Thermodynamic Inequalities Research Articles

Wasserstein distance in speed limit inequalities for Markov jump processes

The role of the Wasserstein distance in the thermodynamic speed limit inequalities for Markov jump processes is investigated. We elucidate the nature of the Wasserstein distance in the thermodynamic speed limit inequality from three different perspectives by resolving three remaining problems. First, we derive a unified speed limit inequality for a general weighted graph, which reproduces both the conventional speed limit inequality and the trade-off relation between current and entropy production as its special case. Second, we treat the setting where the tightest bound with the Wasserstein distance has not yet been obtained and investigate why such a bound is out of reach. Third, we compare the speed limit inequalities for Markov jump processes with the L1-Wasserstein distance and for overdamped Langevin systems with the L2-Wasserstein distance, and argue that these two have different origins, despite their apparent similarity.

Entropy Bounds for Rotating AdS Black Holes.

We propose novel thermodynamic inequalities that apply to stationary asymptotically anti-de Sitter (AdS) black holes. These inequalities incorporate the thermodynamic volume and refine the reverse isoperimetric inequality. To assess the validity of our conjectures, we apply them to a wide range of analytical black hole solutions, observing compelling evidence in their favor. Intriguingly, our findings indicate that these inequalities may also apply for black holes of nonspherical horizon topology, as we show their validity as well for thin asymptotically AdS black rings.

Fluctuation theorems and thermodynamic inequalities for nonequilibrium processes stopped at stochastic times.

We investigate the thermodynamics of general nonequilibrium processes stopped at stochastic times. We propose a systematic strategy for constructing fluctuation-theorem-like martingales for each thermodynamic functional, yielding a family of stopping-time fluctuation theorems. We derive second-law-like thermodynamic inequalities for the mean thermodynamic functional at stochastic stopping times, the bounds of which are even stronger than the thermodynamic inequalities resulting from the traditional fluctuation theorems when the stopping time is reduced to a deterministic one. Numerical verification is carried out for three well-known thermodynamic functionals, namely, entropy production, free energy dissipation, and dissipative work. These universal equalities and inequalities are valid for arbitrary stopping strategies, and thus provide a comprehensive framework with insights into the fundamental principles governing nonequilibrium systems.

Stability and dynamics of bubble comprising carbon dioxide and air

The stability and dynamics of a single partial-equilibrium bubble comprising carbon dioxide (CO2) gas and air were theoretically investigated. The partial equilibrium bubbles, which are typically generated by bubbling in CO2 aqueous solutions initiated by pre-existing fine air bubbles, were assumed to be in equilibrium with the surrounding liquids with respect to the pressure and chemical potential of CO2 but not with respect to the chemical potential of air. The stability of the bubble was formulated using thermodynamic inequalities and two types of potential energy that became minimum and maximum at stable and unstable bubble radii, respectively. The dynamics of the bubble were studied by numerically solving the time evolution equation for the bubble radius. In addition, the stability and dynamics of a single complete-equilibrium bubble comprising CO2 gas and air are discussed.

Modelling of Electro-Viscoelastic Materials through Rate Equations.

Models of dielectric solids subject to large deformations are established by following a thermodynamic approach. The models are quite general in that they account for viscoelastic properties and allow electric and thermal conduction. A preliminary analysis is devoted to the selection of fields for the polarization and the electric field; the appropriate fields are required to comply with the balance of angular momentum and to enjoy the Euclidean invariance. Next, the thermodynamic restrictions for the constitutive equations are investigated using a wide set of variables allowing for the joint properties of viscoelastic solids, electric and heat conductors, dielectrics with memory, and hysteretic ferroelectrics. Particular attention is devoted to models for soft ferroelectrics, such as BTS ceramics. The advantage of this approach is that a few constitutive parameters provide a good fit of material behaviour. A dependence on the gradient of the electric field is also considered. The generality and the accuracy of the models are improved by means of two features. The entropy production is regarded as a constitutive property per se, while the consequences of the thermodynamic inequalities are made explicit by means of representation formulae.

