Onsager–Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction IV. Rotating systems

The statistical approach to the nonequilibrium thermodynamics presented in Part I of this paper was generalized to rotating gaseous systems. Basing on the Boltzmann equation written for rotating reference frame the kinetic coefficients satisfying the Onsager–Casimir reciprocity relations are obtained in a general form. An example of the application of this theory is given. Some new phenomena arising only in rotating system are found.     

Reciprocal relations based on the non-stationary Boltzmann equation

The reciprocal relations for open gaseous systems are obtained on the basis of main properties of the non-stationary Boltzmann equation and gas-surface interaction law. It is shown that the main principles to derive the kinetic coefficients satisfying the reciprocal relations remain the same as those used for time-independent gaseous systems [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law single gas, Phys. Rev. 73 (2006) 026110]. First, the kinetic coefficients are obtained from the entropy production expression; then it is proved that the coefficient matrix calculated for time reversed source functions is symmetric. The proof is based on the reversibility of the gas-gas and gas-surface interactions. Three examples of applications of the present theory are given. None of these examples can be treated in the frame of the classical Onsager-Casimir reciprocal relations, which are valid only in a particular case, when the kinetic coefficients are odd or even with respect to the time reversion. The approach is generalized for gaseous mixtures.     

Onsager-Casimir reciprocity relations for a mixture of rarefied gases interacting with a laser radiation

The behavior of a mixture of optically excitable and inactive gases in the field of a laser radiation is considered from the viewpoint of nonequilibrium thermodynamics. The kinetic coefficients satisfying the Onsager-Casimir reciprocity relations are found from general properties of the Boltzmann equation, boundary condition, and terms describing the gas-radiation interaction. Various kinetic phenomena induced by the laser radiation are coupled with corresponding cross effects.     

The Onsager-Casimir relations revisited

We study the fate of the Onsager-Casimir reciprocity relations for a continuous system when some of its variables are eliminated adiabatically. Just as for discrete systems, deviations appear in correction terms to the reduced evolution equation that are of higher order in the time scale ratio. The deviations are not removed by including correction terms to the coarse-grained thermodynamic potential. However, via a reformulation of the theory, in which the central role of the thermodynamic potential is taken over by an associated Lagrangian-type expression, we arrive at a modified form of the Onsager-Casimir relations that survives the adiabatic elimination procedure. There is a simple relation between the time evolution of the redefined thermodynamic forces and that of the basic thermodynamic variables; this relation also survives the adiabatic elimination. The formalism is illustrated by explicit calculations for the Klein-Kramers equation, which describes the phase space distribution of Brownian particles, and for the corrected Smoluchowski equation derived from it by adiabatic elimination of the velocity variable. The symmetry relation for the latter leads to a simple proof that the reality of the eigenvalues of the simple Smoluchowski equation is not destroyed by the addition of higher order corrections, at least not within the framework of a formal perturbation expansion in the time scale ratio.     

Measure of irreversibility and entropy production in open quantum systems

A new concept of a measure of irreversibility for quantum dynamics in open systems is introduced as a suitably regularized substitute for the common notion of entropy production, which, unfortunately, yields infinite values for so many irreversible processes of physical relevance.     

Conversion between kinetic energy and potential energy in the classical nonlocal Boltzmann equation

An important property of the classical Boltzmann equation is that kinetic energy is conserved. This is closely connected to the fact that the Boltzmann equation describes the nonequilibrium properties of an ideal gas. Generalizations of the Boltzmann equation to higher density involve, among other things, allowing the colliding particles to be at different positions. This spatial nonlocality is known to contribute to the density corrections of gas transport properties. For soft potentials such a spatial separation of the particles also leads to a conversion between kinetic and potential energy. In evaluating these effects the classical dynamics of the whole collision trajectory must be taken into account, involving also the time for the collision process. The resulting time nonlocality has usually been reinterpreted in terms of a spatial nonlocality. However, for a homogeneous system this is not possible and only the time nonlocality remains, this then being responsible for the conversion between kinetic and potential energy. This paper aims to clarify these properties of the nonlocal corrections to the classical mechanical Boltzmann collision term. Comments on the corresponding problem for the quantum Boltzmann equation are also made.     

Transport phenomena in a plane shock wave

With the use of the nonpolynomial closure 1/ _z in the Mott-Smith approximation of the solution of the Boltzmann equation, we obtain a value of the density gradient in the limit of a very weak shock wave that is close to the correct value. For the determination of the transverse temperature gradient we calculated the _x ² / _z moment of the Mott-Smith collision integral. The effective values of viscosity and thermal conductivity in the limit of a very weak shock wave were calculated for inverse-power potentials and found to agree almost exactly with the Chapman-Enskog values. Such a comparison can serve as a criterion for the evaluation of different bimodal theories. Various bimodal theories give different values of viscosity and thermal conductivity, but all of them give 33 % too high a value of the Eucken ratio.     

Quantitative complementarity relations in bipartite systems: Entanglement as a physical reality

We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a “triality” relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell’s inequality, arises between single- and bipartite properties.     

On the theory of fluctuations in continuum systems

The probability of occurrence of fluctuations around nonequilibrium steady states is discussed from a kinetic viewpoint. It is shown that in a large class of continuous media it is possible to extend the thermodynamic theory of fluctuations, provided one uses suitable steady-state parameters rather than equilibrium quantities.     

