scispace - formally typeset
Search or ask a question

Showing papers on "Non-equilibrium thermodynamics published in 1970"


BookDOI
01 Jan 1970

517 citations


Journal ArticleDOI
TL;DR: In this article, the Langevin equation was used to derive the Navier-Stokes equations for the Brownian motion of a particle of arbitrary shape, and these terms and their correlation properties are presented, and then used to obtain the Lagrangian Lagrangians for linearized hydrodynamical equations, which were first proposed by Landau and Lifshitz.
Abstract: The velocity of a particle in Brownian motion as described by the Langevin equation is a stationary Gaussian–Markov process. Similarly, in the formulation of the laws of non‐equilibrium thermodynamics by Onsager and Machlup, the macroscopic variables describing the state of a system lead to an n‐component stationary Gaussian–Markov process, which, in addition, these authors assumed to be even in time. By dropping this assumption, the most general stationary Gaussian–Markov process is discussed. As a consequence, the theory becomes applicable to the linearized hydrodynamical equations and suggests that the Navier–Stokes equations require additional fluctuation terms which were first proposed by Landau and Lifshitz. Such terms and their correlation properties are presented, and these equations are then used to derive the Langevin equation for the Brownian motion of a particle of arbitrary shape.

295 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used the direct simulation Monte Carlo method to study the breakdown of translational equilibrium in steady cylindrical and spherical expansions of hard sphere and Maxwell molecules, and extended it to the combined translational and rotational breakdown in a gas of rough sphere molecules.
Abstract: The direct simulation Monte Carlo method has been used to study the breakdown of translational equilibrium in steady cylindrical and spherical expansions of hard sphere and Maxwell molecules. The study of spherical expansions was extended to the combined translational and rotational breakdown in a gas of rough sphere molecules. The breakdown of translational equilibrium in a complete one-dimensional rarefaction wave in a hard sphere gas was also investigated. In all cases, the breakdown of equilibrium was found to coincide with a constant value of the ratio of the logarithmic time derivative of density following the motion of the fluid to the collision frequency in the gas. This value is proposed as an empirical breakdown criterion for use in engineering studies of systems which involve low-density expansions from continuum to highly rarefied conditions. The onset of nonequilibrium was marked by the divergence of the separate kinetic temperatures based on the molecular velocity components parallel and normal to the flow direction. The parallel temperature in a steady expansion gradually froze to a constant value, in qualitative agreement with experiment and with analytical studies employing the BGK model. The rate of decay of the temperature based on the normal velocity components was greater than the isentropic rate for hard sphere molecules, but less than it was for Maxwell molecules.

202 citations


Journal ArticleDOI
TL;DR: In this article, the theory of nonequilibrium phenomena in a dilute gas as described by the Boltzmann equation is extended in order to also include the fluctuations of the distribution function of the molecules.
Abstract: The theory of the nonequilibrium phenomena in a dilute gas as described by the Boltzmann equation is extended in order to also include the fluctuations of the distribution function of the molecules. It is shown that in a first approximation this extension leads to the Landau‐Lifshitz fluctuation terms in the hydrodynamical equations.

114 citations


Journal ArticleDOI
TL;DR: In this paper, the nonequilibrium behavior of argon and helium confined-arc plasmas at atmospheric pressure was analyzed in terms of a two-temperature ambipolar diffusion model.
Abstract: The nonequilibrium behavior of argon and helium confined‐arc plasmas at atmospheric pressure is analyzed in terms of a two‐temperature ambipolar‐diffusion model. Examination of spectroscopic data for the nonquilibrium arc shows that the free and bound electrons are mutually in equilibrium at a common electron temperature but not in equilibrium with the translational degrees of freedom of the heavy particles; the electron density in the outer portions of the arc may be as much as several orders of magnitude greater than its local thermodynamic‐equilibrium value. Integration of the electron continuity equation and the electron and heavy‐particle energy equations, with calculated transport properties and recombination coefficients, yields a satisfactory comparison with the measurements.

