Comment on"The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy"by X. Gao, E. Gallicchio and A.E. Roitberg [J. Chem. Phys. 151, 034113 (2019)]

Content Provider	Semantic Scholar
Author	Gujrati, Purushottam D.
Copyright Year	2020
Abstract	The title of the paper leads to an incorrect conclusion as we show that the equilibrium result of the paper is a special limit of a general result for nonequilibrium systems in internal equilibrium already available in the literature. We also point out some of the limitations of the approach taken by the authors. Gao, Gallicchio and Roitberg (GGR) have suggested in their work [1] that the generalized Boltzmann distribution is the only distribution for which the GibbsShannon entropy S equals the equilibrium (EQ) thermodynamic entropy Seq. In this form, the result is not new as acknowledged by them for common EQ ensembles (NV T0, V μ0T0 and NP0T0) [2] that require n = 3 independent variables; the suffix 0 has been added to the fields as a reminder that they refer to the medium Σ̃, which is always in EQ. (This choice of notation will become useful below when we discuss nonequilibrium (NEQ) systems.) The generalization to arbitrary EQ ensembles (n > 3) is trivially done; see Guggenheim [3]. Therefore, the main contribution of GGR is their claim that the generalized Boltzmann distribution is the only distribution for which S equals Seq; they donot remark that Seq is defined up to a constant but not S. The Gibbs-Shannon entropy S = − ∑ kpk ln pk [4], where k indexes the microstates of the system, is commonly applied to NEQ states. Their claim, therefore, will most certainly force the reader to incorrectly conclude that S is not equal to the thermodynamic entropy S in a NEQ process where S is well defined as is easily verified for a NEQ ideal gas [4] discussed by Landau and Lifshitz [2]. They equate S with S [2, see Eq. (40.7)] as GGR do in their Postulate 2. As S satisfies the second law, they use the entropy maximization (akin to GGR using their Eq. (9) for EQ as Postulate 1; more on this later) to derive the equilibrium distribution (Conclusion in the GGR approach). Indeed, we have also used S = S to identify S in our work [5, 6]. By using entropy maximization, we then obtain the probability distribution (Conclusion) for a special class of NEQ macrostates said to be in internal equilibrium (IEQ); see below. Thus the result by GGR is a special limit of our more general result: the Gibbs-Shannon entropy S also equals the thermodynamic entropy S of NEQ systems that are in IEQ having a generalized Boltzmann distribution. Let X = (E, V, · · · ) denote the set of n (extensive) observables of the system. In EQ, Seq(X) is a state Electronic address: pdg@uakron.edu function of X in a state space SX. Away from EQ, S(X, t) < Seq(X) has an explicit time dependence and approaches Seq(X) from below as the system approaches EQ [5–7]. The existence of S(X, t) is justified by the law of increase of entropy as discussed elsewhere [6]. It is common to use internal variables [8–10] to justify this extra time dependence. Let ξ denote the set of internal variables needed to account for this t-dependence so that S(X, t) can be written as a unique state function S(Z) in an enlarged state space SZ, Z . = X ∪ ξ. Such a state in SZ is identified as an internal equilibrium state (IEQS) for which pk has a special form; see Eq. (3). States that are not in IEQ will have their entropy given by S(Z, t) < S(Z), and in time approaches S(Z) from below. For them, pk and S(Z, t) have explicit time dependence. Evidently, S(Z, t) must be maximized to yield S(Z), which then leads to Eq. (1). Thus, entropy maximizing is equivalent to Postulate 1 as asserted above. The Gibbs fundamental relation follows from S(Z)
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