On the microscopic derivation of mode-mode coupling equations

Alternative results derived on a microscopic basis for the mode-mode coupling kinetic equations are shown to be identical. It is also emphasized that nonlinear kinetic equations for the gross variables describing the system are only suggested but not implied by the corresponding equations obeyed by their dynamical variables. Finally an equivalent closed form for the renormalized transport coefficients is shown to hold in mode-mode coupling theory.Miembro del Colegio Nacional. On sabbatical leave from UAM—Iztapalapa, Mexico.     

Non-linear Burnett order thermal stress coefficient

The formal theory of non-equilibrium statistical is used to obtain the irreversible Burnett order thermal stresses, and time correlation function expressions for both linear and nonlinear transport coefficients are identified. Evaluation of these expressions in the Boltzmann limit shows agreement with the Chapman-Enskog solution of kinetic theory.     

Locally Anisotropic Kinetic Processes and Thermodynamics in Curved Spaces

The kinetic theory is formulated with respect to anholonomic frames of reference on curved spacetimes. By using the concept of nonlinear connection we develop an approach to modelling locally anisotropic kinetic processes and, in corresponding limits, the relativistic nonequilibrium thermodynamics with local anisotropy. This leads to a unified formulation of the kinetic equations on (pseudo) Riemannian spaces and in various higher dimensional models of Kaluza–Klein type and/or generalized Lagrange and Finsler spaces. The transition rate considered for the locally anisotropic transport equations is related to the differential cross section and spacetime parameters of anisotropy. The equations of states for pressure and energy in locally anisotropic thermodynamics are derived. The obtained general expressions for heat conductivity, shear, and volume viscosity coefficients are applied to determine the transport coefficients of cosmic fluids in spacetimes with generic local anisotropy. We emphasize that such local anisotropic structures are induced also in general relativity if we are modelling physical processes with respect to frames with mixed sets of holonomic and anholonomic basis vectors which naturally admits an associated nonlinear connection structure.     

On the invariance of constitutive equations according to the kinetic theory of gases

Iterative techniques for solving the Boltzmann equation in the kinetic theory of gases yield expressions for the stress tensor and heat flux vector that are analogous to constitutive equations in continuum mechanics. However, these expressions are not generally invariant under the Euclidean group of transformations, whereas constitutive equations in continuum mechanics are usually required to be by the principle of material frame indifference. This disparity in invariance properties has led some previous investigators to argue that Euclidean invariance should be discarded as a contraint on constitutive equations. It is proven mathematically in this paper that the results of the Chapman-Enskog iterative procedure have no direct bearing on this issue. In order to settle this question, it is necessary to examine mathematically the effect of superimposed rigid body rotations on solutions of the Boltzmann equation. A preliminary investigation along these lines is presented which suggests that the kinetic theory is consistent with material frame indifference in at least a strong approximate sense provided that the disparity in the time scales of the microscopic and macroscopic motions is extremely large—a condition which is usually a prerequisite for the existence of constitutive equations.On leave from Stevens Institute of Technology, Hoboken, New Jersey 07030.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.     

Onsager relations for transport in inhomogeneous media

The nonlinear equations that describe transport in inhomogeneous media cannot be obtained by a straightforward extension of the known phenomenological equations for homogeneous media. One cannot therefore asserta priori that the Onsager reciprocity relations remain valid. Previously the correct equations have been obtained for three special models using kinetic theory. It is here shown that in these models the Onsager relations do indeed hold, provided that they are formulated with care.     

Coupled Kardar-Parisi-Zhang Equations in One Dimension

Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.     

Description of far-from-equilibrium processes by mean-field lattice gas models

Mean-field kinetic equations are a valuable tool to study the atomic dynamics and spin dynamics of simple lattice gas and Ising models. They can be derived from the microscopic master equation of the system and contain analytical expressions for kinetic coefficients and thermodynamic quantities which are usually introduced phenomenologically. We review several methods to obtain such equations, and discuss applications to the dynamics of order–disorder transitions, spinodal decomposition, and dendritic growth in the isothermal or chemical model. In the case of dendritic growth we show that the mean-field kinetic equations are equivalent to standard continuum equations for this problem and derive expressions for macroscopic quantities, e.g. the surface tension and kinetic coefficients, as functions of the microscopic order parameters. In spinodal decomposition, we focus our attention on the vacancy mechanism, which is a more faithful picture of diffusion in solids than the more widely examined exchange mechanism. We study the interfaces between an unstable mixture and a stable ‘vapour’ phase, and analyse surface modes that lead to specific surface patterns. For order–disorder transitions, studied in the framework of a repulsive two-sublattice model, we derive sets of coupled equations for the mean concentration (a conserved quantity) and for the occupational difference between the two sublattices emerging from the symmetry breaking due to ordering (non-conserved order parameter). These equations are applied to transport in the presence of ordered domains. Finally, we discuss the possibilities of improving the simple mean-field approximation by density functional theories and various forms of the dynamic pair approximation, including the path-probability method.     

Kinetic Approach to Reaction and Diffusion in Many-Particle Systems

A derivation of hydrodynamic equations which describe diffusion in connection with reaction processes in many-particle systems is given on the basis of quantum kinetic theory. For this purpose kinetic equations are derived for a quantum gas with chemical reactions. Following Grad's method in a linear approximation reaction-diffusion equations can be obtained with explicite expressions for the diffusion and reaction rate coefficients. The influence of nonideality effects in reaction-diffusion equations is discussed.     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.     