Generalized Clausius inequalities in a nonequilibrium cold-atom system

Thermodynamic inequalities, such as the Clausius inequality, characterize the direction of nonequilibrium processes. However, the latter result presupposes a system coupled to a heat bath that drives it to a thermal state. Far from equilibrium, the Clausius inequality can be generalized using information-theoretic quantities. For initially isolated systems that are moved from an equilibrium state by a dissipative heat exchange, the generalized Clausius inequality is predicted to be reversed. We here experimentally investigate the nonequilibrium thermodynamics of an initially isolated dilute gas of ultracold Cesium atoms that can be either thermalized or pushed out of equilibrium by means of laser cooling techniques. We determine in both cases the phase-space dynamics by tracing the evolution with position-resolved fluorescence imaging, from which we evaluate all relevant thermodynamic quantities. We confirm the validity of the generalized Clausius inequality for the first process and of the reversed generalized Clausius inequality for the second transformation.

Thermoimpulse as a true extensive measure of heat

The expediency of replacing the Clausius entropy with a more adequate concept of "thermal impulse" as a measure of the amount of disordered motion in the system is substantiated. It is shown that the thermal impulse also exists in nonequilibrium systems, where it can also both increase and decrease. This makes it possible to solve the problem of thermodynamic inequalities, to return the concept of force to thermodynamics, to unify the methods for finding heterogeneous forces, to propose simpler criteria for the equilibrium, evolution and involution of each degree of freedom of the system separately, to substantiate the unity of the laws of transformation of any form of energy, to eliminate the blatant contradiction of thermodynamics with the nature of biological and cosmological evolution, etc. Other advantages of the thermal impulse are also revealed, which facilitate the proof of its existence, its applicability to thermally inhomogeneous media, physical visibility, measurability, ease of eliminating a number of paralogisms of thermodynamics, etc.

Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states

We derive universal thermodynamic inequalities that bound from below the moments of first-passage times of stochastic currents in nonequilibrium stationary states of Markov jump processes in the limit where the two thresholds that define the first-passage problem are large. These inequalities describe a tradeoff between speed, uncertainty, and dissipation in nonequilibrium processes, which are quantified, respectively, with the moments of the first-passage times of stochastic currents, the splitting probability of the first-passage problem, and the mean entropy production rate. Near equilibrium, the inequalities imply that mean first-passage times are lower bounded by the Van't Hoff-Arrhenius law, whereas far from thermal equilibrium the bounds describe a universal speed limit for rate processes. When the current is proportional to the stochastic entropy production, then the bounds are equalities, a remarkable property that follows from the fact that the exponentiated negative entropy production is a martingale.

A time–energy uncertainty relation in chemical thermodynamics

An analogy between the thermodynamic inequalities presented by Nicholson et al. [Nat. Phys. 16, 1211 (2020)] and by Yoshimura and Ito [Phys. Rev. Res. 3, 013175 (2021)] is discussed. As a result, a time–energy uncertainty relation in chemical thermodynamics in terms of Gibbs free energy and chemical potential is derived. It is numerically demonstrated that the uncertainly relation holds in a model system of oscillatory Brusselator reactions. Our result bridges the thermodynamic time–information uncertainty relation and free energy evolution in chemical reactions.

Synthesis of Equilibrium and Non-Equilibrium Thermodynamics

The possibility of synthesis of equilibrium and non-equilibrium thermodynamics (thermostatics and thermokinetics) on classical principles as a deductive and phenomenological theory has been substantiated. This is achieved in the article on the basis of the law of conservation of energy due to its expression through the flows of the energy carrier and the introduction of the missing parameters of the nonequilibrium state of the system. The proposed way of constructing nonequilibrium thermodynamics as a consequence of energodynamics makes it possible to substantiate all the provisions of the theory of irreversible processes (TIP) and classical thermodynamics without involving hypotheses, postulates and considerations of a molecular-kinetic and statistical-mechanical nature. This approach prevents the occurrence of thermodynamic inequalities and opens up the possibility of generalizing TIP to the processes of useful energy conversion, primarily of interest to power engineers, technologists, biophysicists and astrophysicists.

Passivity Deformation Approach for the Thermodynamics of Isolated Quantum Setups

Recently implemented quantum devices such as quantum processors and quantum simulators combine highly complicated quantum dynamics with high-resolution measurements. We present a passivity deformation methodology that sets thermodynamic constraints on the evolution of such quantum devices. This framework enhances the thermodynamic predictive power by simultaneously resolving four of the cardinal deficiencies of the second law in microscopic setups: i) It yields tight bounds even when the environment is microscopic; ii) The ultra-cold catastrophe is resolved; iii) It enables to integrate conservation laws into thermodynamic inequalities for making them tighter; iv) it bounds observables that are not energy-based, and therefore do not appear in the second law of thermodynamics. Furthermore, this framework provides insights to non-thermal environments, correlated environments, and to coarse-graining in microscopic setups. Our findings can be explored and used in physical setups such as trapped ions, superconducting circuits, neutral atoms in optical lattices and more.

