Probability Derivative of Entropy as Information?
Francesco R. Ruggeri · Zenodo (CERN European Organization for Nuclear Research) · 2023
Entropy in the Maxwell-Boltzmann example of a gas with no potential may be mapped into a set of trial runs for a single gas particle with probabilities p(ei). If N, the number of trials, approaches infinite, the probability of a set of runs is Product over i exp{ ln(p(ei)) N p(ei)h}. This is the same for any set of N runs and so probability is 1/ permutations of N particles with n(ei). If ln{number of permutations is maximized} (i.e. probability minimized) subject to a constraint, say Sum over i ei p(ei), then one obtains the Maxwell-Boltzmann distribution. The ln of the number of permutations is proportional to a function called entropy which is associated with a maximization of permutations. We argue, however, that there are two unusual issues with the above approach. First, there is no need to take ln of the number of arrangements when maximizing. This is specifically done to convert a sum into a product, emphasizing the independence of the trial runs which we argue is an important feature. The second is the constraint. We argue that a very specific constraint is needed, namely one containing the information relevant to physical collision which creates the equilibrium. We have argued this in previous notes. The point we wish to make here is that d/d n(ei) Entropy density = -ei/T = information. Thus entropy density is a function whose derivative with respect to n(ei) yields the information of the system. This is not unlike the quantum free particle case for which -id/dx ln(exp(ipx) = p = information. Mathematically d/dn(ei) is supposed to represent a change in the function n(ei), but n is simply a variable and one may argue that d/dn(ei) means function( n(ei)+1) - function(n(ei)). Thus one has a function which when changed by adding one particle minus its original value yields the information of the particle. Given that one considers a particle to be independent, then the argument of ln should display this independence. nln(n) = ln(n (power n)) has this property. If n->n+1, n or n+1 in the base is essentially the same, but there is an extra n+1 due to the exponent. Ln then pulls this out so the extra particle carries an information of -ei/T. We argue that the idea of entropy is linked to a more physical idea of reaction balance which in turn focuses on conservation of energy and probability. If energy ei multiplied by a constant carries the same information as p(ei)p(ej) converted by a function into a sum of probabilities (i.e. using ln), one has an expression linking the information content of p(ei) to -ei/T, the information of the particle. This expression may be integrated by p(ei) (treating ei as a constant) to obtain the entropy on one side and -ei/T p(ei) on the other, i.e. the special constraint. Thus instead of thinking of maximizing arrangements etc, we suggest that one consider probability and information associated with a collision which creates the equilibrium and focus on the independence of the particle which moves from one collision to another.