Showing papers on "Non-equilibrium thermodynamics published in 1978"

Time, structure, and fluctuations.

Abstract: Fundamental conceptual problems that arise from the macroscopic and microscopic aspects of the second law of thermodynamics are considered. It is shown that nonequilibrium may become a source of order and that irreversible processes may lead to a new type of dynamic states of matter called "dissipative structures." The thermodynamic theory of such structures is outlined. A microscopic definition of irreversible processes is given, and a transformation theory is developed that allows one to introduce nonunitary equations of motion that explicitly display irreversibility and approach to thermodynamic equilibrium. The work of the group at the University of Brussels in these fields is briefly reviewed. In this new development of theoretical chemistry and physics, it is likely that thermodynamic concepts will play an ever-increasing role.

Statistical mechanics of a nonlinear stochastic model

Abstract: A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.

The Second Law of Thermodynamics and Cyclic Processes

Abstract: : This paper is concerned with the implications of an inequality proposed recently by Green and Naghdi in regard to the classical statement of the second law of thermodynamics associated with cyclic thermo-mechanical processes. The results obtained are compared with corresponding previous developments by Fosdick and Serrin and by Truesdell who employ the Clausius-Duhem inequality.

Thermodynamics of the Yang-Mills gas

Abstract: The contribution of nonlinear fluctuations (instantons) to the thermodynamics of the Yang-Mills gas at high temperature is estimated.

Thermodynamics at nonequilibrium steady states

Abstract: This work continues a discussion of how to extend equilibrium thermodynamics to nonequilibrium steady states. The extension is based on molecular fluctuations of the extensive variables and gives rise to a state function, called the σ function. The σ function reduces to the entropy at equilibrium and can be constructed from a knowledge of the local equilibrium entropy and the molecular fluctuations. The σ function depends on all the variables characterizing a steady state, including fluxes of the extensive variables and reservoir parameters. The theoretical analysis of nonequilibrium fluctuations predicts that the σ function is related to stability and the kinetics around a steady state just like the entropy is at equilibrium. Calculations of the σ function are outlined for several multicomponent systems of experimental interest. Using the fluctuation–dissipation theory, a generalization of the Clausius inequality is obtained. This leads to a class of extremum principles at steady state for Legendre transformations of the σ function.

The Poisson Representation. II. Two-Time Correlation Functions

Abstract: Basic formulas for the two-time correlation functions are derived using the Poisson representation method The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed

A model of heat conduction

Abstract: We construct a model of a chain of atoms coupled at its ends to two reservoirs at different temperatures. In a weak coupling limit the atoms obey a stochastic evolution law and have an equilibrium state with a uniform temperature gradient along the chain.

Mass and momentum transport in dilute reacting gases

Abstract: A formal kinetic theory and illustrative model calculations are presented for a mixture of four species coupled together by the two chemical reactions A+BC?AB+C. The cross sections for reaction are not restricted to be small and so the reactive collisions can not be treated as if they were small perturbations, but must be dealt with on an equal par with nonreactive events. To describe the dynamics of the reactive collisions we use an extension of the DIPR (direct interaction, product repulsion) model which takes into account rotational–translational exchanges of energy and, in a cruder way, vibrational excitation as well. Because the dynamics of these reactive events are not derived from a Hamiltonian, care must be exercised to insure that the condition of microreversibility is satisfied. A moment method is used to extract from the kinetic equation estimates for the nonequilibrium corrections to the reaction rates and for the changes of viscosity and diffusion coefficients due to the chemical reaction. Calculations reveal that the reaction rates can differ by as much as 80% from estimates based upon the assumption of an equilibrium distribution over reactant states. Furthermore, reaction can alter the coefficients of viscosity and diffusion by 10% or more. It is shown that the Onsager relations for the diffusion coefficients are valid only when chemical equilibrium prevails.