SOLVING THE PROBLEM OF THERMODYNAMIC INEQUALITIES

It is shown that the combined equation of the 1st and 2nd principles of classical thermodynamics does not transform into inequality in the case of irreversible processes, if the external energy exchange of the system is expressed in terms of energy carrier flows. This means that thermodynamic inequalities are generated by attempts to take into account the irreversibility of real (non-static) processes, without taking into account explicitly its reasons - the inhomogeneity of the system and the presence of internal sources not only for entropy, but also for other parameters.On this basis, exact expressions of heat and work in open nonequilibrium systems, as well as their dissipative function, are obtained. The physical meaning of entropy as a thermal impulse and the unprovability of the principle of its increase in the framework of equilibrium systems are revealed. Non-entropy criteria for evolution are proposed and the latter is shown to be incompatible not only with the second law of thermodynamics, but also with the laws of conservation of energy carriers. The elimination of thermodynamic inequalities opens up the possibility of applying the equations of thermodynamics, taking into account energy dissipation, to other fundamental disciplines.

Thermodynamics-based stability criteria for constitutive equations of isotropic hyperelastic solids

A generalized thermodynamic stability criterion for isotropic finite elastic solids is derived using the fundamental balance laws and field equations of continuum mechanics, which is then used to formulate constitutive inequalities for the polynomial form of hyperelastic constitutive equations. Individual thermodynamic constitutive inequalities (called T-C inequalities) are derived for the neo-Hookean, Mooney Rivlin, and three-parameter generalized Rivlin models under three pure homogeneous deformation modes, namely, uniaxial compression, uniaxial tension and shear (simple and pure), and are compared against two commonly used adscititious inequalities, the Baker-Ericksen (B-E) and E-inequalities. The range of stable model constants as defined by the T-C inequalities is represented by a region in an N-dimensional coordinate space (N is the total number of model constants), which is defined as the Region of Stability (ROS). It is shown that the ROS is a function of material deformation and evolves with the limiting strain, shrinking from an initially large region representing the necessary condition of thermodynamic stability to a converged region under infinite limiting strain that is equivalent to the ROS defined by the E-inequalities. By investigating the evolution of the ROS under different deformation modes, the implication of T-C inequalities on the selection of experimental routines and filtering of erroneous test data and model constants is discussed. It is also demonstrated that while the E-inequalities are over-restrictive for hyperelastic materials with small to moderate limiting strains, an observation supported by recent experimental evidence, the B-E inequalities are inaccurate under moderate to large limiting strain conditions. The applicability of the proposed mathematical framework to other hyperelastic strain energy density forms, such as exponential/logarithmic functions, is demonstrated by investigating the thermodynamic stability of the Fung-Demiray model. It is shown that the commonly assumed restriction that the Fung-Demiray model constants must be positive can be relaxed so that some typical material behaviors under small to moderate limiting strains can also be modeled.

Global Passivity in Microscopic Thermodynamics

The main thread that links classical thermodynamics and the thermodynamics of small quantum systems is the celebrated Clausius inequality form of the second law. However, its application to small quantum systems suffers from two cardinal problems: (i) The Clausius inequality does not hold when the system and environment are initially correlated - a commonly encountered scenario in microscopic setups. (ii) In some other cases, the Clausius inequality does not provide any useful information (e.g. in dephasing scenarios). We address these deficiencies by developing the notion of global passivity and employing it as a tool for deriving thermodynamic inequalities on observables. For initially uncorrelated thermal environments the global passivity framework recovers the Clausius inequality. More generally, global passivity provides an extension of the Clausius inequality that holds even in the presences of strong initial system-environment correlations. Crucially, the present framework provides additional thermodynamic bounds on expectation values. To illustrate the role of the additional bounds we use them to detect unaccounted heat leaks and weak feedback operations ("Maxwell's demons") that the Clausius inequality cannot detect. In addition, it is shown that global passivity can put practical upper and lower bounds on the buildup of system-environment correlation for dephasing interactions. Our findings are highly relevant for experiments in various systems such as ion traps, superconducting circuits, atoms in optical cavities and more.

