Transport and evolution of classical and quantum ensembles

2020 ◽  
pp. 292-341
Author(s):  
Sandip Tiwari
Keyword(s):  
Boltzmann Transport Equation ◽  
Planck Equation ◽  
Kolmogorov Equation ◽  
Boltzmann Transport ◽  
Relaxation Time Approximation ◽  
Quantum Mechanical Description ◽  
Probabilistic Evolution ◽  
Fokker Planck Equations ◽  
And Diffusion ◽  
Fokker Planck

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

scholarly journals First and second order optimality conditions for the control of Fokker-Planck equations

2021 ◽  
Vol 27 ◽  
pp. 15
Author(s):  
M. Soledad Aronna ◽  
Fredi Tröltzsch
Keyword(s):  
Control Variable ◽  
Sufficient Conditions ◽  
Planck Equation ◽  
Second Order ◽  
Optimal Controls ◽  
Main Emphasis ◽  
The Cost ◽  
Necessary And Sufficient ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this article we study an optimal control problem subject to the Fokker-Planck equation ∂tρ − ν∆ρ − div(ρB[u]) = 0 The control variable u is time-dependent and possibly multidimensional, and the function B depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic regularization term on the control. For this problem, we prove existence of optimal controls and first order necessary conditions. Main emphasis is placed on second order necessary and sufficient conditions.

Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution

2020 ◽  
Vol 30 (04) ◽  
pp. 685-725 ◽  
Author(s):  
Giulia Furioli ◽  
Ada Pulvirenti ◽  
Elide Terraneo ◽  
Giuseppe Toscani
Keyword(s):  
Steady State ◽  
Wealth Distribution ◽  
Kinetic Equations ◽  
Planck Equation ◽  
Logarithmic Sobolev Inequality ◽  
One Dimensional ◽  
Distribution Of Wealth ◽  
Multi Agent ◽  
Fokker Planck Equations ◽  
Fokker Planck

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

scholarly journals New Iterative Methods for Solving Fokker-Planck Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
A. A. Hemeda ◽  
E. E. Eladdad
Keyword(s):  
Iterative Method ◽  
Lagrange Multipliers ◽  
Planck Equation ◽  
Adomian Polynomials ◽  
Adomian Decomposition ◽  
Iteration Methods ◽  
Linear And Nonlinear ◽  
Fokker Planck Equations ◽  
New Iterative Method ◽  
Fokker Planck

In this article, we propose the new iterative method and introduce the integral iterative method to solve linear and nonlinear Fokker-Planck equations and some similar equations. The results obtained by the two methods are compared with those obtained by both Adomian decomposition and variational iteration methods. Comparison shows that the two methods are more effective and convenient to use and overcome the difficulties arising in calculating Adomian polynomials and Lagrange multipliers, which means that the considered methods can simply and successfully be applied to a large class of problems.

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