Curado \textit{et al.} [Ann. Phys. \textbf{366} (2016) 22] have recently studied the axiomatic structure and the universality of a three-parameter trace-form entropy inspired by the group-theoretical structure. In this work, we study the group-theoretical entropy $S_{a,b,r}$ in the context of the third law of thermodynamics where the parameters $\left\lbrace a,b,r \right\rbrace $ are all independent. We show that this three-parameter entropy expression can simultaneously satisfy the third law of thermodynamics and the three Khinchin axioms, namely continuity, concavity and expansibility only when the parameter $b$ is set to zero. In other words, it is thermodynamically valid only as a two-parameter generalization $S_{a,r}$. Moreover, the restriction set by the third law i.e., the condition $b = 0$, is important in the sense that the so obtained two-parameter group-theoretical entropy becomes extensive only when this condition is met. We also illustrate the interval of validity of the third law using the one-dimensional Ising model with no external field. Finally, we show that the $S_{a,r}$ is in the same universality class as that of the Kaniadakis entropy for $0 < r < 1$ while it has a distinct universality class in the interval $-1 < r < 0$.
We show that the generalized Boltzmann distribution is the only distribution for which the Gibbs-Shannon entropy equals the thermodynamic entropy. This result means that the thermodynamic entropy and the Gibbs-Shannon entropy are not generally equal, but rather than the equality holds only in the special case where a system is in equilibrium with a reservoir.
The quantum thermodynamic functions of a harmonic oscillator coupled to a heat bath through velocity-dependent coupling are obtained analytically. It is shown that both the free energy and the entropy decay fast with the temperature in relation to that of the usual coupling from. This implies that the velocity-dependent coupling helps to ensure the third law of thermodynamics.
Thermodynamic entropy is determined by a heat measurement through the Clausius equality. The entropy then formalizes a fundamental limitation of operations by the second law of thermodynamics. The entropy is also expressed as the Shannon entropy of the microscopic degrees of freedom. Whenever an extension of thermodynamic entropy is attempted, we must pay special attention to how its three different aspects just mentioned are altered. In this paper, we discuss possible extensions of the thermodynamic entropy.
It is shown that the laws of thermodynamics are extremely robust under generalizations of the form of entropy. Using the Bregman-type relative entropy, the Clausius inequality is proved to be always valid. This implies that thermodynamics is highly universal and does not rule out consistent generalization of the maximum entropy method.
The consequences of the opening of a phason gap in incommensurate systems are studied on a simple model, the discrete frustrated $ϕ^4$-model. Analytical considerations and numerical results show that there is a very weak phase transition that can be characterized as third order.
Recently, there has been a considerable progress on the issue of the thermodynamic second law, which is known as the law of entropy increase or irreversibility. In particular, a novel symmetry known as the Gallavotti-Cohen symmetry is found in nonequilibrium (NEQ) fluctuations, which leads to so-called fluctuation theorems. The thermodynamic second law is a simple corollary of fluctuation theorems, from which one can predict quantitatively how often NEQ processes violate the law of entropy increase. Violations disappear in the thermodynamic limit, but can be observed reasonably well in small s…
Bento \textit{et al.} [Phys. Rev. E 91, 022105 (2015)] state that the Tsallis entropy violates the third law of thermodynamics for $q \leq 0$ and $0<q<1$. We show that their results are valid only for $q \geq 1$, since there is no distribution maximizing the Tsallis entropy for the intervals $q \leq 0$ and $0<q<1$ compatible with the system energy expression.
We study the thermodynamic entropy as a Noether invariant in a stochastic process. Following the Onsager theory, we consider the Langevin equation for a thermodynamic variable in a thermally isolated system. By analyzing the Martin-Siggia-Rose-Janssen-de Dominicis action of the Langevin equation, we find that this action possesses a continuous symmetry in quasi-static processes, which leads to the thermodynamic entropy as the Noether invariant for the symmetry.
The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large, but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out …
We propose to construct Markov dynamics with specified characteristics using a cycle decomposition of the space of Markov chains introduced by Cohen and Alpern. As an application of this approach, we derive the minimal entropy production required to generate prescribed thermodynamic currents.
