Comments on the Extensivity of the Boltzmann Entropy

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.