Maxwell-Boltzmann Case and a Math Function S= Sum over i p(ei) f(p(ei)) Such That dE/dT =TdS/dT

The notion of maximizing the number of physical arrangements of N molecules with n(ei) having energy ei such that the total energy is E, leads to the mathematical maximization of: ln(N!/ Sum over i n(ei)!) -b Sum over i ei n(ei), where n(ei)=Np(ei). One only obtains the Maxwell-Boltzmann distribution p(ei) = C(T) exp(-ei/T) if one uses Stirling’s approximation, ln(n!) = nln(n). Nevertheless, the maximization of arrangements is based on physical reality. In (1), it is argued that one may obtain a distribution (including the Maxwell-Boltzmann one) from the first law of thermodynamics, without any notion of maximization, but a priori knowledge of the first law of thermodynamics. Here we consider Eave = Sum over i ei p(ei) and ask if one may find a mathematical function: S = -Sum over i p(ei) f(p(ei)) such that: dEave = T dS. This is a simplified form of the first law of thermodynamics, i.e. one with no work, but we consider the approach as a math based one. In other words, one could derive the first law of thermodynamics for cases of no work in such a manner. We show that this may be done if one considers p(ei) = C(T) p1(ei/T). In other words, there exists an unnormalized function p1(ei/T) which must contain an argument with no units as probability is simply a number. We then show that a particular math solution allows for S= - Sum over i p(ei) ln(p(ei)) such that dE - TdS = 0. This approach uses neither maximization nor knowledge of the first law of thermodynamics. It is simply a math approach which allows for the form dE = TdS with a specific math form for S as a function of p(ei) and p(ei)=C(T)exp(-ei/T). Given the math result, one might suggest that it is a microscopic representation of the first law of thermodynamics for no work, or given knowledge of p(ei) =C(T)exp(-ei/T) from MaxEnt or reaction balance, one might introduce dE=TdS as an equation describing changes due to heat (not work). In other words, MaxEnt or reaction balance leads to this simplified form of the first law of thermodynamics.