Escape-Rate Formalism, Decay to Steady States, and Divergences in the Entropy-Production Rate

According to thermodynamics the irreversible entropy production of diffusive relaxation processes diverges at the boundary to the vacuum, i.e., to a state of vanishing particle density. By means of a multibaker map we point out that this divergence is not present in the spatially discrete dynamics, which brings forth the evolution equations of irreversible thermodynamics in the continuum limit. In addition, we show that the irreversible entropy production of relaxation towards a nonempty steady state is proportional to the decay rate of the thermodynamic system subjected to absorbing boundary conditions. This generalizes results of the escape rate formalism.     

TheH theorem and irreversible thermodynamics

Following a method first introduced by Prigogine, theH theorem is written as the law of increase of entropy for a slightly inhomogeneous gas. It is shown that the local rate of entropy production for such a gas is simply a homogeneous quadratic form of the generalized forces associated with the various irreversible processes with coefficients possessing all the properties of the phenomenological coefficients of irreversible thermodynamics. The local rate of entropy production is explicitly evaluated for a simple monatomic gas and is compared with the corresponding expression of irreversible thermodynamics.     

Dual-phase-lag transfer equations and entropy behavior in relaxational hydrodynamics

Extended thermodynamics of irreversible processes is developed; based on two postulates by which additional variables of the entropy density are dissipative fluxes and material time derivatives of the ordinary thermodynamic variables. Within these theories a more general approximation of entropy production is obtained. As a consequence of the proposed formalism, the constitutive dual-phase-lag equations, as well as equations of the conventional version of extended irreversible thermodynamics are obtained. The behavior of the entropy during oscillatory approach to equilibrium is considered. The proposed theory leads to a strictly monotonic dependency of the entropy on time.     

强迫耗散系统的有序结构和系统的发展(Ⅰ)，最小熵产生原理和有序结构

一个系统的发展总是由不可逆热力过程和非线性动力过程所驱动.将大气动力学方程组同考虑了动能变化的Gibbs关系结合起来构建的熵平衡方程，才能更好地描述大气系统的不可逆热力过程和非线性动力过程.至今非平衡态热力学仅利用Onsager线性唯象关系证明了最小熵产生原理.利用新建立的熵平衡方程和大气动力学方程的性质证明，最小熵产生原理在热力学线性区和非线性区都是普遍成立的.且当热量输送平衡、水汽输送平衡和动量输送平衡时，系统达到不可逆过程最弱的最小熵产生热力学状态.当系统又是动力平衡且无平流时，这种最小熵产生态就是 关键词： 非线性热力学 熵产生 最小熵产生原理 有序结构     

The kinetic foundation of extended irreversible thermodynamics revisited

Extended irreversible thermodynamics (EIT) has been used mainly to study the short-time behavior of fluids and some other systems. It has also been shown how the structure of the equations of motion constructed for the so-called relaxation variables coincides with those obtained by means of Grad's method in kinetic theory. In this work we calculate the generalized entropy from the one-particle distribution function up to 26 moments. We find that the characteristics of such entropy and the equations of motion for the relaxing variables are supported by the kinetic theory. This is not the case for the hierarchical relaxation hypothesis which is used in the applications of EIT to the generalized hydrodynamic regime.On temporary leave at the Universidad Iberoamericana, Mexico.     

Irreversible thermodynamics of fluids

Irreversible thermodynamics of fluids is formulated based on a set of postulates. The theory thus constructed generalizes thermostatics and linear irreversible thermodynamics into the realm of nonlinear irreversible processes. In this theory the extended Gibbs relation and the entropy balance equation appear as a pair of mutually consistent equations under the postulates made. An equivalent theory is also formulated by replacing one of the postulates with another that is basically a variational principle. The variational principle yields the evolution equations for fluxes as the Euler equations that extremize the variational functional postulated. The local form of the extremized variational functional is the entropy balance equation for the irreversible processes in the system. Some further consequences of the theory are also considered. For example, nonequilibrium specific heats are shown to be at least quadratic functions of fluxes and reduce to the equilibrium specific heats in the limit of vanishing fluxes. In order to illustrate an example of possible applications, we have considered nonlinear transport processes in fluids. The connections of the present theory with other theories are discussed.     

