The definition of the thermodynamic entropy in statistical mechanics

Abstract

A definition of the thermodynamic entropy based on the time-dependent probability distribution of the macroscopic variables is developed. When a constraint in a composite system is released, the probability distribution for the new equilibrium values goes to a narrow peak. Defining the entropy by the logarithm of the probability distribution automatically makes it a maximum at the equilibrium values, so it satisfies the Second Law. It is also satisfies the postulates of thermodynamics. Objections to this definition by Dieks and Peters are discussed and resolved.

Probability, Entropy, and Gibbs’ Paradox(es)

“…First, the domain of thermodynamics must be specified, then the role of limited resolution, and the purpose of the theory. The postulates of thermodynamics provide a convenient list of properties that the entropy must satisfy [ 8 , 9 , 10 , 11 , 12 ].…”

“…In the spirit of Boltzmann [ 4 , 5 ], we can define the configurational component of the total entropy of the M systems as the logarithm of the probability distribution in Equation ( 1 ) [ 12 , 13 , 20 , 25 , 26 , 27 , 28 , 29 ]. where is Boltzmann’s constant (first introduced by Planck [ 16 , 30 ]), and X is an arbitrary constant.…”

“…When I first wrote about this way of defining the thermodynamic entropy, I only used exchanges between two systems to illustrate the idea [ 25 , 26 , 27 , 28 ]. Objections to this approach and its extension to many systems were raised [ 31 , 32 , 33 ] and answered [ 12 , 29 ]. For completeness, I have included a sketch of the original argument in Appendix A [ 25 ].…”

“…The value of X does not affect any physical prediction of the theory, which is perhaps made more obvious by emphasizing that all possible thermodynamic systems are involved in the derivation, and M is an enormous (and unknown) number [ 12 ].…”

“…Since , , and X are all constants, the last three terms in Equation ( 6 ) do not play any role in thermodynamic predictions and may be ignored [ 12 ].…”