Multiple Information, Shannon-like Entropy and Thermodynamic Entropy

The Maxwell-Boltzmann distribution is compatible with Shannon’s entropy which in turn is equivalent to thermodynamic entropy. In such a case, ln(P(e)) is information which matches the information e (energy) brought by a particle into a collision. In (1) we considered “altered” probabilities Palt(e) = P(e)/[1-/+P(e)] to describe information associated with a collision of fermions or bosons. In this note we try to reinterpret Palt(e) for fermions or bosons as two pieces of information which combine to create the information e brought into a collision. In principle there could be more terms. It is noted that these terms are linear in P(e). They lead to a sum of Shannon-like entropies i.e. P(e)ln[P(e)] + [1-P(e)] ln[1-P(e)]. We also consider the link with thermodynamic entropy i.e. for fixed parameters: dE = Sum over i ei dP(ei) = T dS. We argue that “entropy” should also have the form Sum over i F(P(ei)) dP(ei). In (2) we showed that one may obtain the form ln(P(ei)) P(ei) from this consideration alone. Here we extend this idea to Shannon-like terms (a+bP(ei)) ln[a+bP(ei)]