Maxwell-Boltzmann Maximum Entropy as a Guarantor of Stability and Order?

Maximum entropy is traditionally considered a signature of maximum disorder within a system with the word “disorder” having apparently a negative connotation. The idea of maximum disorder seems to follow from the following information theory argument (1). Consider a set of probabilities { p(i) }. For a very large number of events N, event i should appear Np(i). Thus the overall product probability is: P= Product over i p(i) [to the power Np(i)]. This may be written as exp(Sum over i N p(i) ln(p(i))). The probability may be written as 1/ number of arrangements thus maximizing - Sum over i p(i)ln(p(i)) maximizes the number of arrangements i.e maximizes disorder. In (2) we showed that one may obtain Shannon’s entropy by seeking a vector dot product S= Sum over i p(i) G(p(i)) such that dS = Sum over i G(p(i)) dp(i). G(p(i)) then becomes ln(p(i)). The benefit of this is that G(p(i)) remains constant for small dp(i) fluctuations. In fact, for the Maxwell-Boltzmann case it may be compared to an average conserved quantity which in general follows the form: Aave = Sum over i A(i) p(i). dAave = 0 (conservation) = Sum over i A(i) dp(i). This has the same form as S so in some cases, such as the Maxwell-Boltzmann (MB) one may link the two directly i.e. A(i) = ln(p(i)) + C. Given that Aave cannot change under changes dp(i), it is stable. If S= - Sum over i p(i) ln(p(i)) also does not change with changing dp(i) (holding T temperature constant and box size as well with A(i) being energy ei) then it too must be stable. In the MB case A(i)=ei i.e. energy of state i. Thus it is stability which we argue is the key idea to entropy and Shannon’s entropy form which under dp(i) mimics a conserved average quantity which is stable i.e. remains unchanged. One does not need to think of disorder or large N in such an approach. Entropy (which mathematically does follow from a maximization of Shannon’s entropy subject to a constraint of a conserved quantity) is equivalent up to a constant to the conserved quantity at least in the MB case. The fact that there is maximum disorder shows we argue that this allows for stability i.e. allows for a kind of restoring mechanism if one changes probabilities i.e. dp(i). This restoring force seems to be linked to two body scattering which follows from ln(prelative(i)) = -ei/T