We have made a simple and natural modification of a recent quantum refrigerator model presented by Cleuren et al. in Phys. Rev, Lett.108, 120603 (2012). The original model consist of two metal leads acting as heat baths, and a set of quantum dots that allow for electron transport between the baths. It was shown to violate the dynamic third law of thermodynamics (the unattainability principle, which states that cooling to absolute zero in finite time is impossible), but by taking into consideration the finite energy level spacing in metals we restore the third law, while keeping all of the orig…
The Third Law of Thermodynamics constrains the phase diagram of{systems with a first-order quantum phase transition. For zero conjugate field, the coexistence curve has an infinite slope at T=0. If a tricritical point exists at T>0, then the associated tricritical wings are perpendicular to the T=0 plane, but not to the zero-field plane. These results are based on the third law and basic thermodynamics only, and are completely general. As an explicit example we consider the ferromagnetic quantum phase transition in clean metals, where a first-order quantum phase transition is commonly observed…
We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a the…
The irreversible currents and entropy production rate of a dilute colloidal suspension are calculated using the linear irreversible thermodynamics and the linear response theory. The \anomalous" or \hidden" entropy recently discussed in the context of the stochastic thermodynamics is fully accounted in these classic frameworks. We show that the two distinct formulations lead to identical results as long as the local equilibrium assumption or, equivalently the linear response theory, is valid.
The concepts of quantity of heat and work are deduced in the non-extensive statistical mechanics context, following steps in parallel to those employed in the extensive statistical mechanics.
The transfer entropy is a well-established measure of information flow, which quantifies directed influence between two stochastic time series and has been shown to be useful in a variety fields of science. Here we introduce the transfer entropy of the backward time series called the backward transfer entropy, and show that the backward transfer entropy quantifies how far it is from dynamics to a hidden Markov model. Furthermore, we discuss physical interpretations of the backward transfer entropy in completely different settings of thermodynamics for information processing and the gambling wi…
Bento \textit{et al.} [Phys. Rev. E 91, 022105 (2015)] recently stated that the Tsallis entropy violates the third law of thermodynamics for $0 < q <1$ in the sub-additive regime. We first show that the division between the regimes $q < 1$ and $q > 1$ is already inherent in the fundamental incomplete structure of the deformed logarithms and exponentials underlying the Tsallis entropy. Then, we provide the complete deformed functions and show that the Tsallis entropy conforms to the third law of thermodynamics for both super-additive $q < 1$ and sub-additive $q > 1$ regimes. Finally, we remark …
Macroscopic nonextensive thermodynamics is studied without recourse to microscopic statistical mechanics. It is shown that if entropy is nonextensive, the concept of physical temperature introduced through the generalized zeroth law of thermodynamics necessarily leads to modifications of the first law of thermodynamics and some of thermodynamic relations including Clausius definition of thermodynamic entropy. It is also shown, by applying this generalized Clausius entropy to a composite nonextensive system, how the nonextensive entropy and the quantity of heat consistently behave in an arbitra…
The thermodynamic stability condition (TSC) of Tsallis' entropy is revisited. As Ramshaw [Phys. Lett. A {\bf 198} (1995) 119] has already pointed out, the concavity of Tsallis' entropy with respect to the internal energy is not sufficient to guarantee thermodynamic stability for all values of $q$ due to the non-additivity of Tsallis' entropy. Taking account of the non-additivity the differential form of the TSC for Tsallis entropy is explicitly derived. It is shown that the resultant TSC for Tsallis' entropy is equivalent to the positivity of the standard specific heat. These results are consi…
A definition of the thermodynamic entropy based on the time-dependent probability distribution of the macroscopic variables is developed. When a constraint in a composite system is released, the probability distribution for the new equilibrium values goes to a narrow peak. Defining the entropy by the logarithm of the probability distribution automatically makes it a maximum at the equilibrium values, so it satisfies the Second Law. It is also satisfies the postulates of thermodynamics. Objections to this definition by Dieks and Peters are discussed and resolved.
The concept of negative temperature has recently received renewed interest in the context of debates about the correct definition of the thermodynamic entropy in statistical mechanics. Several researchers have identified the thermodynamic entropy exclusively with the "volume entropy'' suggested by Gibbs, and have further concluded that by this definition, negative temperatures violate the principles of thermodynamics. We disagree with these conclusions. We demonstrate that volume entropy is inconsistent with the postulates of thermodynamics for systems with non-monotonic energy densities, whil…
We present a pedagogical review of the fundamental concepts in thermodynamics of information, by focusing on the second law of thermodynamics and the entropy production. Especially, we discuss the relationship among thermodynamic reversibility, logical reversibility, and heat emission in the context of the Landauer principle and clarify that these three concepts are fundamentally distinct to each other. We also discuss thermodynamics of measurement and feedback control by Maxwell's demon. We clarify that the demon and the second law are indeed consistent in the measurement and the feedback pro…
We analyze the relation between the nonequilibrium Lagrange multipliers used in information theory and those used in Liu's technique to exploit the entropy inequality. We observe that there is not a complete equivalence between Liu's technique and information theory.
