On the (Boltzmann) entropy of non-equilibrium systems

By Sheldon Goldstein, Joel Lebowitz. 2003

180 citations recorded by the source corpus.

Abstract Boltzmann defined the entropy of a macroscopic system in a macrostate M as the log of the volume of phase space (number of microstates) corresponding to M . This agrees with the thermodynamic entropy of Clausius when M specifies the locally conserved quantities of a system in local thermal equilibrium (LTE). Here we discuss Boltzmann’s entropy, involving an appropriate choice of macro-variables, for systems not in LTE. We generalize the formulas of Boltzmann for dilute gases and of Resibois for hard sphere fluids and show that for macro-variables satisfying any deterministic autonomous evolution equation arising from the microscopic dynamics the corresponding Boltzmann entropy must satisfy an H -theorem.

Subfields: Biomedical Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a6580fc373f63b13e249eb8a3d1ec1ac5f2def4a

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In Biomedical Engineering

In Statistical and Nonlinear Physics

Citations in the corpus

The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics (2005, background): “Garrido, Goldstein, and Lebowitz (45) , see also [ 46 ] argue that whenever a”
Long-time behavior of macroscopic quantum systems (2010, background): “y either time reversal or recurrence. Put very succinctly, this is because, according to the statement, Sincreases in both time directions and may well decrease after reaching its maximum; see, e.g., [20, 37, 21] for further discussion. 26 Theorem 1. (von Neumann’s 1929 QET) Let ε>0, δ>0, and ”
Statistical mechanics of two-dimensional and geophysical flows (2011, background): “Then for a sample of rare initial microscopic conditions (corresponding to a low entropy macrostate), an overwhelming number of the trajectories evolve towards microscopic configurations corresponding to a more probable (higher entropy) macrostate [87].”
Thermodynamics and dynamics of systems with long-range interactions (2010, background): “Moreover, it is a general property that, for a system in which the evolution of themacroscopic phase space density f is described by an autonomous equation, the number of microscopic states corresponding to f has to increase in time [55].”
A Field Guide to Recent Work on the Foundations of Statistical Mechanics. (2008, unknown): “and α i ∩ α j = ⊘ for all i = j and i, j = 1, ..., k. 13 Goldstein (2001) (43), Goldstein and Lebowitz (2004) (57), Lebowitz (1993a, 34; 1993b, 5; 1999, 348).”