![]() 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= kln W, in the microcanonical ensemble have been competing for over 100 years.
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