Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations

Axel Klar; Florian Schneider; Oliver Tse

doi:10.3934/krm.2014.7.509

Article Contents

2014, Volume 7, Issue 3: 509-529. Doi: 10.3934/krm.2014.7.509

This issue Previous Article Hysteretic behavior of a moment-closure approximation for FENE model Next Article A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods

Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations

1.
Department of mathematics, TU Kaiserslautern, and Fraunhofer ITWM Kaiserslautern, 67663 Kaiserslautern
2.
Department of mathematics, TU Kaiserslautern, 67663 Kaiserslautern
3.
Department of Technomathematics, University of Kaiserslautern, 67663 Kaiserslautern

Received: October 2013

Revised: February 2014

Published: September 2014

Abstract / Introduction Related Papers Cited by

Abstract

We consider stochastic dynamic systems with state space $\mathbb{R}^n \times \mathbb{S}^{n-1}$ and associated Fokker--Planck equations. Such systems are used to model, for example, fiber dynamics or swarming and pedestrian dynamics with constant individual speed of propagation. Approximate equations, like linear and nonlinear (maximum entropy) moment approximations and linear and nonlinear diffusion approximations are investigated. These approximations are compared to the underlying Fokker--Planck equation with respect to quality measures like the decay rates to equilibrium. The results clearly show the superiority of the maximum entropy approach for this application compared to the simpler linear and diffusion approximations.

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Mathematics Subject Classification: Primary: 37H10, 60H30, 70H05.

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