Nonequilibrium thermodynamics and information theory: basic concepts and relaxing dynamics**This article belongs to the special issue: Emerging Talents, which features invited work from the best early-career researchers working within the scope of J. Phys. A. This project is part of the Journal of Physics series’ 50th anniversary celebrations in 2017. Bernhard Altaner was selected by the Editorial Board of J.

Thermodynamics is based on the notions of energy and entropy. While energy is the elementary quantity governing physical dynamics, entropy is the fundamental concept in information theory. In this work, starting from first principles, we give a detailed didactic account on the relations between energy and entropy and thus physics and information theory. We show that thermodynamic process inequalities, like the second law, are equivalent to the requirement that an effective description for physical dynamics is strongly relaxing. From the perspective of information theory, strongly relaxing dynamics govern the irreversible convergence of a statistical ensemble towards the maximally non-commital probability distribution that is compatible with thermodynamic equilibrium parameters. In particular, Markov processes that converge to a thermodynamic equilibrium state are strongly relaxing. Our framework generalizes previous results to arbitrary open and driven systems, yielding novel thermodynamic bounds for idealized and real processes.

Weibull thermodynamics: Subexponential decay in the energy spectrum of cosmic-ray nuclei

The spectral number density of cosmic-ray nuclei is shown to be a multiply broken power law with subexponential spectral cutoff. To this end, a spectral fit is performed to data sets covering the 1GeV−1011GeV interval of the all-particle cosmic-ray spectrum. The flux points of the ultra-high energy spectral tail measured with the Telescope Array indicate a Weibull cutoff exp(−(E∕(kBT))σ) and permit a precise determination of the cutoff temperature kBT=(2.5±0.1)×1010 GeV and the spectral index σ=0.66±0.02. Based on the spectral number density inferred from the least-squares fit, the thermodynamics of this stationary non-equilibrium system, a multi-component mixture of relativistic nuclei, is developed. The derivative of entropy with respect to internal energy defines the effective temperature of the nuclei, S,U=1∕Teff,kBTeff≈16.1 GeV, and the functional dependence between the cutoff temperature in the Weibull exponential and the effective gas temperature is determined. The equipartition ratio is found to be U∕(NkBTeff)≈0.30. The isochoric and isobaric heat capacities of the nuclear gas are calculated, as well as the isothermal and adiabatic compressibilities and the isobaric expansion coefficient, and it is shown that this non-equilibrated relativistic gas mixture satisfies the thermodynamic inequalities 0<CV<CP and 0<κS<κTeff.

Engines with ideal efficiency and nonzero power for sublinear transport laws

It is known that an engine with ideal efficiency ($\eta =1$ for a chemical engine and $e = e_{\rm Carnot}$ for a thermal one) has zero power because a reversible cycle takes an infinite time. However, at least from a theoretical point of view, it is possible to conceive (irreversible) engines with nonzero power that can reach ideal efficiency. Here this is achieved by replacing the usual linear transport law by a sublinear one and taking the step-function limit for the particle current (chemical engine) or heat current (thermal engine) versus the applied force. It is shown that in taking this limit exact thermodynamic inequalities relating the currents to the entropy production are not violated.

Elements of thermomechanics of plasticity under complex loading

The problem of the theory of elastic-plastic process is the establishment of criteria for loading and unloading. In this paper, the equations of the process describe the transition from Euclidean metric, normal for the source deviatoric space, to internal metrics, definitely related to the processes of deformations and loadings. This transition allows without additional assumptions to introduce so-called structural parameters characterizing the irreversible behavior of the material and to build a thermomechanics variant, providing a wide range of opportunities to determine all thermodynamic parameters of the model and formulation of the thermodynamic inequalities.

Entropy and its mathematical properties: consequences for thermodynamics

In this work, a comprehensive meaning for en- tropy is provided on the basis of foundations of information theory and statistical thermodynamics. For this purpose, the close relation between missing information and entropy is presented by emphasizing their probabilistic nature. Fur- thermore, the physical implications of the mathematical properties of the entropy function are exploited using the elementary notions of differential and integral calculus. Particularly, it is evidenced that the usual thermodynamic inequalities found in many textbooks of physical chemistry are direct consequences of the concavity of entropy. The aim of this work is to show that many concepts presented in textbooks of physical chemistry can be obtained in a simple and mathematically clear way.

About Conditions of Equilibrium and Stability of Thermodynamic Systems

The clear statement of a problem on balance and stability of thermodynamic systems is given. In these frameworks a number of important thermodynamic inequalities, a condition of phase equilibrium process is received. The critical phenomena are considered