66 citations



Journal ArticleDOI
TL;DR: In this paper, the authors present a description of rate sensitive plastic materials which agrees with events on the microscopic level and with macroscopic experiments and which is internally consistent, using the framework of the thermodynamics with internal state variables formulated recently by Coleman and Gurtin.
Abstract: The basic objective is the presentation of a description of rate‐sensitive plastic materials which agrees with events on the microscopic level and with macroscopic experiments and which is internally consistent We utilize the framework of the thermodynamics with internal state variables formulated recently by Coleman and Gurtin The internal state variables in the present theory are considered to be average quantities related to properties of the crystal defects in the material Rate‐dependent phenomena, (eg, a time‐dependent stress‐strain relation, creep, etc) are studied in detail The theory is illustrated with a simple example

53 citations


Journal ArticleDOI
TL;DR: In this paper, the authors review the recent controversy concerning relativistic thermodynamics and discuss the advantages of each approach and compare them with the advantages and disadvantages of the other approaches.
Abstract: In this paper we review the recent controversy concerning relativistic thermodynamics. The diversity of viewpoints can be attributed to the numerous ways in which thermodynamic quantities are defined. The advantages of each approach are discussed.

47 citations


Journal ArticleDOI
TL;DR: In this article, the thermodynamic treatment of the interfacial region corresponding to two fluid phases in contact is discussed and the features of this analysis which are reviewed include the classical treatment of Gibbs and the extensions to this treatment which are due to Buff.
Abstract: The thermodynamic treatment of the interfacial region corresponding to two fluid phases in contact is discussed. The features of this analysis which are reviewed include the classical treatment of Gibbs and the extensions to this treatment which are due to Buff. It is shown that these extensions are essential if the logical structure of the analysis is to be regarded as complete.

