspects of the generation of steady-state currents by electron cyclotron waves are explored. A numerical solution of the Fokker-Planck equation is used to verify the theory of Fisch and Boozer and to extend their results into the nonlinear regime. Relativistic effects on the current generated are discussed. Applications to steady-state tokamak reactors are considered

Rational approximations to solutions of linear differential equations.

Science.gov (United States)

Chudnovsky, D V; Chudnovsky, G V

1983-08-01

Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.

Nernst Effect in Magnetized Plasmas

OpenAIRE

Joglekar, Archis S.; Thomas, Alexander G. R.; Ridgers, Christopher P.; Kingham, Robert J.

2015-01-01

We present nanosecond timescale Vlasov-Fokker-Planck-Maxwell modeling of magnetized plasma transport and dynamics in a hohlraum with an applied external magnetic field, under conditions similar to recent experiments. Self-consistent modeling of the kinetic electron momentum equation allows for a complete treatment of the heat flow equation and Ohm's Law, including Nernst advection of magnetic fields. In addition to showing the prevalence of non-local behavior, we demonstrate that effects such...

Kinetic modeling of Nernst effect in magnetized hohlraums

OpenAIRE

Joglekar, A. S.; Ridgers, Christopher Paul; Kingham, R J; Thomas, A. G. R.

2016-01-01

We present nanosecond time-scale Vlasov-Fokker-Planck-Maxwell modeling of magnetized plasma transport and dynamics in a hohlraum with an applied external magnetic field, under conditions similar to recent experiments. Self-consistent modeling of the kinetic electron momentum equation allows for a complete treatment of the heat flow equation and Ohm's law, including Nernst advection of magnetic fields. In addition to showing the prevalence of nonlocal behavior, we demonstrate that effects such...

Stochastic methods for the description of multiparticle production

International Nuclear Information System (INIS)

Carruthers, P.

1984-01-01

Dynamical questions in the evolution of excited hadronic matter are reviewed, with emphasis on KNO scaling and its possible violation. It is suggested that the KNO distributions is described by a stochastic evolution of the Fokker-Planck type related to underlying field theory by coupled rate equations approximated by Langevin equations with noise. Refined correlation analysis of data, especially the use of intensity interferometry techniques, is recommended for data analysis. 26 references

A structural self-regulation of functioning macromolecules

International Nuclear Information System (INIS)

Khristoforov, L.N.

1998-01-01

An approach to describing the functional structural changes of macromolecules processing the flows of low-mass agents is formulated. The latter appear as a source of a discrete noise whose defining parameters depend on structural variables. We derive a forward evolution equation and then, by adiabatic elimination, effective Fokker-Planck's equation for the structural modes. Within the dichotomous case, we discuss noise-induced nonequilibrium phase transitions reflecting the regulatory role of the structural subsystem

Non-linear effects in the Boltzmann equation

International Nuclear Information System (INIS)

Barrachina, R.O.

1985-01-01

The Boltzmann equation is studied by defining an integral transformation of the energy distribution function for an isotropic and homogeneous gas. This transformation may be interpreted as a linear superposition of equilibrium states with variable temperatures. It is shown that the temporal evolution features of the distribution function are determined by the singularities of said transformation. This method is applied to Maxwell and Very Hard Particle interaction models. For the latter, the solution of the Boltzmann equation with the solution of its linearized version is compared, finding out many basic discrepancies and non-linear effects. This gives a hint to propose a new rational approximation method with a clear physical meaning. Applying this technique, the relaxation features of the BKW (Bobylev, Krook anf Wu) mode is analyzed, finding a conclusive counter-example for the Krook and Wu conjecture. The anisotropic Boltzmann equation for Maxwell models is solved as an expansion in terms of the eigenfunctions of the corresponding linearized collision operator, finding interesting transient overpopulation and underpopulation effects at thermal energies as well as a new preferential spreading effect. By analyzing the initial collision, a criterion is established to deduce the general features of the final approach to equilibrium. Finally, it is shown how to improve the convergence of the eigenfunction expansion for high energy underpopulated distribution functions. As an application of this theory, the linear cascade model for sputtering is analyzed, thus finding out that many differences experimentally observed are due to non-linear effects. (M.E.L.) [es

Multigroup discrete ordinates solution of Boltzmann-Fokker-Planck equations and cross section library development of ion transport

International Nuclear Information System (INIS)

Prinja, A.K.

