Liouville{close_quote}s theorems, Gibbs{close_quote} entropy, and multifractal distributions for nonequilibrium steady states
- Department of Applied Science, University of California at Davis/Livermore, Hertz Hall, Livermore, California 94550 (United States)
Liouville{close_quote}s best-known theorem, {dot f}({l_brace}q,p{r_brace},t)=0, describes the incompressible flow of phase-space probability density, f({l_brace}q,p{r_brace},t). This incompressible-flow theorem follows directly from Hamilton{close_quote}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{endash}particle and particle{endash}box collisions are sufficiently mixing, the long-time-averaged value {l_angle}f{r_angle} approaches, in a {open_quotes}coarse-grained{close_quotes} sense, Gibbs{close_quote} equilibrium microcanonical probability density, f{sub eq}, from which all equilibrium properties follow, according to Gibbs{close_quote} 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{close_quote}s compressible phase-space-flow theorem{emdash}in the original {dot f}{ne}0 form{emdash}applies. I illustrate and contrast Liouville{close_quote}s two theorems for two simple nonequilibrium systems, in each case considering both stationary and time-dependent cases. Gibbs{close_quote} 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 {open_quotes}multifractal.{close_quotes} The nonequilibrium analog of Gibbs{close_quote} entropy, S{equivalent_to}{minus}k{l_angle}lnf{r_angle}, diverges, to {minus}{infinity}, in such a case. Gibbs{close_quote} classic remedy for such entropy errors was to {open_quotes}coarse-grain{close_quotes} the probability density{emdash}by averaging over finite cells of dimensions {product}{Delta}qthinsp{Delta}p. 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{close_quote} entropy which depends, logarithmically, upon the grain size. The two Liouville{close_quote}s theorems, their applications to Gibbs{close_quote} entropy, and to the grain-size dependence of that entropy, are clearly illustrated here with simple example problems. {copyright} {ital 1998 American Institute of Physics.}
- OSTI ID:
- 641602
- Journal Information:
- Journal of Chemical Physics, Vol. 109, Issue 11; Other Information: PBD: Sep 1998
- Country of Publication:
- United States
- Language:
- English
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