37 citations


Journal ArticleDOI
TL;DR: Perturbation and variational approaches to equation of state and other equilibrium thermodynamic properties of simple liquids and phase transitions, vapor-liquid and liquid-solid, are reviewed and discussed in this article.
Abstract: Perturbation and variational approaches to equation of state and other equilibrium thermodynamic properties of simple liquids and phase transitions, vapor-liquid and liquid-solid, are reviewed and discussed. count for the lack of convergence of the virial equation for the liquid state. Probably the most widely used equation of state today, indeed the standard of reference often used when proposing new equations of state, is the Benedict-Webb-Rubin equation. Both the form of this equation and the recipes for estimating mixture behavior from known behavior of its components are based upon virial expansion. While the virial expansion has been valuable in treating gases and gaseous mixtures, its failure to converge in the liquid state forces one to look for a more complete description, either empirically or theoreticaliy. The theoretical problem has been approached in several ways, and these are reviewed briefly below. The main emphases of this review are perturbation and variational approaches to evaluate the partition function, hence thermodynamic properties and equation of state, because these approaches show considerable promise for practical application. As a point of reference to the reader, this review is based largely on references available to the authors as of June 1969. Interpretive Approaches (2) Interpretive approaches to the equation of state start from an approximate description of the structure of the system. These approaches are called “lattice” theories because the structure customarily is assumed to resemble the regular lattice structure of crystalline solids. Assumptions regarding structure must be guided both by physical reality and by the ability to calculate the partition function resulting from the assumed structure. For crystalline solids, these requirements are harmonious because solids are known to have regular structures which are disturbed only slightly by thermal motions and such regular, more or less static, structures lead to a partition function which can be evaluated. Liquids, on the other hand, offer a serious challenge to the viability of interpretive approaches because their structure, if it can be called that, is continually changing and can be visualized only on an instantaneous basis. T o account for physical reality in liquids, a static average over the instantaneous structure suggests itself and such an approach may lead to a satisfactory “lattice” theory for liquids. This is true only if one can solve the mathematical problems in the form of very difficult and complicated combinatorial versions (17), which is still a difficult task. Predictive Approaches (58) Predictive approaches place emphasis, initially, on the process by which the intermolecular forces determine the structure, in the hope that a correct mathematical description of this process will lead to equations whose solutions describe the actual structure. Theories of this class are called “distribution function” theories because the equations involve distribution functions specifying the probability of finding sets of molecules in particular configurations. The three theories, customarily called Born-Green (BG) (12), hypernetted chain (HNC), and PercusYevick (PY) (52), of which any discussion of the theory of fluids must take account, had very different origins, and each requires different initial assumptions to get tractable results. The first, BG, which was proposed in 1936, requires the direct assumption that the triplet distribution function can be expressed as a product of pair distribution function for the three pairs involved, that is, the superposition approximation. Although the HNC and PY theories bypass direct assumptions regarding the triplet distribution function in their initial formulation, attempts to improve these theories ultimately lead to approximation or calculation of the triplet distribution function. The PY theory, while capable of producing an analytic and approximately correct equation of state for hard spheres (62, 65), must be improved before it is equally successful for more realistic systems. Rice and coworkers (67) have replaced the superposition approximation in the BG theory with a more adequate approximation, but extensive calculations have demonstrated that none of the predictive theories are capable of predicting the properties of simple liquids and phase transitions at the present time. The triplet distribution function, complicated and unknown, stands in the way of successful predictive approaches to the equation of state and other equilibrium properties of dense gases and liquids. Perturbation and Variational Approaches Variational and perturbation approaches, the subjects of this review, are neither predictive nor interpretive theories. They are mathematical * means of expanding the configurational partition function of an original system around a relatively simple reference system whose properties are known. The reference system can be: a real substance whose properties are known (55, 69) or can be a hypothetical system, such as hard spheres, as long as relationships are available to describe its behavior. This review is concerned only with the second approach where properties of the reference system can be calculated from a statistical mechanic theory. That is, by defining a particular intermolecular potential function for the reference system, one must be able to calculate other properties of the reference system such as Helmholtz free energy, compressibility, and the radial distribution function. The reference system should resemble the original system as close as possible to speed the convergence of the power series of the partition function of the original system, and could be the result of a predictive or an interpretive theory. The equation of state for a gas at very high temperatures is determined largely by the repulsive forces acting between molecules, that is, a real gas at very high temperatures behaves much like an assembly of hard spheres. Thus, it is reasonable to expect that the equation of state a t somewhat lower temperatures could be obtained by treating the attractive forces as perturbations about the repulsive forces. If the perturbational approach is valid, the equation of state for gases at moderate temperatures will be of the following form: VOL. 6 2 NO. 8 A U G U S T 1 9 7 0 13 Perturbation and Variational Approaches to Equilibrium Thermodynamics of Gases, Liquids, and Phase Transitions

32 citations


Journal ArticleDOI
R. Ghez1
TL;DR: In this paper, it was shown that the temperatures and concentrations may be found by solving a coupled system of parabolic differential equations with radiation-type boundary conditions at the interface, and the authors applied the methods of irreversible thermodynamics to heat flow and mass transport on interfaces.

Journal ArticleDOI
TL;DR: In this article, it was shown that the nonequilibrium statistical operator which was obtained previously by one of us (D.N.Z.) can be derived from the requirement that the information entropy of the system must have an extremum with a set of additional conditions.