1995-08-01

We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S N angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes

Asymptotic properties for half-linear difference equations

Czech Academy of Sciences Publication Activity Database

Cecchi, M.; Došlá, Z.; Marini, M.; Vrkoč, Ivo

2006-01-01

Roč. 131, č. 4 (2006), s. 347-363 ISSN 0862-7959 R&D Projects: GA ČR(CZ) GA201/04/0580 Institutional research plan: CEZ:AV0Z10190503 Keywords : half-linear second order difference equation * nonoscillatory solutions * Riccati difference equation Subject RIV: BA - General Mathematics

Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the Voltage-Current relation in neurobiological microdomains

Science.gov (United States)

Cartailler, J.; Schuss, Z.; Holcman, D.

2017-01-01

The electro-diffusion of ions is often described by the Poisson-Nernst-Planck (PNP) equations, which couple nonlinearly the charge concentration and the electric potential. This model is used, among others, to describe the motion of ions in neuronal micro-compartments. It remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane. The purpose of this paper is to construct an asymptotic approximation to the solution of the stationary PNP equations in a d-dimensional ball (d = 1 , 2 , 3) in the limit of large total charge. In this geometry the PNP system reduces to the Liouville-Gelfand-Bratú (LGB) equation, with the difference that the boundary condition is Neumann, not Dirichlet, and there is a minus sign in the exponent of the exponential term. The entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson's equation. These differences replace attraction by repulsion in the LGB equation, thus completely changing the solution. We find that the voltage is maximal in the center and decreases toward the boundary. We also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly, as in classical capacitance theory. This logarithmic singularity is obtained for d = 3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait. These results are used to derive the relation between the outward current and the voltage in a dendritic spine, which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck. This is a fundamental microdomain involved in neuronal communication. We compute the escape rate of an ion from the steady density in a ball, which models a neuronal spine head, to a small absorbing window in the sphere. We

Students' errors in solving linear equation word problems: Case ...

African Journals Online (AJOL)

The study examined errors students make in solving linear equation word problems with a view to expose the nature of these errors and to make suggestions for classroom teaching. A diagnostic test comprising 10 linear equation word problems, was administered to a sample (n=130) of senior high school first year Home ...

Computational physics program of the National MFE Computer Center

International Nuclear Information System (INIS)

Mirin, A.A.

1985-12-01

The development of numerical models for plasma phenomena and magnetic confinement devices is discussed. The multidimensional Fokker-Planck and transport codes are applied to toroidal mirror and compact toroid devices. Linear and nonlinear resistive magnetohydrodynamics in two and three dimensions are used in the investigation of various fusion devices. 362 refs., 4 tabs

New Equating Methods and Their Relationships with Levine Observed Score Linear Equating under the Kernel Equating Framework

Science.gov (United States)

Chen, Haiwen; Holland, Paul

2010-01-01

In this paper, we develop a new curvilinear equating for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score equating. In fact, by applying both the kernel equating framework and the mean preserving linear transformation of…

Theoretical aspects of some collective instabilities in high-energy particle storage rings

International Nuclear Information System (INIS)

Ruggiero, F.

1986-01-01

After an introduction to single-particle dynamics, based on a unified Hamiltonian treatment of betatron and synchrotron oscillations, we consider two examples of collective instabilities which can limit the performances of high-energy storage rings: the transverse mode coupling instability, due to wake fields, and the incoherent beam-beam instability. Special emphasis is placed on the localization of the interactions between particles and surrounding structures, such as the accelerating RF cavities. We derive an exact invariant for the linearized synchrotron motion and, starting from the Vlasov equation, we discuss the coherent synchro-betatron resonances caused by localized impedance. Under suitable assumptions, we show that the effect of the beam-beam kicks in electron-positron machines can be described by new diffusive terms in a ''renormalized'' Fokker-Planck equation and is therefore equivalent to an additional source of noise for the betatron oscillations. (orig.)

One and one-half dimensional model of the EBT reactor

International Nuclear Information System (INIS)

Klein, H.H.; Bathke, C.G.

