Classical Non-equilibrium Thermodynamics Research Articles

Analytical Thermodynamics

This paper proposes a theory that bridges classical analytical mechanics and nonequilibrium thermodynamics. Its intent is to derive the evolution equations of a system from a stationarity principle for a suitably augmented Lagrangian action. This aim is attained for homogeneous systems, described by a finite number of state variables depending on time only. In particular, it is shown that away from equilibrium free energy and entropy are independent constitutive functions.

К построению релятивистских гидродинамических уравнений в искривленном пространстве-времени, записанных в абстрактных геометрических обозначениях

This paper presents a macroscopic derivation of the system of equations of relativistic dissipative hydrodynamics in an abstract geometrical form, which does not depend on the chosen space-time coordinates and reduces to equations valid in both flat and curved space-time. The above procedure, serving as a link between the non-gravitational laws of physics and gravitation, is based on Einstein's equivalence principle which guarantees that the laws of physics written in abstract geometrical representation have the same form both in flat and curved spacetime. A modern formulation of relativistic hydrodynamics based on both classical nonequilibrium thermodynamics (first-order theory) and expanded irreversible thermodynamics (second-order theory) is proposed. The theory presented has both purely conceptual and applied significance. In particular, this theory has applications in such important fields of knowledge as nuclear physics, astrophysics and cosmology. For example, in viscous cosmological models, bulk viscosity acts as a cause of dissipation, which has a significant impact on the processes in the Universe.

Non-equilibrium thermodynamics of quantum bipartite system

In quantum information and quantum computation, a bipartite system provides a basic few-body framework for investigating significant properties of thermodynamics and statistical mechanics. A Hamiltonian model for a bipartite system is introduced to analyze the important role of interaction between bipartite subsystems in quantum non-equilibrium thermodynamics. We illustrate discrimination between such quantum thermodynamics and classical few-body non-equilibrium thermodynamics. By proposing a detailed balance condition of the bipartite system, we generally investigate the properties of the entropy and heat of our model, as well as the relation between them.

A thermodynamics-based hyperelastic-plastic coupled model unified for unbonded and bonded soils

A hyperelastic-plastic coupled constitutive model unified for bonded and unbonded soils is developed in this paper based on thermodynamics. An elastic potential function applicable for different kinds of soils is proposed to derive a hyperelastic model accounting for the pressure- and -density dependency, the stress-induced anisotropy and the bonding effects as well as their couplings with plasticity. From the perspective of elastic stability, state boundary and failure surfaces of different soils can be naturally predicted by the hyperelasticity without any additional definitions and parameters. Based on the classical nonequilibrium thermodynamics, novel plastic constitutive relations are derived and naturally coupled with the hyperelasticity. As a result, elasto-plastic coupling features such as the dissipative history effect on elastic stiffness, the cyclic shear behavior, the degradation of shear modulus under small strain conditions, the stress-induced anisotropy of plastic behavior and the cohesion degradation can be reproduced. The model is well validated by predicting the undrained/drained monotonic and cyclic shear behavior of unbonded and bonded sands, providing useful insights into their critical state behavior, irreversible shear-dilation/contraction and effects of bonding and cohesion degradation. It is also shown that the cohesion degradation in different shearing stages to a large extent determines both the monotonic and cyclic behavior of bonded soils.

On the Fokker–Planck approach to the stochastic thermodynamics of reactive systems

Stochastic thermodynamics is an extension of classical nonequilibrium thermodynamics that aims at including the effect of fluctuations occurring at small scales. In many cases, the starting point of such approaches is a Fokker–Planck equation for the probability density associated to the underlying stochastic variables. It is known that for chemically reactive systems, the usual equation for entropy production derived in this framework is not consistent with its macroscopic counterpart. It moreover also differs from the expression obtained with the master equation, from which the aforementioned Fokker–Planck equation derives. In the present work, we obtain a Fokker–Planck based alternative formula for entropy production in reactive systems that converges to the correct expression in the thermodynamic limit and is moreover consistent with expressions obtained with the master-equation based approach.

New look at an old principle: an alternative formulation of the theorem of minimum entropy production

We formulate a direct generalization of the Prigogine’s principle of minimum entropy production, according to a new isoperimetric variation principle by classical non-equilibrium thermodynamics. We focus our attention on the possible mathematical forms of constitutive equations. Our results show that the Onsager’s reciprocity relations are consequences of the suggested variation principle. Furthermore, we show by the example of the thermo-diffusion such reciprocity relations for diffusion tensor, which are missing in Onsager’s theory. Our theorem applied to the non-linear constitutive equations indicates the existence of dissipation potential. We study the forms of general reciprocity with the dissipation potential. This consideration results in a weaker condition than Li-Gyarmati-Rysselberhe reciprocity has. Furthermore, in the case of electric conductivity in the magnetic field, our theorem shows the correct dependence of the Onsager’s kinetic coefficient by the axial vector of magnetic induction. We show in general that the evolution criterion of the global entropy production is a Lyapunov-function, and so the final stationer state is independent of the initial, time-independent boundary conditions.

