S. Goldstein and Joel L. Lebowitz
On the (Boltzmann) Entropy of Nonequilibrium Systems
(47K, Tex)

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 ${\cal H}$-theorem.