Journal ArticleDOI
TL;DR: The magnetic and thermodynamic properties of a 3.700 cm−diam spherical single crystal of CuSO4·5H2O have been investigated down to 0.02°K with fields along the β magnetic axis as discussed by the authors.
Abstract: The magnetic and thermodynamic properties of a 3.700‐cm‐diam spherical single crystal of CuSO4·5H2O have been investigated down to 0.02°K with fields along the β magnetic axis. Thirty‐eight magnetizations from unknown low temperatures to reference conditions were performed under adiabatic conditions, and 35 were essentially isentropes. These have been used to determine the thermodynamic temperature and heat capacity, both in and out of magnetic fields, without heat introduction. A temperature reference at 0.035°K has been suggested. It has been pointed out that proving a process to be isentropic is no guarantee that “frozen‐in” entropy and energy situations have not developed during the isentropic process, thus conserving nonequilibrium entropy. Entropy conserved in this way cannot be used as an equilibrium variable in determining temperature. It is evident that detailed methods of investigation designed to detect such nonequilibrium magnetic structural situations will be desirable. When “frozen‐in” entropy situations develop on proven isentropes, as is inevitable in many cases, the isentropic transfer of the entropy values to unknown low temperatures will not suffice to determine the thermodynamic temperature by use of heat introduction and the equation T = dq / dS. Earlier magnetothermodynamic data have been supplemented by measurements between 0.4 and 4.2°K with a field of 33 kG.

Journal ArticleDOI
TL;DR: In this article, the authors present a generalisation of classical gas dynamics called relaxation gas dynamics, which is called Relaxation Gas Dynamics (GLD) and is defined as the dynamics of gases that are in local thermodynamic equilibrium everywhere in the flow field.
Abstract: Classical gas dynamics is by definition the dynamics of gases that are in local thermodynamic equilibrium everywhere in the flow field so that their thermodynamic behaviour is completely specified by two independent variables of state, for example by pressure p and density p . The fundamental assumption of local thermodynamic equilibrium is not valid in many flow situations of practical importance, and this fact has given rise to a revision and generalisation of classical gas dynamics. This generalisation is usually called “relaxation gas dynamics”. Ten years ago J. Ackeret mentioned this subject briefly at the end of the Daniel and Florence Guggenheim Memorial Lecture on The Role of Entropy in the Aerospace Sciences. He gave that lecture at the 2nd International Congress of the Aeronautical Sciences in Zurich (12th September 1960) and he said on that occcasion: “Non-equilibrium states are of growing importance, for strictly speaking, exact (thermodynamic) equilibrium occurs only relatively seldom in gas dynamics. However, a rational theory of nonequilibrium states must necessarily be extremely complicated….

Journal ArticleDOI
TL;DR: In this paper, the development of the thermodynamics of irreversible processes is outlined with a review of recent work and a discussion of the application of these concepts to physicochemical, biological and hydrodynamic phenomena.
Abstract: The development of the thermodynamics of irreversible processes is outlined with a review of recent work and a discussion of the application of these concepts to physicochemical, biological and hydrodynamic phenomena. The extension of local thermodynamics to include a theory of stability and of fluctuations receives attention. The author concludes with remarks on the stability properties of chemical reactions in open systems and comments on the possible implications of results in the interpretation of fundamental biological phenomena.


