The way from microscopic many-particle theory to macroscopic hydrodynamics

TL;DR

This paper systematically derives macroscopic hydrodynamic equations from microscopic many-particle theories using statistical mechanics, highlighting the transition from exact nonlocal equations to approximate local forms with fluctuations and thermodynamic irreversibility.

Contribution

It provides a detailed, step-by-step derivation of hydrodynamics from microscopic models, including the treatment of fluctuations and the connection to stochastic equations.

Findings

Derivation of hydrodynamic equations with reversible, dissipative, and fluctuating terms.

Identification of how time-inversion invariance is broken and the second law emerges.

Modification of the fluctuation theorem with an additional term.

Abstract

Starting from the microscopic description of a normal fluid in terms of any kind of local interacting many-particle theory we present a well defined step by step procedure to derive the hydrodynamic equations for the macroscopic phenomena. We specify the densities of the conserved quantities as the relevant hydrodynamic variables and apply the methods of non-equilibrium statistical mechanics with projection operator techniques. As a result we obtain time-evolution equations for the hydrodynamic variables with three kinds of terms on the right-hand sides: reversible, dissipative and fluctuating terms. In their original form these equations are completely exact and contain nonlocal terms in space and time which describe nonlocal memory effects. Applying a few approximations the nonlocal properties and the memory effects are removed. As a result we find the well known hydrodynamic equations of a normal fluid with Gaussian fluctuating forces. In the following we investigate if and how the time-inversion invariance is broken and how the second law of thermodynamics comes about. Furthermore, we show that the hydrodynamic equations with fluctuating forces are equivalent to stochastic Langevin equations and the related Fokker-Planck equation. Finally, we investigate the fluctuation theorem and find a modification by an additional term.