Maximization of Entropy is More General Than Reaction Balance

 In the Maxwell-Boltzmann (MB) case, one may obtain the probability distribution by maximizing Shannon’s entropy - Sum over i p(ei) ln(p(ei)) subject to the constraints Sum over i ei p(ei) = Eave and Sum over i p(ei)=1, or by reaction balance: ei+ej = ek+el with p(ei)p(ej) = p(ek)p(el). This might lead to the notion that the two approaches are essentially equivalent, i.e. reaction balance is maximizaiton of entropy. For the particular case of ln(p(ei)) = -ei/T, they are, but in general, they are not, we argue.

   As noted, reaction balance suggests that ei is the only information in the problem (as T is linked to Eave). Certainly, writing p(ei)p(ej) as the probability for an interaction to occur between ei and ej is not the most general expression. It is an expression based on the least amount of information, namely that if one has an ei and ej present, they will interact. This allows one to write ln(p(ei)) = -ei/T.  It is not just the case that only ei appears on the RHS, it is also the case that it only appears on the LHS, i.e. there is no other parameter. What would happen if another parameter existed, i.e there was extra information? We suggest that ln(p(ei)) may be generalized to F(p(ei,k)), where k is an extra parameter. F is the inverse of p, so in principle if p contains k, F may also contain k in such a way that F(p) removes p. If one uses F = ln(p), however, then ln does not constain k and hence cannot remove this extra parameter. This is the case for which there cannot be any more information than -ei/T. 

   The general process of maximization of entropy subject to constraints seems to be the following, we argue. One first approximates a physical system by average moments of ei, namely ei power 1 and ei power 0. This, however, gives no information on what distribution should appear because one may have many that satisfy Sum over i p(ei) = 1 and Sum over i ei p(ei) = Eave. In this approach there are two pieces of information, p(ei) and the moments used. One might suggest that one might generalize and write an average function  Sum over i  p(ei) F(p(ei)).

  This function must describe the system and so F(p(ei)) is a function of ei and should only represent the same moments as the constraints, i..e F(p(ei)) = a ei + b. F is the inverse of p(ei) and so something interesting occurs. One may have information beyond ei contained in F(p(ei)), i.e in F and p separately as this information may cancel out. If one wishes to minimize information, one may argue that d/dp (p F(p)) = d/dp (a1 ei p(ei) ) + a2 p(ei)). This leads to a second condition:   p dF/dp = 1. Again if an extra parameter appears in the form of a power law, then p dF/dp can cancel the effects of this extra parameter (extra information) and if F=1/k p power k, the p dF/dp shows no trace of the parameter. Thus, one may create a special function F(p) which has more information than simply ei information (unlike the Maxwell-Boltzmann case) and still describe a physical system in terms of average moments of ei. By writing p(ei) F(p(ei) and minimizing information, one may obtain explicit forms for two solutions of p(ei), namely the MB one and a power law (i.e. the Tsallis case). The Tsallis case, however, cannot be represented by reaction balance because p(ei) p(ej) = p(ek) p(el) ((1)) for ei+ej = ek+el assumes no other information than ei and this is not the case for a Tsallis power law which involves an extra piece of information, namely the power law parameter. One cannot have the power law parameter disappear from ((1)). Formally, if one considers ((1)) with p(ei,k), then ln(p(ei,k) = -ei/T and this means that k must disappear because it is not defined by this equation. Having any allowable k is the same as saying one may have many solutions p(ei,k) for k1,k2, k3 etc. This is no different than picking multiple solutions which satisfy Sum over i p(ei) =1 and Sum over i ei p(ei) = Eave.