Related papers: The Maxwell-Boltzmann Distribution is not the Equi…
We consider the mean field Fokker-Planck equation subject to nonlinear no-flux boundary conditions, which necessarily arise when subjecting a system of Brownian particles interacting via a pair potential in a bounded domain. With the…
Combining analytical and numerical methods, we study within the framework of the homogeneous non-linear Boltzmann equation, a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We use the new…
This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how…
The problem of anomalous diffusion in momentum space is considered for plasma-like systems on the basis of a new collision integral, which is appropriate for consideration of the probability transition function (PTF) with long tails in…
Statistical invariance of Wiener increments under SO(n) rotations provides a notion of gauge transformation of state-dependent Brownian motion. We show that the stochastic dynamics of non gauge-invariant systems is not unambiguously…
We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space…
We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by…
We consider different generalizations of the Fokker-Planck-equation devised to describe Levy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, Levy statistics can emerge…
We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special…
In this work we consider the classical non-linear Boltzmann equation, where the unknown is the distribution function $f$, which depends on the time $t$, the vector $\mathbf{x}$ (the position of a molecule) and its velocity $\mathbf{\xi}$.…
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…
We discuss a general class of nonlinear mean-field Fokker-Planck equations [P.H. Chavanis, Phys. Rev. E, 68, 036108 (2003)] and show their applications in different domains of physics, astrophysics and biology. These equations are…
Traditionally, the quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasi-probability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum…
We consider a scalar QED model for the frictional force and the momentum fluctuations of a polarizable particle in thermal equilibrium with radiation or in hyperbolic motion in a vacuum. In the former case the loss of particle kinetic…
Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by…
In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To…
The velocity distribution of inelastic granular gas is examined numerically on two dimensional hard disk system in nearly elastic regime using molecular dynamical simulations. The system is prepared initially in the equilibrium state with…
In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous…
The most rigorous physical description of non-equilibrium gas dynamics is rooted in the numerical solution of the Boltzmann equation. Yet, the large number of degrees of freedom and the wide range of both spatial and temporal scales render…
The quantum Boltzmann equation, or Fokker-Planck equation, has been used to successfully explain a number of experiments in semiconductor optics in the past two decades. This paper reviews some of the developments of this work, including…