We numerically determine the entropy for heat-conducting states, which is connected to the so-called excess heat considered as a basic quantity for steady-state thermodynamics in nonequilibrium. We adopt an efficient method to estimate the entropy from the bare heat current and find that the obtained entropy agrees with the familiar local equilibrium hypothesis well. Our method possesses a wider applicability than local equilibrium and opens a possibility to compare thermodynamic properties of complex systems with those in the local equilibrium. We further investigate the entropy for heat-cond…
A question that is currently highly debated is whether the microcanonical entropy should be expressed as the logarithm of the phase volume (volume entropy, also known as the Gibbs entropy) or as the logarithm of the density of states (surface entropy, also known as the Boltzmann entropy). Rather than postulating them and investigating the consequence of each definition, as is customary, here we adopt a bottom-up approach and construct the entropy expression within the microcanonical formalism upon two fundamental thermodynamic pillars: (i) The second law of thermodynamics as formulated for qua…
To the student of thermodynamics the most difficult subject is entropy. In this paper we examine the actual, practical application of entropy to two simple systems, the homogeneous slab with fixed boundary values of the temperature, and an isolated atmosphere in the presence of the static gravitational field. The first gives valuable insight into the nature of entropy that is subsequently applied to the second system. It is a basic tenet of thermodynamics that the equilibrium of an extended, homogeneous and isolated system is characterized by a uniform temperature distribution and it is a stro…
To investigate the consequences of component confinement such as at a glass transition and the well-known energy or enthalpy gap (between the glass and the perfect crystal at absolute zero, see text), we follow our previous approach [Phys. Rev. E 81, 051130 (2010)] of using the second law applied to an isolated system Σ_0 consisting of the homogeneous system Σ and the medium Σ. We establish on general grounds the continuity of the Gibbs free energy G(t) of Σ as a function of time at fixed temperature and pressure of the medium. It immediately follows from this and the observed continuity of th…
We introduce axiomatically a complete thermodynamic formalism for a single macromolecule, either with or without detailed balance, in an isothermal ambient fluid based on its stochastic dynamics. With detailed balance, the novel theory yields mesoscopic, nonequilibrium generalizations for entropy and free energy of the macromolecule. Mesoscopic energy and entropy fluctuate. Expectation of entropy production equals zero if and only if the macromolecule is at thermal equilibrium, in which we show entropy still fluctuates but free energy is a constant. The entropy fluctuation a la L.D. Landau, pr…
We revisit the concavity property of the thermodynamic entropy in order to formulate a general proof of the minimum energy principle as well as of other equivalent extremum principles that are valid for thermodynamic potentials and corresponding Massieu functions under different constraints. The current derivation aims at providing a coherent formal framework for such principles which may be also pedagogically useful as it fully exploits and highlights the equivalence between different schemes. We also elucidate the consequences of the extremum principles for the general shape of thermodynamic…
An analysis of the thermodynamic behavior of quantum systems can be performed from a geometrical perspective investigating the structure of the state space. We have developed such an analysis for nonextensive thermostatistical frameworks, making use of the q-divergence derived from Tsallis' entropy. Generalized expressions for operator variance and covariance are considered, in terms of which the fundamental tensor is given.
Within the nonextensive framework, it is shown that zeroth law of thermodynamics determines not only the mapping between Lagrange multipliers and intensive variables, but also the mapping between nonextensive and extensive entropy. The form of constraints decides the form of the extensive entropy, standard averages lead to Boltzmann-Shannon-Gibbs entropy while normalised biased averages lead to Renyi entropy. The mapping between Lagrange multipliers and intensive variables is also discussed in the more general context of composable entropy.
We develop the stochastic approach to thermodynamics based on the stochastic dynamics, which can be discrete (master equation) continuous (Fokker-Planck equation), and on two assumptions concerning entropy. The first is the definition of entropy itself and the second, the definition of entropy production rate which is nonnegative and vanishes in thermodynamic equilibrium. Based on these assumptions we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium, and how the macroscopic laws are derived from the stochastic dynamics. These studies inc…
We study the nonextensive thermodynamics for open systems. On the basis of the maximum entropy principle, the dual power-law q-distribution functions are re-deduced by using the dual particle number definitions and assuming that the chemical potential is constant in the two sets of parallel formalisms, where the fundamental thermodynamic equations with dual interpretations of thermodynamic quantities are derived for the open systems. By introducing parallel structures of Legendre transformations, other thermodynamic equations with dual interpretations of quantities are also deduced in the open…
We consider the meta-equilibrium state of a composite system made up of independent subsystems satisfying the additive form of external constraints, as recently discussed by Abe [Phys. Rev. E {\bf 63}, 061105 (2001)]. We derive the additive entropy $S$ underlying a composable entropy $\tilde{S}$ by identifying the common intensive variable. The simplest form of composable entropy satisfies Tsallis-type nonadditivity and the most general composable form is interpreted as a monotonically increasing funtion $H$ of this simplest form. This is consistent with the observation that the meta-equilibri…
The thermodynamics expected of systems undergoing third order phase transition has been investigated by identifying the orders through the analytic continuation of the functional of the free energy, using Ehrenfest thermodynamic theory. We developed the Euler - Lagrange equations for the order parameter and the vector potential and solved them for the first time using well - known mathematical formulations.
