Alkis Grecos

On the Limits of Validity of the Low Magnetic Reynolds Number Approximation in MHD Natural-Convection Heat Transfer

In the majority of magnetohydrodynamic (MHD) natural-convection simulations, the Lorentz force due to the magnetic field is suppressed into a damping term resisting the fluid motion. The primary benefit of this hypothesis, commonly called the low-R m approximation, is a considerable reduction of the number of equations required to be solved. The limitations in predicting the flow and heat transfer characteristics and the related errors of this approximation are the subject of the present study. Results corresponding to numerical solutions of the full MHD equations, as the magnetic Reynolds number decreases to a value of 10−3, are compared with those of the low-R m approximation. The influence of the most important parameters of MHD natural-convection problems (such as the Grashof, Hartmann, and Prandtl numbers) are discussed according to the magnetic model used. The natural-convection heat transfer in a square enclosure heated laterally, and subject to a transverse uniform magnetic field, is chosen as a case study. The results show clearly an increasing difference between the solutions of the full MHD equations and low-R m approximation with increasing Hartmann number. This difference decreases for higher Grashof numbers, while for Prandtl numbers reaching lower values like those of liquid metals, the difference increases.

Some formal aspects of subdynamics

The theory of subdynamics is formulated assuming the existence of a spectral representation of the collision operator. This approach avoids perturbation schemes; however the presentation is formal. It may be used to develop further the theory as well as a starting point for a rigorous mathematical discussion. The construction of the operators introduced in the theory of subdynamics is presented in detail. Some questions related to the transformation theory leading to the so-called “physical representation” are briefly discussed.

On the relation of dynamics to statistical mechanics

A new conceptual framework for the foundations of statistical mechanics starting from dynamics is presented. It is based on the classification and the study of invariants in terms of the concepts of our formulation of non-equilibrium statistical mechanics. A central role is played by thecollision operator. The asymptotic behaviour of a class of states is determined by the collisional invariants independently of the ergodicity of the system. For this class of states we have an approach to thermodynamical equilibrium.We discuss the existence of classes of states which approach equilibrium. The complex microstructure of the phase space, as expressed by the weak stability concept which was introduced by Moser and others, plays here an essential role. The formalism that we develop is meaningful whenever the “dissipativity condition” for the collision operator is satisfied. Assuming the possibility of a weak coupling approximation, this is in fact true whenever Poincarés theorem on the nonexistence of uniform invariants holds. In this respect, our formalism applies to few body problems and no transition to the thermodynamic limit is required.Our approach leads naturally to a ‘classical theory of measurement’. In particular a precise meaning can now be given to ‘thermodynamic variables’ or to ‘macrovariables’ corresponding to a measurement in classical dynamics.

Kinetic theory in the weak-coupling approximation

Abstract 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.

Classical Markovian kinetic equations: Explicit form and H theorem

The probabilistic description of finite classical systems often leads to linear kinetic equations. A set of physically motivated mathematical requirements is accordingly formulated. We show that it necessarily implies that solutions of such a kinetic equation in the Heisenberg representation, define Markov semigroups on the space of observables. Moreover, a general H-theorem for the adjoint of such semigroups is formulated and proved provided that at least locally, an invariant measure exists. Under a certain continuity assumption, the Markov semigroup property is sufficient for a linear kinetic equation to be a second order differential equation with nonegative-definite leading coefficient. Conversely it is shown that such equations define Markov semigroups satisfying an H-theorem, provided there exists a nonnegative equilibrium solution for their formal adjoint.

On the collision operator in an exactly soluble model

Abstract We study the properties of the collision operator, as defined by the Brussels school, for systems whose hamiltonian is a cyclic matrix (finite case) or a Laurent matrix (infinite case). Explicit calculations in particular examples permit us to illustrate the difference in the behaviour of this operator when long-range interactions are present.

Plasma diffusion and relaxation in a magnetic field

A Fokker-Planck-type kinetic equation modeling a test-particle weakly interacting with an electrostatic plasma, in the presence of a magnetic field B, is analytically solved. Explicit expressions are obtained for variable moments and particle density as a function of time. Diffusion in space is classical, characterized by a particle MSD varying asB � 2 , in agreement with previous results. The standard kinetic-theoretical treatment of electrostatic plasmas is based on Landau-type equations (1), describing the evolution of a distribution function (df) f ðv; tÞ in velocity space. This description needs to be modified in case of a non-uniform df and/or in the presence of an external force field. In previous work (2), a Fokker-Planck-type kinetic equation (FPE) was derived from first principles for a test-particle (charge q, mass m) weakly interacting with a plasma embedded in a uniform magnetic field B. This equation, describing the evolution of the df f ðx; v; tÞ in phase space C ¼f x; vg, has the form:

Stochastic modelling of turbulence and anomalous transport in plasmas

Stochastic modelling of turbulence is considered. Exact and approximate solution procedures based on projection techniques are briefly reviewed. Procedures leading to equations that are local in time are discussed. Two different approximations of this type are presented, whose performances are tested and compared on a simple model case.

Generalized Moyal structures in phase-space, master equations and their classical limit II. Applications to harmonic oscillator models

Abstract The formalism of the phase-space representation of quantum master equations via generalized Wigner transformations developed in a previous paper, is applied to the Lindblad-type kinetic equation, for a quantum harmonic oscillator coupled to an equilibrium bath of oscillators. The resulting equation is derived without introducing the rotating-wave approximation. In the classical limit, the equation reduces to a Fokker–Planck equation, which coincides with the one derived from the corresponding classical Hamiltonian. The formalism is also applied to other oscillator model equations often used in quantum optics.

Generalized Moyal structures in phase-space, master equations and their classical limit

Abstract Generalized Wigner and Weyl transformations of quantum operators are defined and their properties, as well as those of the algebraic structure induced on phase-space, are reviewed. Using such transformations, quantum linear evolution equations are given a phase-space representation; particularly, the general master equation of the Lindblad type generating quantum dynamical semigroups. The resulting expressions are better suited for deriving quantum corrections, taking the classical limit and for a general comparison of classical and quantum systems. We show that under quite general conditions, the classical limit of this master equation exists, is independent of the particular generalized Wigner transformation used and is an equation of the Fokker–Planck type (i.e. with nonnegative-definite leading coefficient) generating a classical Markov semigroup.