A coupled model of diffusional creep of polycrystalline solids based on climb of dislocations at grain boundaries

A continuum theory based on thermodynamics has been developed for modeling diffusional creep of polycrystalline solids. It consists of a coupled problem of vacancy diffusion and mechanics where the vacancy generation/absorption at grain boundaries is driven by grain boundary dislocations climb. The model is stated in terms of general balance laws and completed by the choice of constitutive equations consistent with classical non-equilibrium thermodynamics. The kinetics of diffusional creep is derived from physically-based mechanisms of climb of dislocations at grain boundaries, thus introducing a dependence of diffusional creep on the density and mobility of boundary dislocations. Several representative examples have been solved using the finite element method and assuming representative volume elements made up of an array of regular-shaped crystals. The effect of stress, temperature, grain size, and grain boundary dislocation mobility is analyzed and compared with classical theories of diffusional creep. The simulation results demonstrate the ability of the present model to reproduce the macroscopic stress and grain size dependence observed under both diffusion and interface controlled regimes, as well as the evolution of this dependency with the temperature. In addition, the numerical implementation of the model allows to predict the evolution of microscopic fields through the microstructure.

Stochastic thermodynamics based on an Einstein-Boltzmann definition of fluctuating entropy.

Stochastic thermodynamics is an extension of classical nonequilibrium thermodynamics to small systems, where fluctuations are expected to play an important role. A central difficulty met when developing such an extension is how to define a nonequilibrium fluctuating entropy. Typically, the expression used is based on Gibbs' formula for entropy at equilibrium. In this work, we show that one can construct an alternative framework for stochastic thermodynamics based on an extension of Einstein's formula connecting the probability of fluctuations and entropy around equilibrium states. We compare the two approaches and discuss, in particular, how they lead to different interpretations of what a stochastic entropy and entropy production represent.

Research in entropy wonterland: A review of the entropy concept

The entropy concept was introduced in the mid-nineteenth century by Clausius and has been continually enriched, developed, and interpreted by researchers in many scientific disciplines. The use of entropy in a wide range of fields has led to inconsistencies in its application and interpretation, as summarized by von Neuman ?No one knows what entropy really is.? To resolve this problem, thermodynamics and other scientific disciplines face several crucial questions concerning the entropy concept: (1) What is the physical meaning of entropy? (2) Is entropy a subjective or an objective property? (3) How to apply entropy to living organisms? To answer these questions, this paper describes the roots, the conceptual history of this important concept, as well as the path of development and application in various scientific disciplines, including classical thermodynamics, non-equilibrium thermodynamics, statistical mechanics, and life sciences. <br><br><font color="red"><b> This article has been retracted. Link to the retraction <u><a href="http://dx.doi.org/10.2298/TSCI180730231E">10.2298/TSCI180730231E</a><u></b></font>

Nonequilibrium Thermodynamics and Scale Invariance

A variant of continuous nonequilibrium thermodynamic theory based on the postulate of the scale invariance of the local relation between generalized fluxes and forces is proposed here. This single postulate replaces the assumptions on local equilibrium and on the known relation between thermodynamic fluxes and forces, which are widely used in classical nonequilibrium thermodynamics. It is shown here that such a modification not only makes it possible to deductively obtain the main results of classical linear nonequilibrium thermodynamics, but also provides evidence for a number of statements for a nonlinear case (the maximum entropy production principle, the macroscopic reversibility principle, and generalized reciprocity relations) that are under discussion in the literature.