Foundation of quasilinear irreversible thermodynamics

In the set of axioms of the linear thermodynamics of irreversible processes the linearity law is replaced by a more general relation (8) coupling fluxes and forces and their time derivatives. It is shown that the remaining axioms of the classical irreversible thermodynamics (OnsagerCasimir relations and the entropy production expression) can be left unchanged. It is derived that in the frame of this theory it is possible to picture the Maxwell model of viscoelastic effects and the Debye model of the dielectric relaxation.     

Kinetic theory in the weak-coupling approximation: I. Formal theory and application to classical open systems

A general formalism, where irreversible processes are related to singularities of the resolvent of the Liouville operator, is applied to classical open systems. For a system weakly coupled to a thermal reservoir, a kinetic equation is derived. It is shown that the method leads to equations defining a positivity-preserving semigroup with the Maxwell-Boltzmann distribution as a stationary solution and obeying an H-theory. It is pointed out that these properties are not always shared by irreversible equations obtained as asymptotic approximations of the so-called generalized master equation.     

On a kinetic justification of the generalized nonequilibrium thermodynamics of multicomponent systems

The problem of the kinetic justification of the generalized thermodynamics of nonequilibrium processes using the method of moments for solving the kinetic equation for a multicomponent gas mixture is examined. Generalized expressions are obtained for the entropy density, entropy flux density, and entropy production as functions of an arbitrary number of state variables (moments of the distribution function). Different variants of writing the relations between fluxes and thermodynamic forces are considered, which correspond to the Onsager version for spatially homogeneous systems and, in a more general case, lead to the generalized thermodynamic forces of a complicated form, including derivatives of the fluxes with respect to time and spatial coordinates. Some consequences and new physical effects, following from the obtained equations, are analyzed. It is shown that a transition from results of the method of moments to expressions for the entropy production and the corresponding phenomenological relations of the generalized nonequilibrium thermodynamics is possible on the level of a linearized Barnett approximation of the Chapman–Enskog method.     

Macroscopic phenomenological relations for nonlinear processes in kinetic theory

Nonlinear irreversible processes between states which are not local equilibrium states are investigated by methods of the kinetic theory. The phenomenological equations for the second-order fluxes in a multicomponent mixture are derived, and relations between some of the second-order phenomenological coefficients are established. It is shown that new independent forces appear in the second-order equation, namely the gradients of the chemical potentials. Expressions for the entropy, entropy flux, and entropy source are evaluated. These expressions are related to the phenomenological equations and coefficients, e.g., all the second-order contributions of the forces in the equations for the fluxes can be obtained by differentiation of the expression for the second-order entropy source with respect to the coupled forces.     

Zur Dynamik reagierender Gase

The behaviour of mixtures of reacting gases can be described by means of different principles. In this paper the thermodynamics of irreversible processes is confronted by a method based on the mechanics of particles. Here the conception of the “constituents” as the building elements, composing the several components of the mixture, can be used profitably. Taking into consideration the thermal diffusion and the terms due to reactions in the equations of force, when the mechanical or “dynamical” method is used, and formulating the kinetic quantities more precisely, when the thermodynamics of irreversible processes is regarded, the forces of inertia, hitherto missed, appear in the phenomenological equations, and the two systems can be proved to be equivalent.     

Comment on Keizer's critique on extended irreversible thermodynamics

Keizer's critique on extended irreversible thermodynamics is responded and qualified so as to remove misleading points of his statements. It is particularly pointed out that contrary to his assertion, fluctuating irreversible thermodynamics may be regarded as being included in extended irreversible thermodynamics as a special case, since it is derivable from the latter when the relaxation times of fluxes are comparatively shorter than the hydrodynamic relaxation time and the initial conditions for the evolution equations are random.Work supported by the grants from the Natural Sciences and Engineering Research Council of Canada.     

Thermodynamik der Vorgänge in einfachen fluiden Medien und die Charakterisierung der Thermodynamik irreversibler Prozesse

Thermodynamics of processes in continuous matter has found several treatments: (1) classical thermodynamics of irreversible processes, (2) the nonlinear field theory of mechanics with the incorporation of thermodynamic aspects, (3) the new entropyfree thermodynamics of processes. An important feature of the last theory is the fundamental inequality. It provides a basis for the formulation of constitutive equations, which are discussed for simple thermodynamic fluid materials. Classical thermodynamics of irreversible processes results as a well defined special case with a modification that has been overlooked previously. It is shown by an example that this modification which differentiates between a dynamic and a thermostatic temperature is necessary in order to make classical thermodynamics of irreversible processes consistent.     