The Concepts of Thermodynamics

Abstract: Elementary thermodynamics has been much neglected as an object of inquiry both by mathematicians and engineers. The basic concepts of heat and hotness can, however, be given a precise structure consistent with the foundations of classical continuum mechanics, and it is possible to develop a rigorous notion of absolute temperature and a rigorous discussion of the Clausius inequality. In this view entropy is a derived concept, and the existence of internal energy and entropy must be established for any given class of materials. To complete the fundamental notions of thermodynamics one must in addition consider the problem of stability of rest states, and the question of internally constrained materials. To illustrate the problems involved in the foundations of the thermodynamics we give examples of a material obeying the Clausius inequality which does not have a unique entropy function, and of an unstable material satisfying the Clausius-Duhem inequality which is cooled by the addition of heat. We also examine the thermodynamic structure of an internally constrained material whose density is a function of temperature, and finally show how a heat-conducting compressible Navier-Stokes fluid can be introduced as a thermodynamic entity without using either the notion of internal energy or of entropy.

Dynamics of the Open BCS Model

Abstract: The dynamics of the strong coupling BCS model, considered as an open system interacting with a thermal bath, is solved rigorously and explicitly in the weak coupling limit and in the infinite-volume limit. The BCS system goes from the normal phase to the ordered phase by bifurcation. Fluctuations around trajectories of intensive observables are Gaussian and Markovian. Thermodynamic phases are global attractors in the physical domain. Structural stability is discussed. The model provides an example of a nonequilibrium statistical mechanical system with phase transition whose irreversible macroscopic dynamics can be calculated exactly from the underlying Hamiltonian quantum mechanics.

Ion-Coupled Transport across Biological Membranes

Abstract: Only a rudimentary understanding of the principles of classical thermodynamics is needed to appreciate that the flow of an uncharged substance from a region of lower concentration to a region of higher concentration or the flow of a charged substance from a region of lower electrochemical potential to a region of higher electrochemical potential cannot take place unless the processes responsible for these flows are linked or coupled to a supply of energy. Such flows, loosely referred to as “active” or “uphill,” are commonplace in biological systems and the thrust of many investigations is to identify the immediately responsible source(s) of energy. The problem can perhaps be best stated in terms of the formalism of irreversible or nonequilibrium thermodynamics.(1) Thus, in a system in which there is only a single flow of a substance i we may write the straight phenomenologic coefficient relating the flow to the conjugate force and has units of conductance; R ii relates the force to the flow and has units of resistance. Clearly, in this simple system R ii = 1/L ii . Ohm’s law of current flow, Fick’s first law of diffusion, Fourier’s law of heat flow, and Poiseuille’s law of volume flow are but a few familiar examples of Eq. (1).

