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.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction

We study a generalization of irreversible thermodynamics with nonlocal closing relation. Thus parabolic and hyperbolic models can be described within one single theory. In the 1-d case, Guyer–Krumhansl equation and classical Fourier heat conduction may be obtained, depending on the constitutive assumptions. The thermodynamical restrictions in form of the Clausius–Duhem inequality are studied taking into account an extra flux of entropy corresponding to nonlocal irreversible effects. Numerical solutions to the resulting initial-boundary value problem are calculated and compared with available experimental results.     

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.     

Non-equilibrium statistical mechanics in the general theory of relativity III. Collisional stellar dynamics

This paper assembles and extends earlier results to formulate a coherent theory of relativistic stellar dynamics appropriate for comparatively small systems of stars in which relativistic effects can be important. The structure of the Newtonian theory is outlined, culminating in the “collisional Boltzmann” or Fokker-Planck equation appropriate for an unconfined system of point masses. The theory of relativistic Fokker-Planck equations is then developed for general Lorentz-covariant interactions such as electromagnetism or scalar fields. The basic physical ingredients of Newtonian stellar dynamics are identified, and it is indicated how they can be reformulated relativistically. These considerations are then used to construct a relativistic Fokker-Planck equation appropriate for the evolution of a collection of point mass stars. The analysis is then generalized to allow, both Newtonianly and relativistically, for the effects of direct physical collisions between stars of finite size. By way of conclusion and illustration, the theory is applied to the study of a prototypical dense galactic nucleus which could evolve to contain a massive black hole. The paper ends by enumerating a number of tractable unsolved problems deserving of further consideration.     

On the noise operator approach to quantized dissipative systems

Invoking complex classical coordinates and momenta a consistent Hamiltonian theory suitable for the quantization of dissipative systems has been developed previously. In another paper this formalism has been illustrated on the basis of a simple order parameter equation by means of density operator techniques. This quite naturally calls for a comparison with quantum noise operator techniques. The present paper is an attempt to satisfy these demands. Extensive use will be made of operator ordering techniques and quasi-classical Fokker-Planck equations. As before, a certain incompleteness in the extractable information is clearly exhibited. It will be observed that the two techniques do not produce similar results in a general dynamical state as a consequence of dissipation. However, in the stationary state and within certain approximations both methods do lead to identical conclusions for the order parameters statistics. It will be argued that within the present context in general noise operator techniques are to be favoured.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form. Received 26 February 1999     

The maximum entropy principle for non-equilibrium phase transitions: Determination of order parameters,slaved modes,and emerging patterns

The maximum entropy principle allows one to make guesses on the distribution function of systems by maximizing the information entropy under given constraints. In a previous paper we succeeded to formulate appropriate constraints for systems undergoing nonequilibrium phase transitions, but we had to confine our treatment to the order parameters. In this paper we describe a formalism which does not require any a priori knowledge on the order parameters but rather allows us to determine these as well as the slaved modes and the emerging patterns. The method is applicable also to non-physical systems such as neural nets. Our approach allows us to reconsider the Landau theory of phase transitions from a new point of view. A guess is made on the Fokker-Planck equation underlying the processes which give rise to stationary distribution functions of a single order parameter.     

Nonlocal conserved quantities,balance laws,and equations of motion

A formalism is developed whereby balance laws are directly obtained from nonlocal (integrodifferential) linear second-order equations of motion for systems described by several dependent variables. These laws augment the equations of motion as further useful information about the physical system and, under certain conditions, are shown to reduce to conservation laws. The formalism can be applied to physical systems whose equations of motion may be relativistic and either classical or quantum. It is shown to facilitate obtaining global conservation laws for quantities which include energy and momentum. Applications of the formalism are given for a nonlocal Schrödinger equation and for a system of local relativistic equations of motion describing particles of arbitrary integral spin.     

Stochastic quantization and detailed balance in Fokker-Planck dynamics

The path integral and operator formulations of the Fokker-Planck equation are considered as stochastic quantizations of underlying Euler-Lagrange equations. The operator formalism is derived from the path integral formalism. It is proved that the Euler-Lagrange equations are invariant under time reversal if detailed balance holds and it is shown that the irreversible behavior is introduced through the stochastic quantization. To obtain these results for the nonconstant diffusion Fokker-Planck equation, a transformation is introduced to reduce it to a constant diffusion Fokker-Planck equation. Critical comments are made on the stochastic formulation of quantum mechanics.     

