Dissipation in Hydrodynamics from Micro- to Macroscale: Wisdom from Boltzmann and Stochastic Thermodynamics

Résumé : 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 conjugated currents, as well as a change in suitable nonequilibrium potential that acts as a Lyapunov function in absence of forces. At the microscale, for isolated Hamiltonian systems, we identify the exact identities at the origin of these dissipative contributions. Indeed, 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 entropy production rate. By generalizing the Boltzmann equation to account for boundaries with nonuniform temperature and nonzero velocity, and resorting to the Chapman--Enskog expansion, we reproduce the macroscopic theory we started from. We also show that an equivalent mesoscale description in terms of a linearized Boltzmann equation obeys local detailed balance (LDB) and thus reproduces a ST theory. Our work unambiguously demonstrates the information-theoretical origin of thermodynamic notions of entropy and dissipation in macroscale irreversible thermodynamics.