A statistical mechanical approach to fluid dynamics: Thermodynamically consistent equations for simple dissipative fluids

Abstract

In this paper, a statistical mechanical derivation of thermodynamically consistent fluid dynamical equations is presented for non-isothermal viscous molecular fluids. The coarse-graining process is based on the combination of key concepts from earlier works, including the Dirac-delta formalism of Irving and Kirkwood, the identity of statistical physical ensemble averages by Khinchin, and a first-order Taylor expansion around the leading-order solution of the Chapman-Enskog theory. The non-equilibrium thermodynamic quantities and constitutive relations directly emerge in the proposed coarse-graining process, which results in a completion of the phenomenological theory. We show that (i) the variational form of the thermodynamic part of the reversible stress is already encoded on the level of the Hamiltonian many-body problem, (ii) the dynamics monotonically maximizes the entropy at constant energy, and (iii) that the phenomenological energy balance equation obtained in the adiabatic approximation lacks the contribution of non-local interactions, which is crucial in modelling the gas-liquid transition in near-critical fluids