Abstract
We show that, to generate the statistical operator appropriate for a given system, and as an alternative to Jaynes’ MaxEnt approach, that refers to the entropy S, one can use instead the increments ±S in S. To such an effect, one uses the macroscopic thermodynamic relation that links ±S to changes in i) the internal energy E and ii) the remaining M relevant extensive quantities Ai, i = 1; : : : ;M; that characterize the context one is working with.
Highlights
We wish to address an issue belonging to the foundations of statistical mechanics (SM) by revisiting the role of the entropy S in Jaynes’ SM-formulation [1, 2], based upon the MaxEnt axiom: entropy is to be extremized
As our goal, that one can give Eq (3) the status of an axiom of statistical mechanics! We introduce first a set of new extensive quantities Aν, appropriately related to the Rν and postulate for statistical mechanics that (Axiom (1), the incremental entropy postulate) dE = T dS +
If we expand the resulting equation up to first order in the dpi it is immediately found that the following set of equations ensues [11, 12] (remember that the Lagrange multipliers are identical to minus the generalized pressures Pν of Eq (3)) : (1)
Summary
2. There exist particular states of a system, called the equilibrium ones, that are uniquely determined by E and a set of, say M , extensive (macroscopic) parameters Rν. We introduce first a set of new extensive quantities Aν , appropriately related (see below) to the Rν and postulate for statistical mechanics that (Axiom (1), the incremental entropy postulate) dE = T dS +. The minimum amount of microscopic information that we would have still to add to our axiomatics in order to get all the results of equilibrium statistical mechanics is precisely such relation At this point we will merely conjecture that the following statements might suffice: Axiom (2). One may recognize that Axiom (2) is just a form of Boltzmann’s “atomic” conjecture, pure and simple: macroscopic quantities are statistical averages evaluated using a microscopic probability distribution [5]. The probability variations dpi in turn generate corresponding changes dS, dAν , and dE in, respectively, S, the Aν , and E
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