Comments on the Extensivity of the Boltzmann Entropy

Abstract

In thermodynamics entropy Std is an extensive state function. Its derivation by statistical mechanics following Boltzmann and Gibbs with the famous formula S=kBlnW for a micro-canonical ensemble with N particles, kB the Boltzmann constant, and W the number of accessible micro-states is however in general not extensive unless the Stirling approximation given by lnN! – NlnN + N is used. Furthermore, at the thermodynamic limit with the number of particles N→∞ at constant density the Stirling approximation can not be used to show extensivity because limN→∞ (lnN! – NlnN + N)=∞. Hence, the Boltzmann entropy S as shown here for the ideal gas is neither for a small system with N particles nor at the thermodynamic limit extensive. Thus, if strict extensivity for the entropy is requested the claim of statistical mechanics that the Boltzmann entropy is a microscopic description of its thermodynamic analog is challenged.