Quantum statistics of reaction-diffusion systems

A strictly quantum-statistical theory of inhomogeneous reactions is presented. The treatment is based on the theory of multi-channel reactive scattering. For the configuration probabilities of the reactants a system of reaction-diffusion equations is obtained. The expressions for the diffusion tensor and the reaction-rate coefficients are established in terms of microscopic parameters characteristic of the reacting species.     

On the theory of nonlinear response in a nonideal plasma

A theory of nonlinear response is developed for studying nonlinear phenomena and nonlinear transport processes in nonideal Coulomb systems. Temporal plasma echo and transformation of waves in a nonideal Coulomb system are studied on the basis of the theory of nonlinear response to mechanical perturbations. General constraints imposed on nonlinear response functions are considered, and the model for determining quadratic response functions is formulated. The conditions for the emergence of temporal plasma echo and wave transformation are determined. It is shown that these nonlinear effects in a nonideal plasma can be initiated by ultrashort field pulses. A theory of transport is developed for determining the Burnett transport properties of a nonideal multielement plasma. A procedure is proposed for comparing the phenomenological conservation equations for a charged continuous medium and equations of motion for the operators of corresponding dynamic variables. The Mori algorithm is used for deriving the equations of motion for operators of dynamic variables in the form of generalized Langevin equations. The linearized Burnett approximation is considered in detail. The properties of the matrices of coefficients of higher-order derivatives in the system of conservation equations in the linearized Burnett approximation, which are important for hydrodynamic applications, are discussed. Various versions of the theory of nonlinear response are compared.     

Grad theory and extended thermodynamics

Grad-type approaches introduce an ansatz involving tensor Hermite functions with coefficients expresed in terms of moments of the ansatz. This formalism in usual form yields terms linear in first-order spatial derivatives in kinetic equations for the moments. Such terms disagree with alternative statistical derivations and phenomenological arguments. This disagreement is removed if different ansatzes are used to calculate entropy and moment equations. These are non-unique, and so Grad theory, while providing theoretical expressions for transport coefficients, does not serve uniquely to determine the structure of phenomenological equations.     

Relativistic kinetics of phonon gas in superfluids

The relativistic kinetic theory of the phonon gas in superfluids is developed. The technique of the derivation of macroscopic balance equations from microscopic equations of motion for individual particles is applied to an ensemble of quasi-particles. The necessary expressions are constructed in terms of a Hamilton function of a (quasi-)particle. A phonon contribution into superfluid dynamic parameters is obtained from energy-momentum balance equations for the phonon gas together with the conservation law for superfluids as a whole. Relations between dynamic flows being in agreement with results of relativistic hydrodynamic consideration are found. Based on the kinetic approach a problem of relativistic variation of the speed of sound under phonon influence at low temperature is solved.     

Derivation of the nonlinear hydrodynamic equations using multi-mode techniques

A general mode-mode coupling theory is developed for the microscopic mass, energy and momentum densities of a simple classical fluid. A projection operator method is employed to derive a generalized Langevin equation that contains nonlinearities of all orders with both convective and dissipative terms. A general nonequilibrium ensemble average, which contains local equilibrium as a special case, is employed to derive nonlinear transport equations that are nonlocal in both space and time.The nonlinear Euler and Navier-Stokes equations are recovered using a factorization procedure based on an inverse system size approximation. We show that in the context of mode-mode coupling theory, nonlinearities of all orders must be retained to derive the full nonlinear transport equations. We also slow that the space and time dependent nonequilibrium pressure and transport coefficients are functions of the nonequilibrium mass and internal energy densities. The thermodynamic closure relationships follow as a natural consequence of mode-mode coupling theory. For a system linearly displaced from equilibrium we demonstrate the role of the corrections to our factorization approximation in renormalizing the transport coefficients.     

On the determination of the Burnett kinetic coefficients in nonideal media

The approaches to determining linearized and nonlinear Burnett kinetic coefficients in the media with a strong interaction between particles are discussed. A technique for computing the linearized Burnett kinetic coefficients using long-wave and low-frequency asymptotics for the correlation functions of charged nonideal media is proposed.     

On the magnetization of a thin-walled cylindrical sheath with a magnetic field from the Rayleigh range inside

In terms of the general theory of magnetization of thin-walled sheaths, magnetization of a thin-walled indefinitely long circular cylindrical sheath that is placed in an external magnetic field and experiences the action of an internal magnetic field from the Rayleigh range is considered. It is shown that the solution of the problem can be reduced to solving infinite sets of nonlinear equations in the coefficients of Fourier series representing the components of the magnetization and internal magnetic field strength that are averaged over the thickness of the sheath and tangent to its surface. Approximate solutions to these sets of equations are obtained, and expressions for the coefficients multiplying the first terms of the series are given.     

Nonequilibrium Sources of Hydrodynamic Fluctuations Mutual Influence of Hydrodynamic Motion and Nonequilibrium Fluctuations on the Kinetic Coefficients

From generalized expressions for the entropy production the kinetic coefficients in the hydrodynamic equations are determined taking into account the mutual influence of the hydrodynamic motion and the fluctuations. Nonlinear effective and turbulent kinetic coefficients are obtained. The usefulness of the introduced coefficients is demonstrated for boundary layer and tube flows.