The Physical Basis of the Gibbs-von Neumann entropy

Statistical thermodynamics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental question, \textit{what is thermodynamics?}, has remained unanswered. We answer this question here. Generalized statistical thermodynamics is a variational calculus of probability distributions. It is independent of physical hypotheses but provides the means to incorporate our knowledge, assumptions and...

In this paper we discuss about the validity of the Shannon entropy functional in connection with the correct Gibbs-Hertz probability distribution function. We show that there is no contradiction in using the Shannon-Gibbs functional and restate the validity of information theory applied to equilibrium statistical mechanics. We show that under these assumptions, entropy is always a monotone function of energy, irrespective to the shape of the density of states, leading always ...

We prove that information-theoretic maximum entropy (MaxEnt) approach to canonical ensemble is mathematically equivalent to the classic approach of Boltzmann, Gibbs and Darwin-Fowler. The two approaches, however, "interpret" a same mathematical theorem differently; most notably observing mean-energy in the former and energy conservation in the latter. However, applying the same MaxEnt method to grand canonical ensemble fails; while carefully following the classic approach bas...

I expound and defend the ``bare probabilism'' reading of Gibbsian (i.e. mainstream) statistical mechanics, responding to Frigg and Werndl's recent (BJPS 72 (2021), 105-129) plea: ``can somebody please say what Gibbsian statistical mechanics says?''

We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as $S_d=-\sum_n \rho_{nn}\ln \rho_{nn}$ with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy $S_n=-{\rm Tr}\, \rho\ln\rho$. However, in contrast to $S_n$, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of ...

Thermodynamic entropy, as defined by Clausius, characterizes macroscopic observations of a system based on phenomenological quantities such as temperature and heat. In contrast, information-theoretic entropy, introduced by Shannon, is a measure of uncertainty. In this Letter, we connect these two notions of entropy, using an axiomatic framework for thermodynamics [Lieb, Yngvason, Proc. Roy. Soc.(2013)]. In particular, we obtain a direct relation between the Clausius entropy a...

The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here, we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focussing specifically on whether they satisfy or violate the zeroth, first and second law of thermodyn...

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed number degrees of freedom $N$. The functional form of group entropies is restricted (though not uniquely determined) by assum...

In order to undestand how the complex concept of entropy emerged,we propose a trip towards the past reviewing the works of Clausius, Boltzmann, Gibbs and Planck. In particular, since the Gibbs's work is not very well known, we present a detailed analysis, recalling the three definitions of the entropy that Gibbs gives. May be one of the most important aspect of the entropy is to see it as a thermodynamic potential like the other thermodynamic potentials as proposed by Callen....

Very recently, the validity of the concept of negative temperature has been challenged by several authors since they consider Boltzmann's entropy (that allows negative temperatures) inconsistent from a mathematical and statistical point of view, whereas they consider Gibbs' entropy (that does not admit negative temperatures) the correct definition for microcanonical entropy. In the present paper we prove that for systems with equivalence of the statistical ensembles Boltzma...