1979-01-01

A one-dimensional, time-dependent model is described for plasma particle and energy transport and alpha particle transport coupled with magnetic field evolution in a geometry appropriate to EBT. The transport equations used are derived from exact moments of the Boltzmann equation, and the magnetic field is calculated from Faraday's and Ampere's laws. The set of transport equations is closed by incorporating into them transport coefficiencents derived from the appropriate kinetic equation. Also included in the model is a Fokker-Planck calculation of the alpha particle slowing down and resultant plasma heating

High-order quantum algorithm for solving linear differential equations

International Nuclear Information System (INIS)

Berry, Dominic W

2014-01-01

Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods (where the error over a time step is a high power of the size of the time step) to improve the efficiency. These provide scaling close to Δt 2 in the evolution time Δt. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution. (paper)

Periodic feedback stabilization for linear periodic evolution equations

CERN Document Server

Wang, Gengsheng

2016-01-01

This book introduces a number of recent advances regarding periodic feedback stabilization for linear and time periodic evolution equations. First, it presents selected connections between linear quadratic optimal control theory and feedback stabilization theory for linear periodic evolution equations. Secondly, it identifies several criteria for the periodic feedback stabilization from the perspective of geometry, algebra and analyses respectively. Next, it describes several ways to design periodic feedback laws. Lastly, the book introduces readers to key methods for designing the control machines. Given its coverage and scope, it offers a helpful guide for graduate students and researchers in the areas of control theory and applied mathematics.

The contrasting roles of Planck's constant in classical and quantum theories

Science.gov (United States)

Boyer, Timothy H.

2018-04-01

We trace the historical appearance of Planck's constant in physics, and we note that initially the constant did not appear in connection with quanta. Furthermore, we emphasize that Planck's constant can appear in both classical and quantum theories. In both theories, Planck's constant sets the scale of atomic phenomena. However, the roles played in the foundations of the theories are sharply different. In quantum theory, Planck's constant is crucial to the structure of the theory. On the other hand, in classical electrodynamics, Planck's constant is optional, since it appears only as the scale factor for the (homogeneous) source-free contribution to the general solution of Maxwell's equations. Since classical electrodynamics can be solved while taking the homogenous source-free contribution in the solution as zero or non-zero, there are naturally two different theories of classical electrodynamics, one in which Planck's constant is taken as zero and one where it is taken as non-zero. The textbooks of classical electromagnetism present only the version in which Planck's constant is taken to vanish.

Infinite sets of conservation laws for linear and nonlinear field equations

International Nuclear Information System (INIS)

Mickelsson, J.

1984-01-01

The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the 'coupling constant') the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant under the group of space-time symmetries. As an example, our method is applied to the Korteweg-de Vries equation and to the massless Thirring model. (orig.)

A Comparison between Linear IRT Observed-Score Equating and Levine Observed-Score Equating under the Generalized Kernel Equating Framework

Science.gov (United States)

Chen, Haiwen

2012-01-01

In this article, linear item response theory (IRT) observed-score equating is compared under a generalized kernel equating framework with Levine observed-score equating for nonequivalent groups with anchor test design. Interestingly, these two equating methods are closely related despite being based on different methodologies. Specifically, when…

Critical power for lower hybrid current drive

International Nuclear Information System (INIS)

Assis, A.S. de; Sakanaka, P.H.; Azevedo, C.A. de; Busnardo-Neto, J.

1995-11-01

We have solved numerically the quasilinear Fokker-Planck equation which models the critical power for lower hybrid wave current drive. An exact value for the critical power necessary for current saturation, for tokamak current drive experiments, has been obtained. The nonlinear treatment presented here leads to a final profile for the parallel distribution function which is a plateau only in a part of the resonance region. This form of the distribution function is intermediate between two well known results: a plateau throughout the resonance region for the linear strong-source regime, D wave >> D coll and no plateau at all in the resonance region the linear weak-source regimen, D wave coll . The strength of the external power source and the value of the dc electric field are treated as given parameters in the integration scheme. (author). 24 refs, 6 figs

Dual exponential polynomials and linear differential equations

Science.gov (United States)

Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

2018-01-01

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

Nonstationary stochastic charge fluctuations of a dust particle in plasmas.

