Maximization of Scattering Arrangements Equivalent to Reaction Balance?

n regular statistical mechanics, the idea of “maximizing” the number of physical arrangements of particles with different energies leads to both Shannon’s entropy and the canonical distribution (1). This “maximization of arrangements”, it seems, is treated as a fundamental idea of nature. In a series of notes, we argued that the equilibrium distribution for general cases as well as for the Maxwell-Boltzmann case may be obtained from a time reversal balance of elastic two body collisions. In a previous note (2), we argued one may map collisions (e.g. e1+e2=e3+e4) into M!/ (m1! m2! …) where m1 is the number of reactions of particles with energy m1. (For the Maxwell-Boltzmann case, then number of reactions is proportional to the number of particles, but this is not the general case.) If e1 and e2 appear next to each other in the factorial arrangement, they may react. In (2), however, we did not argue why one needs to “maximize” the scattering arrangements. Instead, we showed maximization leads to the same result as reaction balance. In this note, we try to investigate why maximization is the same reaction balance.