  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.