Science.gov (United States)

Shotorban, B

2011-06-01

Stochastic charge fluctuations of a dust particle that are due to discreteness of electrons and ions in plasmas can be described by a one-step process master equation [T. Matsoukas and M. Russell, J. Appl. Phys. 77, 4285 (1995)] with no exact solution. In the present work, using the system size expansion method of Van Kampen along with the linear noise approximation, a Fokker-Planck equation with an exact Gaussian solution is developed by expanding the master equation. The Gaussian solution has time-dependent mean and variance governed by two ordinary differential equations modeling the nonstationary process of dust particle charging. The model is tested via the comparison of its results to the results obtained by solving the master equation numerically. The electron and ion currents are calculated through the orbital motion limited theory. At various times of the nonstationary process of charging, the model results are in a very good agreement with the master equation results. The deviation is more significant when the standard deviation of the charge is comparable to the mean charge in magnitude.

Transport theory of dissipative heavy-ion collisions

International Nuclear Information System (INIS)

Norenberg, W.

1979-01-01

The lectures present the formulation of a transport theory, the derivation of a practicable transport equation (Fokker-Planck equation) and the evaluation of transport coefficients for dissipative (or deeply inelastic) heavy-ion collisions. The applicability of the theoretical concept is tested with remarkable success in the analyses of various experimental information (mass transfer, angular-momentum dissipation and energy loss). Some critical remarks on the present situation of transport theories are added. Future developments are outlined. (author)

Transport theory of dissipative heavy-ion collisions

International Nuclear Information System (INIS)

Noerenberg, W.

1979-03-01

The lectures present the formulation of a transport theory, the derivation of a practicable transport equation (Fokker-Planck equation) and the evaluation of transport coefficients for dissipative (or deeply inelastic) heavyion collisions. The applicability of the theoretical concept is tested with remarkable success in the analyses of various experimental informations (mass transfer, angular-momentum dissipation and energy loss). Some critical remarks on the present situation of transport theories are added. Future developments are outlined. (orig.) [de

Gas-induced friction and diffusion of rigid rotors

Science.gov (United States)

Martinetz, Lukas; Hornberger, Klaus; Stickler, Benjamin A.

2018-05-01

We derive the Boltzmann equation for the rotranslational dynamics of an arbitrary convex rigid body in a rarefied gas. It yields as a limiting case the Fokker-Planck equation accounting for friction, diffusion, and nonconservative drift forces and torques. We provide the rotranslational friction and diffusion tensors for specular and diffuse reflection off particles with spherical, cylindrical, and cuboidal shape, and show that the theory describes thermalization, photophoresis, and the inverse Magnus effect in the free molecular regime.

Projection operator techniques in nonequilibrium statistical mechanics

International Nuclear Information System (INIS)

Grabert, H.

1982-01-01

This book is an introduction to the application of the projection operator technique to the statistical mechanics of irreversible processes. After a general introduction to the projection operator technique and statistical thermodynamics the Fokker-Planck and the master equation approach are described together with the response theory. Then, as applications the damped harmonic oscillator, simple fluids, and the spin relaxation are considered. (HSI)

Charmonia enhancement in quark-gluon plasma with improved description of c-quarks phase distribution

International Nuclear Information System (INIS)

Gossiaux, Pol Bernard; Guiho, Vincent; Aichelin, Joerg

2005-01-01

We present a dynamical model of heavy quark evolution in the quark-gluon plasma (QGP) based on the Fokker-Planck equation. We then apply this model to the case of central ultra-relativistic nucleus-nucleus collisions performed at RHIC and estimate the component of J/ψ production (integrated and differential) stemming from c-c-bar pairs that are initially uncorrelated

Path integrals for inertialess classical particles under-going rapid stochastic trembling. I

International Nuclear Information System (INIS)

Bezak, V.

