Thermodynamics with generalized ensembles: The class of dual orthodes

We address the problem of the foundation of generalized ensembles in statistical physics. The approach is based on Boltzmann's concept of orthodes. These are the statistical ensembles that satisfy the heat theorem, according to which the heat exchanged divided by the temperature is an exact differential. This approach can be seen as a mechanical approach alternative to the well established information-theoretic one based on the maximization of generalized information entropy. Our starting point are the Tsallis ensembles which have been previously proved to be orthodes, and have been proved to interpolate between canonical and microcanonical ensembles. Here we shall see that the Tsallis ensembles belong to a wider class of orthodes that include the most diverse types of ensembles. All such ensembles admit both a microcanonical-like parametrization (via the energy), and a canonical-like one (via the parameter $\beta$). For this reason we name them ``dual''. One central result used to build the theory is a generalized equipartition theorem. The theory is illustrated with a few examples and the equivalence of all the dual orthodes is discussed.Comment: 20 pages, 4 figures. Minor improvement

Quantum Fluctuation Relations for Ensembles of Wave Functions

New quantum fluctuation relations are presented. In contrast with the the standard approach, where the initial state of the driven system is described by the (micro)canonical density matrix, here we assume that it is described by a (micro)canonical distribution of wave functions, as originally proposed by Schr\"odinger. While the standard fluctuation relations are based on von Neumann measurement postulate, these new fluctuation relations do not involve any quantum collapse, but involve instead a notion of work as the change in expectation of the Hamiltonian.Comment: 12 pages, 1 figure. Added illustrative example in v2. Accepted for publication in New Journal of Physic

Complementary expressions for the entropy-from-work theorem

We establish an expression of the entropy-from-work theorem that is complementary to the one originally proposed in [P. Talkner, P. Hanggi and M. Morillo, arXiv:0707.2307]. In the original expression the final energy is fixed whereas in the present expression the initial energy is fixed.Comment: 2 Page

On the Limiting Cases of Nonextensive Thermostatistics

We investigate the limiting cases of Tsallis statistics. The viewpoint adopted is not the standard information-theoretic one, where one derives the distribution from a given measure of information. Instead the mechanical approach recently proposed in [M. Campisi, G.B. Bagci, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.09.081], is adopted, where the distribution is given and one looks for the associated physical entropy. We show that, not only the canonical ensemble is recovered in the limit of $q$ tending to one, as one expects, but also the microcanonical ensemble is recovered in the limit of $q$ tending to minus infinity. The physical entropy associated with Tsallis ensemble recovers the microcanonical entropy as well and we note that the microcanonical equipartition theorem is recovered too. We are so led to interpret the extensivity parameter q as a measure of the thermal bath heat capacity: $q=1$ (i.e. canonical) corresponds to an infinite bath (thermalised case, temperature is fixed), $q=-\infty$ (microcanonical) corresponds to a bath with null heat capacity (isolated case, energy is fixed), intermediate $q's$ (i.e. Tsallis) correspond to the realistic cases of finite heat capacity (both temperature and energy fluctuate).Comment: 5 pages, 2 figure

Comment on "Experimental Verification of a Jarzynski-Related Information-Theoretic Equality by a Single Trapped Ion" PRL 120 010601 (2018)

The target paper presents an experimental verification of a "Jarzynski-related" equality. We show that the latter equality is in fact not related to the Jarzynski equality.Comment: 1 pag