Liouville’s theorems, Gibbs’ entropy, and multifractal distributions for nonequilibrium steady states

Liouville’s best-known theorem, ḟ({q,p},t)=0, describes the incompressible flow of phase-space probability density, f({q,p},t). This incompressible-flow theorem follows directly from Hamilton’s equations of motion. It applies to simulations of isolated systems composed of interacting particles, whether or not the particles are confined by a box potential. Provided that the particle–particle and particle–box collisions are sufficiently mixing, the long-time-averaged value 〈f〉 approaches, in a “coarse-grained” sense, Gibbs’ equilibrium microcanonical probability density, feq, from which all equilibrium properties follow, according to Gibbs’ statistical mechanics. All these ideas can be extended to many-body simulations of deterministic open systems with nonequilibrium boundary conditions incorporating heat transfer. Then Liouville’s compressible phase-space-flow theorem—in the original ḟ≠0 form—applies. I illustrate and contrast Liouville’s two theorems for two simple nonequilibrium systems, in each case considering both stationary and time-dependent cases. Gibbs’ distributions for incompressible (equilibrium) flows are typically smooth. Surprisingly, the long-time-averaged phase-space distributions of nonequilibrium compressible-flow systems are instead singular and “multifractal.” The nonequilibrium analog of Gibbs’ entropy, S≡−k〈lnf〉, diverges, to −∞, in such a case. Gibbs’ classic remedy for such entropy errors was to “coarse-grain” the probability density—by averaging over finite cells of dimensions ∏Δq Δp. Such a coarse graining is effective for isolated systems approaching equilibrium, and leads to a unique entropy. Coarse graining is not as useful for deterministic open systems, constrained so as to describe stationary nonequilibrium states. Such systems have a Gibbs’ entropy which depends, logarithmically, upon the grain size. The two Liouville’s theorems, their applications to Gibbs’ entropy, and to the grain-size dependence of that entropy, are clearly illustrated here with simple example problems.