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

Note: April 13, 2023. In Parts I and II, when taking derivatives, probabilities are unnormalized. In Part I, we examined how a recipe for MaxEnt (maximum entropy) usually includes the constraint Sum over i ei P(ei/T) and possibly a second constraint Sum over i P(ei). This leads to a recipe for the entropy such that dS/dP = a+bei/T ((A)) where a and b are constants, S being a function of P(ei/T). We saw that in both the Maxwell-Boltzmann and Power Law distributions, dS/dP is strictly a function of ln(P), which is called information in information theory. We then differentiated ((A)) with respect to 1/T to see what distributions appear if S in a function of ln(P) alone and saw the emergence of the MB and Power Law solutions. In this note, we consider the Zwanzig differential equation (from (1)) which includes random noise in the form of a variance function D(x,p). The power law distribution is a solution of this equation. We examine the Zwanzig equation to see exactly what condition needs to hold for any general solution (for a constant coefficient of friction for simplicity) and find the relation: 1= D d/dy ln(P) ((B)) where y = a+bei/T. Thus ln(P) and its derivative which appears in Fisher’s information are key pieces. Instead of differentiating the MaxEnt equation ((A)) by 1/T, we argue that differentiation by ei/T leads to ((B)) i.e. the key solution portion of the Zwanzig equation. Thus we link maximum entropy (albeit a derivative of this equation) to the solution of the Zwanzig equation for the MB and Power Law distributions