A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

Abstract

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.