Dissipation in hydrodynamics from micro- to macroscale: wisdom from Boltzmann and stochastic thermodynamics

Abstract

We show that macroscopic irreversible thermodynamics for viscous fluids can be derived from exact information-theoretic thermodynamic identities valid at the microscale. Entropy production, in particular, is a measure of the loss of many-particle correlations in the same way in which it measures the loss of system-reservoirs correlations in stochastic thermodynamics (ST). More specifically, we first show that boundary conditions at the macroscopic level define a natural decomposition of the entropy production rate (EPR) in terms of thermodynamic forces multiplying their conjugate currents, as well as a change in suitable nonequilibrium potential that acts as a Lyapunov function in the absence of forces. Moving to the microscale, we identify the exact identities at the origin of these dissipative contributions for isolated Hamiltonian systems. We then show that the molecular chaos hypothesis, which gives rise to the Boltzmann equation at the mesoscale, leads to a positive rate of loss of many-particle correlations, which we identify with the Boltzmann EPR. By generalizing the Boltzmann equation to account for boundaries with nonuniform temperature and nonzero velocity, and resorting to the Chapman–Enskog expansion, we recover the macroscopic theory we started from. Finally, using a linearized Boltzmann equation we derive ST for dilute particles in a weakly out-of-equilibrium fluid and its corresponding macroscopic thermodynamics. Our work unambiguously demonstrates the information-theoretical origin of thermodynamic notions of entropy and dissipation in macroscale irreversible thermodynamics.

Keywords

• stochastic thermodynamics
• nonequilibrium thermodynamics
• Navier–Stokes equations
• information theory