Local equations of state in nonequilibrium heterogeneous physicochemical systems

Equations describing local thermal and caloric equations of state in heterogeneous systems at any degree of their states’ deviation from equilibrium are derived. The state of a system is described by equations of the transfer of mixture components; these generalize the equations of classical non-equilibrium thermodynamics for strongly nonequilibrium processes. The contributions from reactions and external fields are taken into account. The equations are derived using the lattice gas model with discrete molecular distributions in space (on a scale comparable to molecular dimensions) and continuous molecular distributions (at short distances inside cells) during their translational and vibrational motions. For simplicity, it is assumed that distinctions between the sizes of mixture components are small. Contributions from potential functions of intermolecular interaction (of the Lennard-Jones type) to some coordination spheres are considered. The theory provides a unified description of the dynamics of distributions of concentrations and pair functions of mixture components in three aggregate states, and at their interfaces. Universal expressions for the local components of the pressure tensor and internal energy inside multicomponent bulk phases and at their interfaces are obtained. Local components of the pressure tensor and the internal energy are universally expressed through local unary and pair distribution functions (DFs) in any nonequilibrium state. The time evolution of the unary and pair DFs themselves is determined from the derived system of equations of mass, momentum, and energy transfer that ensure the transition of the system from a strongly nonequilibrium state to both the local equilibrium state described within traditional nonequilibrium thermodynamics and the complete thermodynamic equilibrium state postulated by classical thermodynamics.

Coherent Application of a Contact Structure to Formulate Classical Non-Equilibrium Thermodynamics

This contribution presents an outline of a new mathematical formulation for Classical Non-Equilibrium Thermodynamics (CNET) based on a contact structure in differential geometry. First a non-equilibrium state space is introduced as the third key element besides the first and second law of thermodynamics. This state space provides the mathematical structure to generalize the Gibbs fundamental relation to non-equilibrium thermodynamics. A unique formulation for the second law of thermodynamics is postulated and it showed how the complying concept for non-equilibrium entropy is retrieved. The foundation of this formulation is a physical quantity, which is in non-equilibrium thermodynamics nowhere equal to zero. This is another perspective compared to the inequality, which is used in most other formulations in the literature. Based on this mathematical framework, it is proven that the thermodynamic potential is defined by the Gibbs free energy. The set of conjugated coordinates in the mathematical structure for the Gibbs fundamental relation will be identified for single component, closed systems. Only in the final section of this contribution will the equilibrium constraint be introduced and applied to obtain some familiar formulations for classical (equilibrium) thermodynamics.

Thermodynamic restrictions on linear reversible and irreversible thermo-electro-magneto-mechanical processes.

A unified thermodynamic framework for the characterization of functional materials is developed. This framework encompasses linear reversible and irreversible processes with thermal, electrical, magnetic, and/or mechanical effects coupled. The comprehensive framework combines the principles of classical equilibrium and non-equilibrium thermodynamics with electrodynamics of continua in the infinitesimal strain regime.In the first part of this paper, linear Thermo-Electro-Magneto-Mechanical (TEMM) quasistatic processes are characterized. Thermodynamic stability conditions are further imposed on the linear constitutive model and restrictions on the corresponding material constants are derived. The framework is then extended to irreversible transport phenomena including thermoelectric, thermomagnetic and the state-of-the-art spintronic and spin caloritronic effects. Using Onsager's reciprocity relationships and the dissipation inequality, restrictions on the kinetic coefficients corresponding to charge, heat and spin transport processes are derived. All the constitutive models are accompanied by multiphysics interaction diagrams that highlight the various processes that can be characterized using this framework.

Molecular kinetic theory of strongly nonequilibrium processes of mass, momentum, and energy transfer: Local equilibrium criteria

Consequences of the complete system of transfer equations of the properties (momentum, energy, and mass) of particles and their pairs are considered under local equilibrium conditions with regard to the Bogoliubov hierarchy of relaxation times between the first and second distribution functions (DFs) and distinctions in the characteristic relaxation times of particle momentum, energy, and mass. It is found that even under the local equilibrium condition in the Bogoliubov hierarchy of relaxation times between the first and second DFs, pair correlations are maintained between all dynamic variables (velocity, temperature, and density) whose values are proportional to the gradients of transferable properties. A criterion is introduced requiring there be no local equilibrium condition upon reaching the critical value at which the description of the transfer process becomes incorrect in classical nonequilibrium thermodynamics. External forces are considered in the equations for strongly nonequilibrium processes. Along with allowing for intermolecular potentials, it becomes possible to discuss the concept of passive forces (introduced in thermodynamics by Gibbs) from the standpoint of the kinetic theory. It is shown that use of this concept does not reflect modern representations of real processes.

