The basic idea of a microscopic understanding of Thermodynamics is to derive its main features from a microscopic probability distribution. In such a vein, we investigate the thermal statistics of quasi-probabilities's semi-classical analogs in phase space for the important case of quadratic Hamiltonians, focusing attention in the three more important instances, i.e., those of Wigner, $P$-, and Husimi distributions. Introduction of an effective temperature permits one to obtain a unified thermodynamic description that encompasses and unifies the three different quasi-probability distributions. This unified description turns out to be classical.
Systems with interacting degrees of freedom play a prominent role in stochastic thermodynamics. Our aim is to use the concept of detached path probabilities and detached entropy production for bipartite Markov processes and elaborate on a series of special cases including measurement-feedback systems, sensors and hidden Markov models. For these special cases we show that fluctuation theorems involving the detached entropy production recover known results which have been obtained separately before. Additionally, we show that the fluctuation relation for the detached entropy production can be us…
For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defini…
By considering an explicit nonequilibrium model, we analyze the statistics of the irreversible work, $w_{\rm irr}$, and irreversible entropy production, $Δ_i s$, within the stochastic energetics framework. Restating the second law of thermodynamics as a function of $w_{\rm irr}$, we introduce the explicit probability of violating the canonical form of that second law for a different set of parameters and initial conditions of the model. Moreover, we study the irreversible entropy production along the same lines, since it can be cast as a generalization of the irreversible work. From an informa…
We discuss peculiar aspects of the first law of thermodynamics for systems characterized by the presence of meta-equilibrium quasi-stationary states for which the pertinent phase/configuration spaces is generally inhomogeneous. As a consequence, the naive additivity requirement for thermodynamic quantities ceases to be satisfied.
We discuss the role of contextuality within quantum fluctuation theorems, in the light of a recent no-go result by Perarnau \emph{et al}. We show that any fluctuation theorem reproducing the two-point-measurement scheme for classical states either admits a notion of work quasi-probability or fails to describe protocols exhibiting contextuality. Conversely, we describe a protocol that smoothly interpolates between the two-point measurement work distribution for projective measurements and Allahverdyan's work quasi-probability for weak measurements, and show that the negativity of the latter is …
Clausius introduced, in the 1860s, a thermodynamical quantity which he named {\it entropy} $S$. This thermodynamically crucial quantity was proposed to be {\it extensive}, i.e., in contemporary terms, $S(N) \propto N$ in the thermodynamic limit $N \to\infty$. A decade later, Boltzmann proposed a functional form for this quantity which connects $S$ with the occurrence probabilities of the microscopic configurations (referred to as {\it complexions} at that time) of the system. This functional is, if written in modern words referring to a system with $W$ possible discrete states, $S_{BG}=-k_B \s…
The general formalism for the nonextensive statistics based on the Landsberg-Vedral entropy was derived. The formula for the first law of thermodynamics and the exact relations of the thermodynamic quantities to their ensemble averages were obtained. It was found that under the transformation $q\to 2-q$ the probabilities of microstates of the nonextensive statistics based on the Landsberg-Vedral entropy formally resemble the corresponding probabilities of the Tsallis statistics with escort probabilities. However, the nonextensive statistics with the Landsberg-Vedral entropy does not require in…
We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities's (QP) in phase space for the important case of quadratic Hamiltonians. We consider the three more important OPs: 1) Wigner's, $P$-, and Husimi's. We show that, for all of them, the ensuing semiclassical entropy is a function {\it only} of the fluctuation product $Δx Δp$. We ascertain that {\it the semi-classical analog of the $P$-distribution} seems to become un-physical at very low temperatures. The behavior of several other information quantifiers reconfirms such an assertion in manifold ways. We …
For small thermodynamic systems in contact with a heat bath, we determine the free energy by imposing the following two conditions. First, the quasi-static work in any configuration change is equal to the free energy difference. Second, the temperature dependence of the free energy satisfies the Gibbs-Helmholtz relation. We find that these prerequisites uniquely lead to the free energy of a classical system consisting of $N$-interacting identical particles, up to an additive constant proportional to $N$. The free energy thus determined contains the Gibbs factorial $N!$ in addition to the phase…
We determine the quasistationary distribution of Floquet-state occupation probabilities for a parametrically driven harmonic oscillator coupled to a thermal bath. Since the system exhibits detailed balance, and the canonical representatives of its quasienergies are equidistant, these probabilities are given by a geometrical Boltzmann distribution, but its quasitemperature differs from the actual temperature of the bath, being affected by the functional form of the latter's spectral density. We provide two examples of quasithermal engineering, i.e., of deliberate manipulation of the quasistatio…
Low-temperature-differential (LTD) Stirling heat engines are able to operate with a small temperature difference between low-temperature heat reservoirs that exist in our daily lives, and thus they are considered to be an important sustainable energy technology. The author recently proposed a nonlinear dynamics model of an LTD kinematic Stirling heat engine to study the rotational mechanism of the engine [Y. Izumida, EPL \textbf{121}, 50004 (2018)]. This paper presents our study of the nonequilibrium thermodynamics analysis of this engine model, where a load torque against which the engine doe…
It is now well established that there is no lower bound for the energy dissipated during a computation. The relevance of the zero-energy limit is unclear, however, because it entails computations that are unreliable or infinitely slow, or both. In contrast the thermodynamic costs of deterministic computations that complete in finite time is less well understood. We propose a model of universal computation based on Markovian stochastic processes. While strictly deterministic computation is not possible in such systems, we show that the scaling of the energy consumption in relation to the accura…
The thermodynamic properties of superconducting electrons are usually studied by means of the quasi-particles distribution; but in this approach, the ground state energy and the dependence of the chemical potential on the electron density cannot be determined. In order to solve these problems, the thermodynamic potentials are derived by means of the Bogoliubov-Valatin formalism. The thermodynamic potentials can be obtained by computing the free energy of a gas of quasi-particles, whose energy spectrum is conditional on the gap function. However, the nontrivial dependence of the gap on the temp…
Motivated by black holes thermodynamics, we consider the zeroth law of thermodynamics for systems whose entropy is a quasi-homogeneous function of the extensive variables. We show that the generalized Gibbs-Duhem identity and the Maxwell construction for phase coexistence based on the standard zeroth law are incompatible in this case. We argue that the generalized Gibbs-Duhem identity suggests a revision of the zeroth law which in turns permits to reconsider Maxwell's construction in analogy with the standard case. The physical feasibility of our proposal is considered in the particular case o…
Using the quasi-equilibrium Helmholtz energy (qHE), defined as the thermodynamic work in a quasi-static process, we investigate the thermal properties of both an isothermal process and a transition process between the adiabatic and isothermal states (adiabatic transition). Here, the work is defined by the change in energy from a steady-state to another state under a time-dependent perturbation. In particular, the work for a quasi-static change is regarded as thermodynamic work. We employ a system--bath model that involves time-dependent perturbations in both the system and the system--bath int…
For soft matter to form quasicrystals an important ingredient is to have two characteristic lengthscales in the interparticle interactions. To be more precise, for stable quasicrystals, periodic modulations of the local density distribution with two particular wavenumbers should be favored, and the ratio of these wavenumbers should be close to certain special values. So, for simple models, the answer to the title question is that only these two ingredients are needed. However, for more realistic models, where in principle all wavenumbers can be involved, other wavenumbers are also important, s…
