Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

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.