On the (Boltzmann) Entropy of Nonequilibrium Systems

Details On the (Boltzmann) Entropy of Nonequilibrium Systems On the (Boltzmann) Entropy of Nonequilibrium Systems

2003-04-10

arxiv.org/abs/cond-mat/0304251v2

Physica D193 (2004) 53-66

Physics

Condensed Matter

Statistical Mechanics

31 pages, in Tex, authors' e-mails: oldstein@math.rutgers.edu, lebowitz@math.rutgers.edu

Scientific paper

10.1016/j.physd.2004.01.008

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 ${\cal H}$-theorem.

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