By Jacques Yvon
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DymÀ2,dt, as explained below in Eq. 1).  The ratios la/lc and lb/lc cannot depend on the variables cited in ; otherwise sc would also depend on them, in contradiction to . ,ymÀ2; otherwise, la in Eq. 2) would have to depend on these same coordinates and in such a manner as to cancel out from la/lc, for consistency with assertions  and . However, la cannot possibly be allowed to depend on the y coordinates of system B. ,xnÀ2. ,ymÀ2. For, according to , these variables are missing from lc, whence the ratios la/lc and lb/lc must also be free of these variables, to remain consistent with [1–3].
1) into the exact differential for the function of state ds. However, the physical results of interest to experimentalists tend to be hidden behind a thicket of relatively sophisticated mathematical manipulations. For that reason, the Carathe´odory’s theory has been relegated to Chapter 9. You are urged to consult that exposition for a more rigorous approach to the subject. Meanwhile, let us examine the consequences of adopting Eq. 1). In the present context, the second law statement amounts to an affirmation that the requirements of Carathe´odory’s theory are met only by a study of reversible processes.
For a brief review of the fundamental concepts, see Chapter 10 and consult texts on statistical thermodynamics for an in-depth exposition. ). Quantum mechanics furnishes the latter information in terms of the elementary statistics to which the particles are subject, and via the energy levels εi and their associated degeneracies gi among which the particles are distributed. Each such distribution corresponds to a possible macroscopic manifestation of the system. At equilibrium, it is the most probable distribution that prevails overwhelmingly as compared to any other configuration, and that corresponds to the maximum entropy of the macroscopic system.