Modelling of thermal diffusion and thermodynamic restrictions

Upon a review of the balance equations for a mixture, diffusion fluxes are shown to satisfy an evolution equation which, with slight differences, describes diffusion relative to a constituent or relative to the barycentric reference. As a consequence, the particular case of binary mixtures is considered with the view of establishing a connection with the current models. The fact that the diffusion flux is governed by a balance (evolution) equation is quite unusual in the literature since customary models are based on constitutive equations (Fick’s law, Soret and Dufour models). This in turn motivates the analysis for a more appropriate thermodynamic setting. The second law for the whole mixture, along with some constitutive assumptions in classical (nonequilibrium) thermodynamics, is shown to provide the constitutive equations which are currently applied in the literature. Instead, the model of diffusion flux through a balance equation proves to be fully consistent with rational thermodynamics and the assumption that the functions, relative to a single constituent, depend on variables pertaining to that constituent whereas interaction terms are allowed to depend also on quantities pertaining to the other constituents.

Effective stress in saturated soil: a granular solid hydrodynamics approach

Effective stress is one of the most fundamental concepts in soil mechanics. A comprehensive interpretation of its physical essence is important for a better understanding and prediction of the mechanical behaviour of the soil. The granular solid hydrodynamics approach is used in this study to establish a more generalised effective stress principle for saturated soils. In this approach, the stress formulation is theoretically derived from the constraints between conservation laws and thermodynamic principles, in which the energies in soils are quantitatively described by the classical non-equilibrium thermodynamics. Five physical mechanisms of the soil deformation, including the elastic potential, the elastic relaxation, the granular fluctuation, the viscosity and the effect of bound water, are taken into account in the derivation of the stress formulation. Consequently, the effective stress derived in this study is classified into four categories. The first is elastic effective stress that is determined by the elastic potential energy and is limited by a stress state boundary line. The second is granular effective stress that is relevant to the granular fluctuation and is very important to the unrecoverable soil deformation. The third is bound effective stress which is a result of defining the bound water as an independent phase of the soil system. It is an important driving force in the non-isothermal unrecoverable soil deformation induced by the conversion of bound water to free water. The last—viscous stress—is another important part of effective stress.

Contact Geometry of Mesoscopic Thermodynamics and Dynamics

The time evolution during which macroscopic systems reach thermodynamic equilibrium states proceeds as a continuous sequence of contact structure preserving transformations maximizing the entropy. This viewpoint of mesoscopic thermodynamics and dynamics provides a unified setting for the classical equilibrium and nonequilibrium thermodynamics, kinetic theory, and statistical mechanics. One of the illustrations presented in the paper is a new version of extended nonequilibrium thermodynamics with fluxes as extra state variables.

Fully coupled modeling of magneto-mechanical hysteresis through ‘thermodynamic’ compatibility

The paper proposes a phenomenological model for magneto-elastic interactions in materials with hysteresis, where both mechanical and magnetic variables are fully coupled. The approach allows one to provide a tool for the description of the energy conversion mechanism in the presence of losses, for example, in harvesting devices. The present approach takes into account exchange between mechanical and magnetic energies, as well as dissipation due to hysteresis phenomena. Using the concept of hysteresis potentials, it is shown that the model is consistent with classical nonequilibrium thermodynamics. The effectiveness of the approach is demonstrated by comparison with experimental data.

The modification of entropy production by heat condution in non-equilibrium thermodynamics

The entropy production is expressed as the product of the generalized force (driving force) and generalized flux, which plays a central role in classical non-equilibrium thermodynamics. This expression has shortcomings in two aspects: first, the decomposition into generalized fluxes and forces is arbitrary to some extent; more importantly, the entropy production is negative value calculated in heat wave propagation, which breaks the second law. In this paper, we carry out analyses based on the thermomass theory and show that the entropy production is induced by the dissipation of thermomass energy during heat condution. The generalized force of entropy production is not driving force but resistive force, having a unit of force in Newton’s mechanics. The modified expression for entropy production not only guarantees its positiveness in propagation of heat waves consistent with the extended irreversible thermodynamics, but also avoids the arbitrariness of decomposition.

Interface model for non-equilibrium evaporation

A microscopic interface condition for condensing/evaporating interfaces is developed by combining a velocity dependent condensation probability [T. Tsuruta, H. Tanaka, T. Masuoka, Int. J. Heat Mass Transfer 42 (1999) 4107] and Maxwell type interface conditions with accommodation. Using methods from kinetic theory, macroscopic interface conditions for mass and energy transport across the phase boundary are derived. This model only applies to simple substances, where diffusive effects in the bulk phases are not present. The results are compared to classical non-equilibrium thermodynamics. The interface conditions are considered for the limit of small deviation from equilibrium, and the corresponding Onsager coefficients are computed. These results are useful as boundary conditions for non-equilibrium evaporation and condensation problems, as done previously by our group [M. Bond, H. Struchtrup, Phys. Rev. E 70 (2004) 061605].