  The Gibbs entropy of a macroscopic classical system is a function of a
probability distribution over phase space, i.e., of an ensemble. In contrast,
the Boltzmann entropy is a function on phase space, and is thus defined for an
individual system. Our aim is to discuss and compare these two notions of
entropy, along with the associated ensemblist and individualist views of
thermal equilibrium. Using the Gibbsian ensembles for the computation of the
Gibbs entropy, the two notions yield the same (leading order) values for the
entropy of a macroscopic system in thermal equilibrium. The two approaches do
not, however, necessarily agree for non-equilibrium systems. For those, we
argue that the Boltzmann entropy is the one that corresponds to thermodynamic
entropy, in particular in connection with the second law of thermodynamics.
Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue
that the individualist (Boltzmannian) concept of equilibrium is supported by
the recent works on thermalization of closed quantum systems.

Statistical Mechanics