![]() ![]() In another manuscript, Campisi 9 starts from the volume entropy and derives the δ-function in energy as the phase space density for the microcanonical ensemble by maximizing the entropy subject to normalization and energy constraints and an assumption that the dynamics is ergodic. Campisi 1 also showed that the equipartition theorem is satisfied if and only if the temperature is T G. On the other side, Campisi 1 showed that the volume entropy, but not the surface entropy, satisfies the generalized Helmholtz theorem dS = ( dE + PdV)/ T, where T is the Gibbs/Hertz temperature, T G (defined below), on condition that the system is ergodic. Gross and Kenney 8 favored the surface energy in describing phase transitions and because it includes only the states accessible to the system, in agreement with ideas of entropy using information theory, whereas the volume entropy includes inaccessible states. 7 favored the volume entropy because it obeys equipartition. Among many who have contributed to the argument, Hertz 5 favored the volume entropy because it is an adiabatic invariant, as did Rugh. Gibbs 4 was the first, in 1902, to compare the utility of these two definitions. 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. 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. In a fourth example, a collection of two-level atoms, the Boltzmann/Planck entropy is in somewhat better agreement with canonical ensemble results. ![]() 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. 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. ![]() 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). 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). Two different definitions of entropy, S = k ln W, in the microcanonical ensemble have been competing for over 100 years. ![]()
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