1978-01-01

Feynman path integrals are studied in reference to the Fokker-Planck (Smoluchowski) equation. Examples are presented including the motion of an inertialess classical charged particle between electrodes in plate and cylindrical capacitors with charges fluctuating rapidly as Gaussian white-noise stochastic processes. Another example concerns magnetodiffusion of a charged particle in an non-polarized electromagnetic beam characterized by a white-noise spectrum. (author)

Expectation-based intelligent control

International Nuclear Information System (INIS)

Zak, Michail

2006-01-01

New dynamics paradigms-negative diffusion and terminal attractors-are introduced to control noise and chaos. The applied control forces are composed of expectations governed by the associated Fokker-Planck and Liouville equations. The approach is expanded to a general concept of intelligent control via expectations. Relevance to control in livings is emphasized and illustrated by neural nets with mirror neurons

Relaxation of ion energy spectrum just after turbulent heating pulse in TRIAM-1 tokamak

International Nuclear Information System (INIS)

Nakamura, Kazuo; Hiraki, Naoji; Nakamura, Yukio; Itoh, Satoshi

1982-01-01

The temporal evolution and spatial profile of the ion energy spectrum just after the application of a toroidal current pulse for turbulent heating are investigated experimentally in the TRIAM-1 tokamak and also numerically using the Fokker-Planck equation. The two-component ion energy spectrum formed by turbulent heating relaxes to a single one within tausub(i) (the ion collision time). (author)

Relaxation of ion energy spectrum just after turbulent heating pulse in TRIAM-1 tokamak

Energy Technology Data Exchange (ETDEWEB)

Nakamura, Kazuo; Hiraki, Naoji; Nakamura, Yukio; Itoh, Satoshi [Kyushu Univ., Fukuoka (Japan). Research Inst. for Applied Mechanics

1982-07-01

The temporal evolution and spatial profile of the ion energy spectrum just after the application of a toroidal current pulse for turbulent heating are investigated experimentally in the TRIAM-1 tokamak and also numerically using the Fokker-Planck equation. The two-component ion energy spectrum formed by turbulent heating relaxes to a single one within tausub(i) (the ion collision time).

Exact non-linear equations for cosmological perturbations

Energy Technology Data Exchange (ETDEWEB)

Gong, Jinn-Ouk [Asia Pacific Center for Theoretical Physics, Pohang 37673 (Korea, Republic of); Hwang, Jai-chan [Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu 41566 (Korea, Republic of); Noh, Hyerim [Korea Astronomy and Space Science Institute, Daejeon 34055 (Korea, Republic of); Wu, David Chan Lon; Yoo, Jaiyul, E-mail: jinn-ouk.gong@apctp.org, E-mail: jchan@knu.ac.kr, E-mail: hr@kasi.re.kr, E-mail: clwu@physik.uzh.ch, E-mail: jyoo@physik.uzh.ch [Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, Universität Zürich, CH-8057 Zürich (Switzerland)

2017-10-01

We present a complete set of exact and fully non-linear equations describing all three types of cosmological perturbations—scalar, vector and tensor perturbations. We derive the equations in a thoroughly gauge-ready manner, so that any spatial and temporal gauge conditions can be employed. The equations are completely general without any physical restriction except that we assume a flat homogeneous and isotropic universe as a background. We also comment briefly on the application of our formulation to the non-expanding Minkowski background.

Finite Orbit Width Features in the CQL3D Code

Energy Technology Data Exchange (ETDEWEB)

Petrov, Y. V.; Harvey, R., E-mail: petrov@compxco.com [CompX, Del Mar (United States)

2012-09-15

Full text: The CQL3D Fokker-Planck equation solver is being upgraded to allow for the Finite-Orbit- Width (FOW) capabilities, which will provide an accurate description for a neoclassical transport, losses to the walls, and transfer of particles, momentum, and heat to the scrape-off layer. Two different options are discussed for implementing the FOW capabilities. In one option, the Fokker-Planck equation is solved for the distribution function of orbits centered around given flux surface; in the other, the equation is solved for the local distribution function at the outer-most point of flux surface at the midplane. Both options use a fast lookup table that allows characterization of orbits without actually tracing them. The lookup table, in effect, performs mapping from the Constants-Of-Motion space onto the (R{sub o}, u{sub o}, {theta}{sub o}) computational space on the midplane. The FOW modifications have been implemented for the formations of neutral beam source, RF quasilinear diffusion operator, particle diagnostics and collisional operator, and internal boundary conditions are being refined. Initial test runs show that in general, the FOW modifications result in a broader profiles of power absorption and RF-driven current, and accurate description of the loss cone. (author)

Blackbody Radiation and the Loss of Universality: Implications for Planck's Formulation and Boltzman's Constant

Directory of Open Access Journals (Sweden)

Robitaille P.-M.

