Relationship Between Energy, Entropy and the Distribution Normalization Factor

  In a previous note (1), we discussed the equivalence of  -T Sum over e f(e/T) ln(f(e/T) i.e. temperature times entropy and internal energy =  Sum over e e f(e) up to a constant related to the natural log of the normalization constant of f(e/T) (divided by V, the volume). We also noted that distributions, including the Maxwell-Boltzmann (MB), Juttner and Kaniadakis and the Tsallis, to an extent, made use of function-inverse function pairs. (E.g. for the MB or Juttner distribution  ln is the inverse of exp(-e/T).) In this note, we wish to examine the relationship between internal energy and entropy in more detail, in particular because F=Helmholtz free energy = E-TS and Gibbs free energy = E-TS+PV and we argue that E-TS= TN ln[ 1/V (m/(6.28T)1.5] for the MB case.