The time-local view of nonequilibrium statistical mechanics. I. Linear theory of transport and relaxation

The various approaches to nonequilibrium statistical mechanics may be subdivided into convolution and convolutionless (time-local) ones. While the former, put forward by Zwanzig, Mori, and others, are used most commonly, the latter are less well developed, but have proven very useful in recent applications. The aim of the present series of papers is to develop the time-local picture (TLP) of nonequilibrium statistical mechanics on a new footing and to consider its physical implications for topics such as the formulation of irreversible thermodynamics. The most natural approach to TLP is seen to derive from the Fourier-Laplace transform ) of pertinent time correlation functions, which on the physical sheet typically displays an essential singularity at z= and a number of macroscopic and microscopic poles in the lower half-plane corresponding to long- and short-lived modes, respectively, the former giving rise to the autonomous macrodynamics, whereas the latter are interpreted as doorway modes mediating the transfer of information from relevant to irrelevant channels. Possible implications of this doorway mode concept for socalled extended irreversible thermodynamics are briefly discussed. The pole structure is used for deriving new kinds of generalized Green-Kubo relations expressing macroscopic quantities, transport coefficients, e.g., by contour integrals over current-current correlation functions obeying Hamiltonian dynamics, the contour integration replacing projection. The conventional Green-Kubo relations valid for conserved quantities only are rederived for illustration. Moreover, may be expressed by a Laurent series expansion in positive and negative powers ofz, from which a rigorous, general, and straightforward method is developed for extracting all macroscopic quantities from so-called secularly divergent expansions of as obtained from the application of conventional many-body techniques to the calculation of . The expressions are formulated as time scale expansions, which should rapidly converge if macroscopic and microscopic time scales are sufficiently well separated, i.e., if lifetime (memory) effects are not too large.     

The equivalence between Forward and Backward Boltzmann Equations in multi-component medium

The author generalized the propagator function theory introduced first by Sigmund, and gave an explicitly proof of an equivalence between Forward and Backward Boltzmann Equations in a multi-component medium by using the generalized propagator function theory.     

The runaway effect in a Lorentz gas

The Lorentz gas of charged particles in a constant and uniform electric field is studied. The gas flows through the medium of immobile, randomly distributed scatterers. Particles with velocity v suffer collisions with frequency proportional to ¦v¦ ⁿ . Forn < 0 runaway of the gas is forced by the field: the mean velocity of the flow increases without bounds. By a simple physical argument an integral relation is established between the probability of collisionless motion and the velocity distribution. It is then shown that whenn < –1 a fraction of particles moves as if the scattering centers were absent. The detailed discussion of this uncollided runaway is presented. Some qualitative features of the velocity distribution are illustrated on rigorous solutions in one dimension.     

A computational model for direction relations between spatial objects in GIS

Direction relations between spatial objects have been a research focus in different fields like spatial reasoning, spatial query, GIS, CAD/CAM, and AI. A novel model named Radial model is proposed to obtain both quantitative and qualitative direction relations based on the principle of the ray transmitting as a straight line. The framework of the proposed model is composed of reference point, reference, and target object. Theoretically, we group direction relations into three cases (case I: point/point; case II: point/line and point/area; case III: line/line, line/area and area/area) for calculation based on point-set theory. Additionally, we employ the model in calculating direction relations on the vector and raster spatial data model in GIS.     

Nonlinear competition between asters and stripes in filament-motor systems

A model for polar filaments interacting via molecular motor complexes is investigated which exhibits bifurcations to spatial patterns. It is shown that the homogeneous distribution of filaments, such as actin or microtubules, may become either unstable with respect to an orientational instability of a finite wave number or with respect to modulations of the filament density, where long-wavelength modes are amplified as well. Above threshold nonlinear interactions select either stripe patterns or periodic asters. The existence and stability ranges of each pattern close to threshold are predicted in terms of a weakly nonlinear perturbation analysis, which is confirmed by numerical simulations of the basic model equations. The two relevant parameters determining the bifurcation scenario of the model can be related to the concentrations of the active molecular motors and of the filaments, respectively, which both could be easily regulated by the cell.     

The onset of instabilities in nonequilibrium systems

The nonlinear master equation previously proposed by Malek-Mansour and Nicolis is applied to the analysis of unstable transitions leading to temporally or spatially organized patterns. Thecorrelation length of the destabilizing fluctuations is determined, and a number of striking analogies with equilibrium phase transitions are pointed out.     

The relations between agent performances and their intellective abilities in mix-games

This paper studies the relations between agent performances and their intellective abilities in mix-games in which there are two groups of agents: one group plays a minority game and the other plays a majority game. These two groups have different historical memories and different time horizons. It is found that these relations are greatly influenced by the configurations of historical memories of the two groups.     

The boundary value problem in fermion systems

The half-space boundary value problem for fermions near zero temperature in plane geometry is solved for diffuse boundary scattering by numerically constructing the spatial propagator in terms of the eigenfunctions of a generalized eigenvalue problem for the linearized Uehling-Uhlenbeck collision integral. The slip length is calculated for several interparticle scattering laws and compared with a relaxation time ansatz result and the experimental values for normal fluid³He. It is shown that the nonsingular part of the collision operator is relatively compact to the singular part.