Quantum Moment Hydrodynamics and the Entropy Principle

This paper presents how a non-commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models. Our starting point is the moment method, which consists in integrating the quantum Liouville equation with respect to momentum p against a given vector of monomials of p. Like in the classical case, the so-obtained moment system is not closed. Inspired from Levermore's procedure in the classical case,⁽²⁶⁾ we propose to close the moment system by a quantum (Wigner) distribution function which minimizes the entropy subject to the constraint that its moments are given. In contrast to the classical case, the quantum entropy is defined globally (and not locally) as the trace of an operator. Therefore, the relation between the moments and the Lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators (instead of purely local ones in the classical case). In the present paper, we discuss some practical aspects and consequences of this nonlocal feature.     

Entropy Production Rate of Non-equilibrium Systems from the Fokker-Planck Equation

The entropy production rate of non-equilibrium systems is studied via the Fokker-Planck equation. This approach, based on the entropy production rate equation given by Schnakenberg from a master equation, requires information on the transition rate of the system under study. We obtain the transition rate from the conditional probability extracted from the Fokker-Planck equation and then derive a new and more operable expression for the entropy production rate. A few examples are presented as applications of our approach.     

Linear alternative to the Boltzmann equation without linearization: III. Impurity scattering and electric field

Our previously reported projection formalism leading to a linear alternative to the Boltzmann (-Uehling-Uhlenbeck) kinetic equation is here applied to standard elastic scattering on impurities in solids and external electric field. In the former case in which the Boltzmann equation is linear, we prove that our formalism reproduces this equation. In the dc as well as ac electric field and for any scattering mechanism, it is verified that inclusion of the field reduces (upon neglecting of the field dependence of scattering processes) in the lowest order to inclusion of just the standard field driving term.     

General transformation theory of Lagrangian mechanics and the Lagrange group

The general transformation theory of Lagrangian mechanics is revisited from a group-theoretic point of view. After considering the transformation of the Lagrangian function under local coordinate transformations in configuration spacetime, the general covariance of the formalism of Lagrange is discussed. Next, the group of Lagrange (for alln-dimensional Lagrangian systems) is introduced, and some important features of this group, as well as of its action on the set of Lagrangians, are briefly examined. Only finite local transformations of coordinates are considered here, and no variational transformation of the action is required in this study. Some miscellaneous examples of the formalism are included.     

Entropy production estimates for Boltzmann equations with physically realistic collision kernels

We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL ¹ convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL ¹ neighborhood of equilibrium.     

Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations

Consequences of the connection between nonlinear Fokker-Planck equations and entropic forms are investigated. A particular emphasis is given to the feature that different nonlinear Fokker-Planck equations can be arranged into classes associated with the same entropic form and its corresponding stationary state. Through numerical integration, the time evolution of the solution of nonlinear Fokker-Planck equations related to the Boltzmann-Gibbs and Tsallis entropies are analyzed. The time behavior in both stages, in a time much smaller than the one required for reaching the stationary state, as well as towards the relaxation to the stationary state, are of particular interest. In the former case, by using the concept of classes of nonlinear Fokker-Planck equations, a rich variety of physical behavior may be found, with some curious situations, like an anomalous diffusion within the class related to the Boltzmann-Gibbs entropy, as well as a normal diffusion within the class of equations related to Tsallis’ entropy. In addition to that, the relaxation towards the stationary state may present a behavior different from most of the systems studied in the literature.     

Some New Features of Linear Theory for Describing Non-linear Diffusion

The non-linear flux equation, the non-linear Fokker-Planck equation (or Smoluchowski equation), and the non-linear Langiven equation are the basicequations for describing particle diffusion in non-ideal system subjected totime-dependent external fields. Nevertheless, the exact solution of thoseequations is still a challenge because of their inherent complexity of thenon-linear mathematics. Li et al. found that, based on the defined apparentvariables, the non-linear Fokker-Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or local equilibrium assumption. In this paper, some new features of the theory were found: (i) The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; (ii) The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker-Planck equation, can be derived more consistently from the microscopic molecular statistical theory; (iii) In the theory, thenon-linear Langiven equation also bears a linear form; (iv) For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.