Boltzmann and statistical mechanics

摘要

The beauty of the 11-theorem was that it derived in one swoop both aspects of the Second Law: first the (irreversible) approach to thermal equilibrium and then, from the value of the H-function in equilibrium, the connection between the H-function and Clausius' entropy S: H = -const. S + const. Only later forced by Loschmidt's Reversibility Paradox [8] and Zermelo's Recurrence Paradox [9], as the Ehrenfests were to call them [10], did Boltzmann clearly state the probabilistic nature of the Stosszahl Ansatz, viz. that the Stosszahl Ansatz and the H-theorem only held for disordered states of the gas and that these states were much more probable than the ordered ones, since the number of the first far exceeded that of the second. In the paper itself, however, this is never mentioned; it is as if the Ansatz was self evident. Therefore, Boltzmann did not derive here the Second Law purely from mechanics alone either and till the present day, no mechanical derivation of the Boltzmann equation exists, although the Stosszahl Ansatz must ultimately be derivable from the mechanics of a very large number N of particles, i.e., from "large N-dynamics".