Journal ArticleDOI
TL;DR: In this paper, the breakdown of translational equilibrium in steady cylindrical and spherical expansions was studied for both hard sphere and Maxwell molecules, and the results gave qualitative support to one of the major predictions of the BGK theory.
Abstract: An application of the direct simulation Monte Carlo method, which is claimed to give a solution of the full Boltzmann equation. The breakdown of translational equilibrium in steady cylindrical and spherical expansions was studied for both hard sphere and Maxwell molecules. The study of spherical expansions was extended to the combined translational and rotational breakdown in a gas of rough sphere molecules. The breakdown of translational equilibrium in a complete one-dimensional rarefaction wave in a hard sphere gas was also studied. The application of the method generally followed standard procedures, except for the unsteady flow, in which a Lagrangian system of cells moving with the fluid was used in place of the usual Eulerian cells fixed in physical space. The onset of nonequilibrium was marked by the divergence of the separate kinetic temperatures based on the molecular velocity components parallel and normal to the flow direction. The region of simulation of the steady expansion generally covered only a restricted range of Mach number in the vicinity of the point of breakdown. However, several runs were made over a wide range of Mach number and the temperatures were plotted against radius r in order to obtain an over-all picture of the process. The results gave qualitative support to one of the major predictions of the BGK theory— that the freezing of the parallel temperature Tx occurs gradually over a wide range of Mach number and is much less rapid for Maxwell molecules than for hard sphere molecules. In the case of the normal temperature Tn, there is a qualitative difference between the results for the two molecular models. The Monte Carlo calculation for the Maxwell molecules gave a Tn curve that remains above the r~ / 3 continuum curve and could well be consistent with the r" prediction of the BGK theory. However, for the simulation with hard sphere molecules, the Tn curve falls below the continuum curve. A rarefaction parameter P was defined by the ratio of the logarithmic time derivative of density p, following the motion of the fluid, to the collision frequency v in the gas. That is, The departure of the temperature ratio Tx/Tn from unity gives the best indication of translational breakdown, and it was plotted against P for all the steady expansion runs. These involved two geometries, three molecular models and Mach numbers ranging from 2 to 20. In all cases, the breakdown of equilibrium occurred at a value of P of approximately 0.04. If the hard sphere formula for collision frequencyin terms of viscosity coefficient is substituted into Eq. (1), this leads to the following empirical breakdown criterion for steady expansions



Journal ArticleDOI
TL;DR: In this article, a general problem for the determination of the generalized fluxes and the conjugate generalized forces describing irreversible processes in nonequilibrium thermodynamics is considered, and it is shown that for linear systems the solution of this problem is also a solution of a minimum problem and of a maximum problem.
Abstract: A general problem is considered for the determination of the generalized fluxes and the conjugate generalized forces describing irreversible processes in nonequilibrium thermodynamics. It is shown that for linear systems the solution of this problem is also the solution of a minimum problem and of a maximum problem. In certain cases the functional which is minimized is the rate of entropy production, while in other cases the functional which is maximized is minus this rate. Thus, in these cases Prigogine's principle of the minimum rate of entropy production is valid. For certain dynamical systems it is also shown that the functional in the minimum problem is a decreasing function of time, while for other systems the functional in the maximum problem is an increasing function of time. The results are applied to a system of chemical components in which various chemical reactions, diffusion, and heat conduction are occurring. Similar results are obtained for special nonlinear systems of this kind and for cer...

Journal ArticleDOI
TL;DR: In this article, the stagnation-point state is found to be in a narrow range bounded on one side by the state obtained in an equilibrium flow, and the other bound, called the frozen limit, is far removed from the state of an identically frozen flow (infinite relaxation times).
Abstract: : In nonequilibrium inviscid blunt-body flows, the state of the gas at the stagnation point is known to be in thermodynamic equilibrium for all finite relaxation times. The dependence of this state on the nonequilibrium processes and body geometry is investigated for the most general conditions. The stagnation-point state is always found to be in a narrow range bounded on one side by the state obtained in an equilibrium flow. The other bound, called the frozen limit, is far removed from the state obtained in an identically frozen flow (infinite relaxation times). For certain state variables, the frozen-limit value lies outside the range determined by frozen and equilibrium flow. Significant errors are found in several published predictions of the stagnation-point state, resulting from the nonanalytic approach to equilibrium in nearly frozen flow. The two bounds on the pressure are expressible in terms of the normal shock density ratios for equilibrium and frozen flow. The actual pressure for an arbitrary flow situation is found to depend only on the shock nose radius and the relation between density and time in the relaxation zone behind a normal shock wave. If the density law is given by a single relaxation model, a closed form expression for the pressure is obtained. The analysis is carried out for both plane and axisymmetric flows, and is also valid for nonequilibrium free-stream conditions. (Author)