On the Onsager Variation Principle and its Generalization for Nonlinear Irreversible Thermodynamics

The Onsager variation principle is examined from the viewpoint of the thermodynamic analogue of the D'Alembert principle in mechanics when the irreversible processes are linear and thus the system is near equilibrium. The thermodynamic D'Alembert principle is shown to be a precursor to the Onsager variation principle. The thermodynamic D'Alembert principle is then generalised to the cases of nonlinear irreversible processes occurring removed from equilibrium and a generalised form of the Onsager variation principle is obtained under some restricting conditions. The restricted variation principle so deduced has an accompanying exact differential form generalising the Clausius entropy differential (equilibrium Gibbs relations) and contains in it the essence of the thermodynamics of irreversible processes in systems where non-linear transport processes occur. An example is given for the nonlinear dissipation function in the variation functional. The evolution equations for fluxes are shown to yield those known in the literature.     

Response to Z. Banach's comments on the modified moment method and irreversible thermodynamics

In this paper the author responds to the comments on the modified moment method and irreversible thermodynamics made by Z. Banach [Physica A 145 (1987) 105]. In this paper Banach suggests a variational method in which the Lagrange multipliers are determined from the constraints alone by disregarding the entropy balance equation. It is shown that since this method does not yield an extended Gibbs relation consistent with the entropy balance equation or the H-theorem, there is no irreversible thermodynamics formalism afforded by the method. Consequently, his criticism cannot be supported from the viewpoint of irreversible thermodynamics. It is also pointed out that neither is there a mathematical and physical support for the criticism he makes on the cumulant expansion for the Boltzmann collision integral.     

Entropy, entropy flux and entropy rate of granular materials

The aim of this work is to analyze the entropy, entropy flux and entropy rate of granular materials within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the entropy density rate. From the knowledge of a non-equilibrium distribution function-valid for processes closed to equilibrium-it is obtained that the entropy density rate is proportional to the internal energy density rate divided by the temperature, while the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular material is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the entropy density rate and entropy flux are valid in general for processes close to equilibrium of granular materials, where linearized constitutive equations hold.     

Der Drehimpulssatz in der Thermodynamik der irreversiblen Prozesse

The conservation law of the angular momentum has not been given sufficient attention in thermodynamics of irreversible processes. It plays an important part, however, when irreversible processes in the presence of electromagnetic fields are examined and the intrinsic angular momentum, due to spins of electrons and nuclei and electron orbits in atoms and molecules, is taken account of. Then the thermodynamic properties of matter with the macroscopic intrinsic angular momentum as an additional extensive parameter can be developed. Thermodynamics of irreversible processes is completed by the fact that a characteristic irreversible process is now associated also with the conservation law of angular momentum and Bloch's equation of nuclear induction is thus obtained in a straightforward way as one of the phenomenological equations.     

Beyond the navier-stokes regime in hydrodynamics

In this letter we show that if one suitably modifies the conventional linear relationship between forces and fluxes in irreversible thermodynamics, then all the higher-order hydrodynamic equations, namely, the Burnett, super-Burnett, etc. equations, are consistent with both the entropy balance equation and the local equilibrium assumption. Some implications of this result are also pointed out.     

Magnetic relaxation in a spin-1 Ising model near the second-order phase transition point

The magnetic relaxation of a spin-1 Ising model with bilinear and biquadratic interactions is formulated within the framework of statistical equilibrium theory and the thermodynamics of irreversible processes. Using a molecular-field expression for the magnetic Gibbs energy, the magnetic Gibbs energy produced in the irreversible process is calculated and time derivatives of the dipolar and quadrupolar order parameters are treated as fluxes conjugate to their appropriate generalized forces in the sense of Onsager theory. The kinetic equations are obtained by introducing kinetic coefficients that satisfy the Onsager relation. By solving these equations an expression is derived for the dynamic or complex magnetic susceptibility. From the real and imaginary parts of this expression, magnetic dispersion and absorption factor are calculated and analyzed near the second-order phase transition.     

A minimal dissipation type-based classification in irreversible thermodynamics and microeconomics

We formulate the problem of finding classes of kinetic dependencies in irreversible thermodynamic and microeconomic systems for which minimal dissipation processes belong to the same type. We show that this problem is an inverse optimal control problem and solve it. The commonality of this problem in irreversible thermodynamics and microeconomics is emphasized.Received: 10 June 2003, Published online: 24 October 2003PACS: 05.70.Ln Nonequilibrium and irreversible thermodynamics - 89.65.Gh Economics; econophysics, financial markets, business and management