Nonequilibrium Thermodynamics of the Nonlinear Equations of Chemical Kinetics

Abstract: The well known rate equations of chemical kinetics are used in order to check the validity of two commonly postulated extensions of the Onsager reciprocal relations into the nonlinear domain. It is shown that these two nonlinear extensions are invalid, but that there exists a generalized dissipation potential which uniquely determines the dissipative aspects of the problem. Introduction The Onsager or, more generally, the Onsager-Casimir relations have now become an accepted and well-understood part of the science of nonequilibrium thermodynamics; their consistency with experimental results has recently been carefully evaluated by D. G. Miller [ 1 ], and there is very little doubt that they represent a proven (and useful) law of thermodynamics in all cases when linear relations between the dissipative fluxes Ja and their conjugate generalized forces X<* may be assumed. On the other hand, particularly in reaction kinetics and in the study of dissipation in solid continua, linear force-flux relations prove to be inadequate and the question arises whether an extension of the Onsager reciprocal relations into the non-linear domain can be discovered. At the present time it is possible to distinguish two proposals for such a possible extension. In each case an essential aspect of the linear theory is formulated in general terms and it is postulated that it continues to be valid in the extended domain. The first proposal seems to have appeared in a number of publications and cannot with certainty be attributed to a particular author. Its essence is to postulate that the matrix of derivatives of the fluxes with respect to the generalized forces is symmetric. This is a \"direct\" generalization of the relations first derived by L. Onsager [2] which is, evidently, equivalent to assume that the fluxes are derivable from a J. Non-Equilib. Thermodyn., Vol. 3, 1978, No. 3 154 J. Bataille, D. G. B. Edelen, J. Kestin potential function of dissipation (as distinct from the dissipation function, which is the product of the entropy production by temperature). The second proposal was made by H. Ziegler [3, 4] who postulated that the nvector of fluxes, Ja, is orthogonal to the surface of constant entropy production. This postulate has been inspired by the fact that it is a rigorous consequence of the theory based on linear, phenomenological force-flux relations. Furthermore, the postulate is reminiscent of the orthogonality principle which followed from D. C. Drucker's [5] stability postulate in plasticity theory. The two proposals are not independent of each other because they can be reduced to the statement that the linear differential form 2 Ja dXa is integrabie in that it admits the existence of an integrating factor, a fact also implied in the work of Li [7]. In the present paper, which has been motivated by an exchange of correspondence with J. Meixner and which expands on a suggestion made by him, we propose to examine the validity of each of the two possibilities in the field of chemical kinetics. On the one hand, it is clear that the relations between the fluxes (reaction rates) and their conjugate forces (affinities) are, generally speaking, non-linear. On the other hand, chemical kinetics is a mature science whose physical validity has been amply proven by comparison with experiment. Thus, chemical kinetics can serve as a counter-example, if none of the proposals are generally verified in its domain. Idelly, it could serve to discriminate between the two alternatives or just to suggest that either of the two proposals, if true, can, at best, enjoy limited applicability, L e., for a class of systems only. In Section 3 we discuss in detail the form which should be given to the equations of chemical kinetics in order to make them consistent wich the practice of thermodynamics of irreversible processes and so to enable us to submit them to such a verification. Section 4 shows that neither of the two proposals outlined earlier proves to be acceptable in general, that is, consistent with the preceding under the most general conditions. Nevertheless, it is shown in Section 5 that even though a potential function of dissipation does not exist, it is possible to show that there exists a generalized dissipation potential which uniquely determines the dissipative terms in the problem. If attention is centered on processes during which the concentrations can be varied independently and at will, such as in a suitably designed open system, it becomes possible to design a class of paths or processes along which the symmetry relations are satisfied (Section 6). The paper ends with a discussion of two illustrative examples which help to clarify the more abstract considerations. L Notation Consider a single phase fluid mixture consisting of n chemically distinct species [MJ, [M2],. . ., [Mn] of molecular mass M l 3 M2 , . . ., Mn, respectively, that undergo r < n — 1 independent chemical reactions. It is assumed that these r independent In the study of viscoelastic solid continua this is referred to as the viscoelastic potential; it also appeared as the so-called plastic potential in some papers [6]. J. Non-Equilib. Thermodyn., Vol. 3, 1978, No. 3 Nonlinear equations of chemical kinetics 1 55 reactions describe the detailed reaction mechanisms rather than just the overall reactions; i. e., we consider both the fast and the slow reaction mechanisms of the system under study. Let i>ak (a = 1 , . . ., r = reaction index, k = 1 , . . ., n = species index) denote the standard stoichiometric coefficients by which the reactions are specified. Then the stoichiometric equations of the system read

Non-equilibrium thermodynamics of a discontinuity surface

Abstract: A comprehensive axiomatic system of the non-equilibrium thermodynamics of a surface of discontinuity is formulated. The general formulation of the axioms is applied to the problem of an n -component visco-elastic isotropic non-polar mixture. The effect of mass currents from one phase to the other on the formulation of the hypothesis of local equilibrium is discussed.

Origin of the Landau-Lifshitz hydrodynamic fluctuations in nonequilibrium systems and a new method for reducing the Boltzmann equation

Abstract: The Landau-Lifshitz fluctuating fluxes in fluctuating hydrodynamics are derived from the deterministic Boltzmann equation with the aid of a reduction method developed by Fujisaka and Mori. Thus it is shown that the hydrodynamic fluctuations innonequilibrium systems are generated by the reduction of variables from theΜ-space distribution function to its five momentum moments, i.e., the hydrodynamic variables. This differs from the Bixon-Zwanzig and Fox-Uhlenbeck theories, in which the Landau-Lifshitz fluctuating fluxes are derived from the molecular fluctuating force in thestochastic Boltzmann-Langevin equation, which is, however, negligible in nonequilibrium systems. Thus the present method improves the Chapman-Enskog reduction method so as to include the hydrodynamic fluctuations generated by the reduction of variables.

