Microscopic Derivation of Non-Markovian Thermalization of a Brownian Particle

Abstract

In this paper, the first microscopic approach to the Brownian motion is developed in the case where the mass density of the suspending bath is of the same order of magnitude as that of the Brownian (B) particle. Starting from an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematicaly via the multiple time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e. the relaxation towards the Maxwell distribution in velocity space. In contradistinction to the Fokker-Planck equation, the derived new evolution equation is non-local both in time and in velocity space, owing to correlated recollision events between the fluid and particle B. In the long-time limit, it describes a non-markovian generalized Ornstein-Uhlenbeck process. However, in spite of this complex dynamical behaviour, the Stokes-Einstein law relating the friction and diffusion coefficients is shown to remain valid. A microscopic expression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the mean-free in the gas is small compared to the radius of particle B.

arXiv comment28 pages, no figure, submitted to Journal of Statistical Physics

Cited by in corpus (10)

• Nanofluidics, from bulk to interfaces
• Molecular hydrodynamics from memory kernels
• Likelihood-based non-Markovian models from molecular dynamics
• Solving the Fokker-Planck kinetic equation on a lattice
• Dynamical density functional theory for orientable colloids including inertia and hydrodynamic interactions
• Langevin Equation for the Rayleigh model with finite-ranged interactions
• Non-equilibrium statistical mechanics of the heat bath for two Brownian particles
• Anisotropic active Brownian particle with a fluctuating propulsion force
• A non-local quasi-equilibrium state in the Bhatnagar-Gross-Krook Boltzmann equation for thermo-hydrodynamics: Conservation laws, the Boltzmann H-theorem, and the fluctuation-dissipation theorem
• Non-Markovian quantum Brownian motion: a non-Hamiltonian approach