Non-commutative Geometry and Kinetic Theory of Open Systems

The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space $M$, they are at most of the 2nd order. For open systems interacting with a bath at canonical equilibrium they have a particular form of an equation of a generalized Fokker-Planck type. We show that it is possible to obtain them as Liouville equations of Hamiltonian dynamics on $M$ with a particular non-commutative differential structure, provided certain geometric in character, conditions are fulfilled. To this end, symplectic geometry on $M$ is developped in this context, and an outline of the required tensor analysis and differential geometry is given. Certain questions for the possible mathematical interpretation of this structure are also discussed.Comment: 22 pages, LaTe

Dissipative Properties of Quantum Systems

We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large “volume,” becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration

Kinetic theory and ergodic properties

It is often assumed that the justification of kinetic theory lies in ergodic theory. From the properties of the collision operator, which plays a basic role in our kinetic description of dynamical systems, we show that this is not the case. We deduce that the asymptotic behavior of a class of states and observables is determined by the collisional invariants, independently of the ergodicity of the system. The relation between our conclusion and the stability concepts for classical Hamiltonian systems, introduced by Moser and others, is briefly indicated

MHD natural convection in a laterally and volumetrically heated square cavity

A numerical study is presented of unsteady two-dimensional natural convection of an electrically conducting fluid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. The flow is characterized by the external Rayleigh number, Ra-E, determined from the temperature difference of the side walls, the internal Rayleigh number, Raj, determined from the volumetric heat rate, and the Hartmann number, Ha, deter-mined from the strength of the imposed magnetic field. Starting from given values of Ra-E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra-1/Ra-E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multicellular flow up to a threshold value, Ra-1,Ra-cr, Of the internal Rayleigh number depending on Ha. Oscillatory multicellular flow fields were observed for S values up to 100 for the range 10(5)-10(6) of Ra-E studied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Hartmann number is not found to be straightforward connected with the resulting flow patterns in the present flow configuration. (c) 2005 Elsevier Ltd. All rights reserved

Magnetohydrodynamic natural convection in a vertical cylindrical cavity with sinusoidal upper wall temperature

A series of numerical simulations were performed in order to study liquid metal MHD natural convection in a vertical cylindrical container with a sinusoidal temperature distribution at the upper wall and the other surfaces being adiabatic. Starting from the basic hydrodynamic case, the effect of vertical (axial) and horizontal magnetic fields is assessed. Depending on the magnitude of the Rayleigh and Hartmann numbers, both turbulent and laminar (azimuthally symmetric or not) flows are observed. The results show that the increase of Rayleigh number promotes heat transfer by convection while the increase of Hartmann number favors heat conduction. The vertical magnetic field reduces the Nusselt number more than the horizontal. The circulation patterns for the most convective cases are confined close to the top corner of the container with the simultaneous formation of a secondary flow pattern at the bottom corner, while for the more conductive cases only one circulation pattern exists covering the entire domain. (c) 2008 Elsevier Ltd. All rights reserved

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