The quantum thermodynamic property of the fractional damping system is investigated extensively. A fractional power-law decaying entropy function is revealed which presents another evidence for the validity of the third law of thermodynamics in the quantum dissipative region. Several non-trivial characters are excavated such as that the entropy varies from a non-linear diverging function to a semi-linear decaying function of the fractional exponent as the temperature tends to absolute zero.
Conventional wisdom holds that the von Neumann entropy corresponds to thermodynamic entropy, but Hemmo and Shenker (2006) have recently argued against this view by attacking von Neumann (1955) and his argument. I argue that Hemmo and Shenker's arguments fail due to several misunderstandings: about statistical-mechanical and thermodynamic domains of applicability, about the nature of mixed states, and about the role of approximations in physics. As a result, their arguments fail in all cases: in the single-particle case, the finite particles case, and the infinite particles case.
Thermodynamics is usually developed starting from entropy and the maximum entropy principle. We investigate here to what extent one can replace entropy with relative entropy which has several advantages, for example in the context of local quantum field theory. We find that the principle of maximum entropy can be replaced by a principle of minimum expected relative entropy. Various ensembles and their thermodynamic potentials can be defined through relative entropy. We also show that thermal fluctuations are in fact governed by a relative entropy. Furthermore we reformulate the third law of th…
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermodynamic entropy f…
The first in a long series of papers by John T. Lewis, G. W. Ford and the present author, considered the problem of the most general coupling of a quantum particle to a linear passive heat bath, in the course of which they derived an exact formula for the free energy of an oscillator coupled to a heat bath in thermal equilibrium at temperature T. This formula, and its later extension to three dimensions to incorporate a magnetic field, has proved to be invaluable in analyzing problems in quantum thermodynamics. Here, we address the question raised in our title viz. Nernst's third law of thermo…
It is argued that a typical many body energy eigenstate has a well defined thermodynamic entropy and that individual eigenstates possess thermodynamic characteristics analogous to those of generic isolated systems. We examine large systems with eigenstate energies equivalent to finite temperatures. When quasi-static evolution of a system is adiabatic (in the quantum mechanical sense), two coupled subsystems can transfer heat from one subsystem to another yet remain in an energy eigenstate. To explicitly construct the entropy from the wave function, degrees of freedom are divided into two unequ…
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 and the Shannon entropy, or its generalisation to quantum systems, the von Neumann entropy. More generally, we find that …
Similarity between quantum mechanics and thermodynamics is discussed. It is found that if the Clausius equality is imposed on the Shannon entropy and the analogue of the heat quantity, then the value of the Shannon entropy comes to formally coincide with that of the von Neumann entropy of the canonical density matrix, and pure-state quantum mechanics apparently transmutes into quantum thermodynamics. The corresponding quantum Carnot cycle of a simple two-state model of a particle confined in a one-dimensional infinite potential well is studied, and its efficiency is shown to be identical to th…
Copula Entropy (CE) is a recently introduced concept for measuring correlation/dependence in information theory. In this paper, the theory of CE is introduced and the thermodynamic interpretation of CE is presented with N-particle correlated systems in equilibrium states.
We provide a quantum statistical basis for (a)a characterisation of a complete set of thermodynamic variables and (b) the differentiability of the entropy function of these variables
The configurational entropy of ice is calculated by thermodynamic integration from high to low temperatures. We use Monte Carlo simulations with a simple energy model which reproduces the Bernal-Fowler ice rules. This procedure is found to be precise enough to give reliable values for the residual entropy s_th of different ice phases in the thermodynamic limit. First, we check it for a two-dimensional ice model. Second, we calculate s_th for ice Ih, and compare our result with those previously given in the literature. Third, we obtain s_th for ice VI, for which we find a value clearly higher t…
We present our numerical study on quantum thermodynamics of the resonant level model subjected to non-equilibrium condition as well as external driving. Following our previous work on non-equilibrium quantum thermodynamics (Phys. Rev. B 101, 184304 [2020]), we expand the density operator into a series of power in the driving speed, where we can determine the non-adiabatic thermodynamic quantities. Particularly, we calculate the non-equilibrium entropy production rate as well as higher order non-adiabatic corrections to the energy and/or population. In the limit of weak system-bath coupling, ou…