2009-10-01

Full Text Available Through the reevaluation of Kirchhoff's law (Robitaille P.M.L. IEEE Trans. Plasma Sci., 2003, v.31(6, 1263-1267, Planck's blackbody equation (Planck M. Ann. der Physik, 1901, v.4, 553-356 loses its universal significance and becomes restricted to perfect absorbers. Consequently, the proper application of Planck's radiation law involves the study of solid opaque objects, typically made from graphite, soot, and carbon black. The extension of this equation to other materials may yield apparent temperatures, which do not have any physical meaning relative to the usual temperature scales. Real temperatures are exclusively obtained from objects which are known solids, or which are enclosed within, or in equilibrium with, a perfect absorber. For this reason, the currently accepted temperature of the microwave background must be viewed as an apparent temperature. Rectifying this situation, while respecting real temperatures, involves a reexamination of Boltzman's constant. In so doing, the latter is deprived of its universal nature and, in fact, acts as a temperature dependent variable. In its revised form, Planck's equation becomes temperature insensitive near 300K, when applied to the microwave background.

Inverse scattering solution of non-linear evolution equations in one space dimension: an introduction

International Nuclear Information System (INIS)

Alvarez-Estrada, R.F.

1979-01-01

A comprehensive review of the inverse scattering solution of certain non-linear evolution equations of physical interest in one space dimension is presented. We explain in some detail the interrelated techniques which allow to linearize exactly the following equations: (1) the Korteweg and de Vries equation; (2) the non-linear Schrodinger equation; (3) the modified Korteweg and de Vries equation; (4) the Sine-Gordon equation. We concentrate in discussing the pairs of linear operators which accomplish such an exact linearization and the solution of the associated initial value problem. The application of the method to other non-linear evolution equations is reviewed very briefly

HESS Opinions: Linking Darcy's equation to the linear reservoir

Science.gov (United States)

Savenije, Hubert H. G.

2018-03-01

In groundwater hydrology, two simple linear equations exist describing the relation between groundwater flow and the gradient driving it: Darcy's equation and the linear reservoir. Both equations are empirical and straightforward, but work at different scales: Darcy's equation at the laboratory scale and the linear reservoir at the watershed scale. Although at first sight they appear similar, it is not trivial to upscale Darcy's equation to the watershed scale without detailed knowledge of the structure or shape of the underlying aquifers. This paper shows that these two equations, combined by the water balance, are indeed identical provided there is equal resistance in space for water entering the subsurface network. This implies that groundwater systems make use of an efficient drainage network, a mostly invisible pattern that has evolved over geological timescales. This drainage network provides equally distributed resistance for water to access the system, connecting the active groundwater body to the stream, much like a leaf is organized to provide all stomata access to moisture at equal resistance. As a result, the timescale of the linear reservoir appears to be inversely proportional to Darcy's conductance, the proportionality being the product of the porosity and the resistance to entering the drainage network. The main question remaining is which physical law lies behind pattern formation in groundwater systems, evolving in a way that resistance to drainage is constant in space. But that is a fundamental question that is equally relevant for understanding the hydraulic properties of leaf veins in plants or of blood veins in animals.

Stochastic processes and applications diffusion processes, the Fokker-Planck and Langevin equations

CERN Document Server

Pavliotis, Grigorios A

2014-01-01

This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated.                 The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to eq...

ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

Science.gov (United States)

Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

2018-04-01

In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

Integration of differential equations by the pseudo-linear (PL) approximation

International Nuclear Information System (INIS)

Bonalumi, Riccardo A.

1998-01-01

A new method of integrating differential equations was originated with the technique of approximately calculating the integrals called the pseudo-linear (PL) procedure: this method is A-stable. This article contains the following examples: 1st order ordinary differential equations (ODEs), 2nd order linear ODEs, stiff system of ODEs (neutron kinetics), one-dimensional parabolic (diffusion) partial differential equations. In this latter case, this PL method coincides with the Crank-Nicholson method

Electron-cyclotron-resonant-heated electron distribution functions

International Nuclear Information System (INIS)

Matsuda, Y.; Nevins, W.M.; Cohen, R.H.

