MaxEnt, ln(Probability) and the Maxwell-Boltzmann and Power Law Distributions

Addition: Reference (1) is: Ran, G. and Du, J. Are power law distributions an equilibrium distribution or a stationary nonequilibrium distribution? (2013 or later) https://arxiv.org/pdf/1403.5441.pdf The idea of extremizing entropy (written as a function of P(e/T)=probability, where e is single particle energy and T temperature, with respect to P(e/T) including the constraint Sum over i ei P(ei/T)=constant leads to a recipe for entropy. This recipe involves solving P(e/T) for e/T and then integrating with respect to dP(e/T) so that d/dP from the extremization cancels the integration procedure. This recipe for entropy is thus based on: dS/dP = e/T or more generally on dS/dP= a+be/T where a and b are constant and the extra constraint Sum over i P(ei)=1 is used. In general, S(P), but we argue that in both the Maxwell-Boltzmann and Power Law Cases, dS/dP is strictly a function of ln(P), which in information theory is called information. We consider how this is linked to taking the derivative d/d (1/T) of dS/dP= e/T or its generalization