A comparison of Boltzmann and Gibbs definitions of microcanonical entropy for small systems

Two different definitions of entropy, S = k ln W, in the microcanonical ensemble have been competing for over 100 years. The Boltzmann/Planck definition is that W is the number of states accessible to the system at its energy E (also called the surface entropy). The Gibbs/Hertz definition is that W is the number of states of the system up to the energy E (also called the volume entropy). These two definitions agree for large systems but differ by terms of order N−1 for small systems, where N is the number of particles in the system. For three analytical examples (a generalized classical Hamiltonian, identical quantum harmonic oscillators, and the spinless quantum ideal gas), neither the Boltzmann/Planck entropy nor heat capacity is extensive because it is always proportional to N − 1 rather than N, but the Gibbs/Hertz entropy is extensive and, in addition, gives thermodynamic quantities, which are in remarkable agreement with canonical ensemble calculations for systems of even a few particles. In a fourth example, a collection of two-level atoms, the Boltzmann/Planck entropy is in somewhat better agreement with canonical ensemble results. Similar model systems show that temperature changes when two subsystems come to thermal equilibrium are in better agreement with expectations for the Gibbs/Hertz temperature than for the Boltzmann/Planck temperature, except when the density of states is decreasing. I conclude that the Gibbs/Hertz entropy is more useful than the Boltzmann/Planck entropy for comparing microcanonical simulations with canonical molecular dynamics simulations of small systems.