Contributions to Non-Equilibrium Thermodynamics. I. Theory of Hydrodynamical Fluctuations

TLDR

Stationary Gaussian–Markov processes underlie both Brownian motion and Onsager–Machlup non‑equilibrium thermodynamics, the latter traditionally assumed to be time‑even. This work generalizes the framework by removing the time‑even assumption to study the most general stationary Gaussian–Markov process. The authors introduce the necessary fluctuation terms and their correlations, then use them to derive a Langevin equation for a Brownian particle of arbitrary shape. Consequently the theory applies to linearized hydrodynamics and shows that Navier–Stokes equations must include Landau–Lifshitz fluctuation terms.

Abstract

The velocity of a particle in Brownian motion as described by the Langevin equation is a stationary Gaussian–Markov process. Similarly, in the formulation of the laws of non-equilibrium thermodynamics by Onsager and Machlup, the macroscopic variables describing the state of a system lead to an n-component stationary Gaussian–Markov process, which, in addition, these authors assumed to be even in time. By dropping this assumption, the most general stationary Gaussian–Markov process is discussed. As a consequence, the theory becomes applicable to the linearized hydrodynamical equations and suggests that the Navier–Stokes equations require additional fluctuation terms which were first proposed by Landau and Lifshitz. Such terms and their correlation properties are presented, and these equations are then used to derive the Langevin equation for the Brownian motion of a particle of arbitrary shape.