1981-01-01

Recent studies at Lawrence Livermore National Laboratory (LLNL) with a bounce-averaged Fokker-Planck code indicate that the energetic electron tail formed by electron-cyclotron resonant heating (ECRH) at the second harmonic is not Maxwellian. We present the results of our bounce-averaged Fokker-Planck code along with some simple analytic models of hot-electron distribution functions

Linear System of Equations, Matrix Inversion, and Linear Programming Using MS Excel

Science.gov (United States)

El-Gebeily, M.; Yushau, B.

2008-01-01

In this note, we demonstrate with illustrations two different ways that MS Excel can be used to solve Linear Systems of Equation, Linear Programming Problems, and Matrix Inversion Problems. The advantage of using MS Excel is its availability and transparency (the user is responsible for most of the details of how a problem is solved). Further, we…

Contribution to the determination of nuclear friction by studying the de-excitation of nuclei in the transient regime

International Nuclear Information System (INIS)

Hassani, S.

1985-01-01

An old idea of Kramers is to consider nuclear fission as a diffusion process in phase space corresponding to the collective variable of fission. The fission width is taken as an escape rate of the system over the barrier potential. The evolution of the distribution of this collective variable and its conjugate is governed by a Fokker-Planck equation. In a quasistationary treatment Kramers obtained a fission rate which differs from the result given by the transition state method by a friction dependent factor. The non quasistationary solution of the Fokker-Planck equation allows to obtain an escape rate that presents a transient regime: from zero it grows and reaches asymptotically the Kramers' value. This time-dependent fission width is included in a formalism that describes the deexcitation of the compound nucleus in order to calculate the neutron multiplicities in competition with fission. A sensitive friction-dependence of the multiplicities is obtained. Using this formalism and comparing the results with data of a recent experiment gives a good agreement; resolving the disagreement between data and the usual statistical model at high energy. A range of values of the friction coefficient is deduced [fr

Hamiltonian theory of the ion cyclotron minority heating dynamics in tokamak plasmas

International Nuclear Information System (INIS)

Becoulet, A.; Gambier, D.J.; Samain, A.

1990-03-01

The question of heating a tokamak plasma by means of electromagnetic waves in the Ion Cyclotron Range of Frequency (ICRF) is considered in the perspective of large RF powers and in the low collisionality regime. In such case the Quasi Linear Theory (QLT) is validated by the Hamiltonian dynamics of the wave particle interaction which exceeds the threshold of the intrinsic stochasticity. The Hamiltonian dynamics is represented by the evolution of a set of three canonical action angle variables well adapted to the tokamak magnetic configuration. This approach allows to derive the RF diffusion coefficient with very few assumptions. The distribution function of the resonant ions is written as a Fokker Planck equation but the emphasis is put on the QL diffusion instead of on the usual diffusion induced by collisions. Then the Fokker Planck equation is given a variational from which a solution is derived in the form of a semi analytical trial function of three parameters: the percentage of resonant particle contained in the tail; an isotropic width ΔT and an anisotropic one ΔP. This solution is successfully tested against real experimental observations. Practically it is shown that in the case of JET the distribution function is influenced by adiabatic barriers which in turn limit the Hamiltonian stochasticity domain within energy values typically in the MeV range. Consequently and for a given ICRF power, the tail energy excursion is lower and its concentration higher than that of a bounce averaged prediction. This may actually be an advantage for machines like JET considering the energy range required to simulate the α-particle behaviour in a relevant fusion reactor

Representation of the Fokker-Planck collision term for Coulomb interaction as series of Legendre polynomials

International Nuclear Information System (INIS)

Almeida Ferreira, A.C. de.

1984-01-01

For problems with azimuthal symmetry in velocity space, the distribution function depends only on the speed and on the pitch angle. The angular dependence of the distribution function is expanded in Legendre polynomials, and the expansions of the collision integrals describing two-body Coulomb interactions in a plasma are determined through the use of the Rosenbluth potentials. The electron distribution function is written as a Maxwellian plus a deviation, and the representation in Legendre polynomials of the electron-electron collision term is given for both its linear and nonlinear part. To determine the representation of the electron-ion collision term it is assumed that the ion distribution is much narrower in velocity space than the electron distribution, and shifted from the origin by a flow velocity. The equations are presented in a form that is suitable for their use in a computer. (Author) [pt

Noise-induced modulation of the relaxation kinetics around a non-equilibrium steady state of non-linear chemical reaction networks.

Directory of Open Access Journals (Sweden)

Rajesh Ramaswamy