Linear nonequilibrium thermodynamic theory of glass transition kinetics

Abstract: A phenomenological nonequilibrium thermodynamic theory of the glass transition is presented in a linear approximation and applied to glass transition kinetics. A linear relationship between ordering parameters and affinities is diagonalized so that each new affinity is proportional to the nonequilibrium portion of a corresponding single ordering parameter. It is shown that fictive temperatures defined for each of these new parameters can themselves serve as ordering parameters, and that the experimental fictive temperatures represent averaged ordering parameter values. A further transformation is made to uncouple the equations of motion, obtaining exponentially relaxing ordering parameters and affinities. The nonequilibrium portions of the dependent thermodynamic variables take the form of convolutions between linear response functions and the time derivatives of the independent variables. It is shown for the first time that the coefficients of expansion of the response functions into exponetials are not ...

On the thermodynamics of non-linear elastic dielectrics

Abstract: A complete non-linear theory of thermoelastic dielectrics including polarization gradient effects is derived by using the two laws of thermodynamics and invariance requirements. Restrictions on the constitutive equations are obtained by using the classical form of the Clausius-Duhem inequality as well as a revised version used by Muller in which the entropy flux is not assumed, a priori, to be the ratio of heat flux to absolute temperature. Results from both formulations are compared.

Variational irreversible thermodynamics of heat and mass transfer in porous solids: new concepts and methods

Abstract: A recently developed variational principle of virtual dissipation along with a new approach to the thermodynamics of open systems is applied to coupled mass and heat transfer in a porous solid containing a fluid. General differential field equations are derived directly from the variational principle. A general energy flux theorem is formu- lated. Vapor-liquid phase transition and capillary condensation are discussed. Field equations for nonequilibrium adsorption are also obtained. Lagrangian equations with generalized coordinates are derived directly from the variational principle without use of the field equations. They provide the foundation of finite-element methods as well as of many other techniques particularly suitable in geothermal systems analysis. 1. Introduction. A principle of virtual dissipation has recently been developed gener- alizing d'Alembert's principle to nonlinear dissipative thermodynamic systems ( 11. This new principle is a natural outgrowth of earlier work providing a variational-Lagrangian formulation of linear thermodynamics (2). Application of this variational principle pro- vides directly both the field equations of general continuous systems as well as the corresponding Lagrangian equations with generalized coordinates. The Lagrangian equa- tions thus obtained constitute a powerful tool for systems analysis of very complex physical and technological systems. They are formulated from basic physical invariants of the system without a priori detailed knowledge of the field differential equations. This is in contrast with current procedures which derive variational principles from the particular differential field equations for each type of problem. An important aspect of the new approach is its unified interdisciplinary nature, which embodies a complete synthesis between mechanics and thermodynamics. This is well illustrated by its application to the nonlinear thermorheology of continua (3) which covers a large category of phenomena. Another'innovation which has considerably enlarged the field of application of the variational technique is the development of an entirely new fundamental approach to the thermodynamics of open systems (4, 5, 61. The concept of "thermobaric potential" replaces Gibbs' chemical potential and bypasses the well-known difficulties associated with it. Another essential concept which has been introduced is that of entropy convection which is given a precise definition. This has led to new results in the theory of chemical reactions (4, 51. Along with the principle of virtual dissipation, these new concepts were applied to the non-isothermal dynamics of Newtonian and non-Newtonian fluids (6).

Derivation of generalized Fourier and Stokes-Newton equations based on the thermodynamics of irreversible processes

Abstract: An extension of Gibbs equation for isotropic fluid systems was proposed by Muller [17] a few years ago. Following the outlines of the classical theory of irreversible processes, Muller derived generalized Fourier and Stokes-Newton laws which, however, did not satisfy the principle of material frame indifference. The purpose of the present paper is to present a modified version of Muller's equations meeting the requirement of material frame indifference. It is also shown to what extent the results are in agreement with the kinetic theory of gases.