Journal ArticleDOI
TL;DR: In this paper, Caratheodory's thermodynamics is developed in a rigorous and quite general form and entropy and temperature are defined in terms of quantities which are more directly measurable, but Pfaffian forms and quasistatic processes do not appear.
Abstract: Classical thermodynamics is developed in a rigorous and quite general form The approach is similar to Caratheodory's in that entropy and temperature are defined in terms of quantities which are more directly measurable, but Pfaffian forms and quasistatic processes do not appear The mathematics used is elementary, apart from a small amount of symbolic logic and a very little topology



Journal ArticleDOI
01 Jun 1970
TL;DR: In this article, density wave analogous to second sound is studied in a gas of magnons and the non-equilibrium theory is based on a Boltzmann equation for magnon-magnon scattering Contrary to the total energy and magnetization, (quasi)-momentum is not strictly conserved.
Abstract: Density waves analogous to second sound are studied in a gas of magnons Quasiparticle interaction is considered for both equilibrium and non equilibrium thermodynamics The non equilibrium theory is based on a Boltzmann equation for magnon-magnon scattering Contrary to the total energy and magnetization, (quasi)-momentum is not strictly conserved In the hydrodynamic regime, the transport equation is reduced to a set of two coupled equations for the magnetization and the local temperature For low temperatures these have diffusive and propagating solutions while for high temperatures, where momentum is dissipated by Umklapp processes, the solutions are only diffusive The magnetization response function and the corresponding spectral function are discussed for various wavenumbers and temperatures


Journal ArticleDOI
TL;DR: In this paper, the first and second laws of thermodynamics are presented without the use of work or heat, and the experimental data necessary to know substances thermodynamically are their internal energy functions, their isotherm functions, and their equations of chemical equilibrium.
Abstract: The paper outlines the current presentation of thermodynamic principles to the combined Part I Engineering students at the University of Hong Kong. By considering all bodies taking part in a process, the first and second laws of thermodynamics are presented without the use of work or heat—terms which cannot be generally defined without anticipating the second law. The experimental data necessary to know substances thermodynamically are their internal energy functions, their isotherm functions, and their equations of chemical equilibrium—all in terms of pressure p. specific volume v, and the degree of advancement of chemical processes c. Thermodynamic temperature functions, affinity functions and entropy functions may be derived from these data. The paper concludes with a discussion of interactions between bodies in terms of work and heat.

Journal ArticleDOI
TL;DR: In this article, the authors give a new verbal formulation of the second law of thermodynamics, which is closely modelled on one which pertains to the derivation of the first law.
Abstract: The article gives a new verbal formulation of the second law of thermodynamics. It is claimed that the physical content of this statement as well as the derivation of the mathematical consequences normally referred to as the first and second parts of the second law are simpler and more easily grasped by beginners than the standard formulations. The argument is so designed as to be closely modelled on one which pertains to the derivation of the mathematical formulation of the first law. 1. MOTIVATION FOR THIS AR11CLE There is little advantage, from the point of view of advancing progress in physics, in reopening the question of the optimal formulation of the second law of thermodynamics. However, a case can be made for returning to this fundamental topic in the interests of those who are engaged in transmitting existing knowledge. Regardless of which primary formulation of the second law is adopted, it is commonly agreed that it must lead, by an easy logical and mathematical derivation, to three statements: (a) There exists a property called entropy, S, which is additive for subsystems and which possess the mathematical properties of a potential. (b) There exists a variable, called thermodynamic temperature, T, which has the mathematical property of being that integrating denominator, among infinitely many, for an element of heat, dQ°, in a reversible process1 which turns the latter into the perfect differential of entropy dS = dQ°/T (Carnot's theorem) (1) The thermodynamic temperature, T, is a unique function of any empirical temperature, t. (c) There exists a quantity called entropy production, 0, which is positive in any irreversible process. In an adiabatic irreversible process between an initial state 1 and a final state 2, we define (2a) and must have 0>0 (2b) All symbols with the superscript ° refer to reversible processes.