Stoßkoeffizienten im Plasma und Gleichgewichtsbetrachtungen

The relation between the collision coeffizients of a reaction and the inverse reaction is calculated for nonequilibrium processes for any kind of distribution functions and cross sections. In the case of equilibrium, this relation equals the Maxwell-Boltzmann factor for transitions between discrete levels, and the Saha expression for transitions between free and discrete levels. In case of nonequilibrium, the relation is a middlevalue expression, depending on the distribution function of electrons. If this is Maxwellian, the middle value expression does not depend on cross sections. For high electron densities data of the plasma approach to equilibrium values though radiation processes are still efficient. The reasons are discussed.     

The Modified Group Expansions for Construction of Solutions to the BBGKY Hierarchy

A solution to the BBGKY hierarchy for nonequilibrium distribution functions is obtained within modified boundary conditions. The boundary conditions take into account explicitly both the nonequilibrium one-particle distribution function as well as local conservation laws. As a result, modified group expansions are proposed. On the basis of these expansions, a generalized kinetic equation for hard spheres and a generalized Bogolubov–Lenard–Balescu kinetic equation for a dense electron gas are derived within the polarization approximation.     

Heating of two-dimensional excitons by nonequilibrium acoustic phonons

The energy distribution functions of two-dimensional excitons in the presence of nonequilibrium acoustic phonons have been calculated for the geometry used in heat-pulse experiments. The results were obtained by solving numerically the kinetic equation for the case where the exciton gas equilibrates with phonons during its lifetime. The cases of the low and high exciton-gas density limits are considered. It is shown that at low exciton-gas densities the distribution does not follow the Boltzmann function and depends on the quantum-well width. A comparison with earlier experimental data is made. Fiz. Tverd. Tela (St. Petersburg) 41, 1707–1711 (September 1999)     

Effect of normal phonon-phonon scattering on the mutual electron-phonon drag and kinetic phenomena in degenerate conductors

The effect of normal phonon-phonon scattering processes on momentum relaxation in a nonequilibrium electron-phonon system is considered. A system of rate equations for the electron and phonon distribution functions has been solved with the inclusion of mutual electron-phonon drag. The kinetic coefficients of conductors have been calculated in a linear approximation in the degeneracy parameter. The effect of normal phonon-phonon scattering processes on the electron-phonon drag and on the kinetic phenomena in conductors with degenerate carrier statistics is analyzed.     

Stochastic dynamics of a bistable reaction system

The method of stochastic dynamics is used for the study of a discrete trimolecular chemical model with internal fluctuations. Several trajectories of the phase point are simulated for the bistable Schlögl model at small volumes. The influence due to the discontinuity of the process on the stationary probability distribution function is analyzed. The transition to the bimodal distribution is shown to be a cusp catastrophe. The location of the cuspoid strongly depends on the reaction volume. Autocorrelation functions are calculated numerically for short and large time scales.     

On a kinetic justification of the generalized nonequilibrium thermodynamics of multicomponent systems

The problem of the kinetic justification of the generalized thermodynamics of nonequilibrium processes using the method of moments for solving the kinetic equation for a multicomponent gas mixture is examined. Generalized expressions are obtained for the entropy density, entropy flux density, and entropy production as functions of an arbitrary number of state variables (moments of the distribution function). Different variants of writing the relations between fluxes and thermodynamic forces are considered, which correspond to the Onsager version for spatially homogeneous systems and, in a more general case, lead to the generalized thermodynamic forces of a complicated form, including derivatives of the fluxes with respect to time and spatial coordinates. Some consequences and new physical effects, following from the obtained equations, are analyzed. It is shown that a transition from results of the method of moments to expressions for the entropy production and the corresponding phenomenological relations of the generalized nonequilibrium thermodynamics is possible on the level of a linearized Barnett approximation of the Chapman–Enskog method.     

Time-displaced correlation functions in an infinite one-dimensional mixture of hard rods with different diameters

Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Perçus, and Sykes, removing some nontypical features of that system.Supported by NSF grant No. MCS 75-21684 A01 (M. A.), NSF grant No. MPS 75-20638 (J. L.), and USAFOSR grant No. 73-2430 B (J. M.)John Guggenheim Fellow on sabbatical leave from Belfer Graduate School of Science, Yeshiva University, New York.     

Initial state dependence of nonlinear kinetic equations: The classical electron gas

The method of nonequilibrium cluster expansion is used to stydy the decay to equilibrium of a weakly coupled inhomogeneous electron gas prepared in a local equilibrium state at the initial time,t=0. A nonlinear kinetic equation describing the long time behavior of the one-particle distribution function is obtained. For consistency, initial correlations have to be taken into account. The resulting kinetic equation-differs from that obtained when the initial state of the system is assumed to be factorized in a product of one-particle functions. The question of to what extent correlations in the initial state play an essential role in determining the form of the kinetic equation at long times is discussed. To that end, the present calculations are compared with results obtained before for hard sphere gases and in general gases with strong short-range forces. A partial answer is proposed and some open questions are indicated.     

Distribution function and electron state density in nonequilibrium plasma created by dye laser

Ultracold nonequilibrium plasma created by a dye laser has been studied by the molecular dynamics method. Electrons and protons in this model of nonequilibrium plasma interacted according to the Coulomb law. In the case of electron-proton interaction and a distance between particles r < a ₀ (Bohr radius), the interaction energy is constant, e ²/a ₀ (e is the charge of electron). An initial proton kinetic energy is set randomly so that the average kinetic energy is 0.01–1 K. Initial full electron energy is also set randomly, but at the same time it is positive; i.e., all the electrons according to our task are located in the continuous spectrum. Average kinetic electron energy per one particle varies from 1 to 50 K. The motion equations in periodical boundary condition for this system have been solved by molecular dynamics method. We have calculated the distribution function in the region near the ionization threshold. The distribution function is being described using electron state density in the nearest neighbor approximation with activity correction.     

Nonlinear transport in the Boltzmann limit

Formal expressions for the irreversible fluxes of a simple fluid are obtained as functionals of the thermodynamic forces and local equilibrium time correlation functions. The Boltzmann limit of the correlation functions is shown to yield expressions for the irreversible fluxes equivalent to those obtained from the nonlinear Boltzmann kinetic equation. Specifically, for states near equilibrium, the fluxes may be formally expanded in powers of the thermodynamic gradients and the associated transport coefficients identified as integrals of time correlation functions. It is proved explicitly through nonlinear Burnett order that the time correlation function expressions for these transport coefficients agree with those of the Chapman-Enskog expansion of the nonlinear Boltzmann equation. For states far from equilibrium the local equilibrium time correlation functions are determined in the Boltzmann limit and a similar equivalence to the Boltzmann equation solution is established. Other formal representations of the fluxes are indicated; in particular, a projection operator form and its Boltzmann limit are discussed. As an example, the nonequilibrium correlation functions for steady shear flow are calculated exactly in the Boltzmann limit for Maxwell molecules.Research supported in part by NSF grant PHY 76-21453.     

Normal quasiparticle scattering and kinetic effects in degenerate semiconductors

The influence of normal processes of electron-electron and phonon-phonon scattering on quasiparticle momentum relaxation in nonequilibrium electron-phonon systems of degenerate semiconductors is investigated. A system of kinetic equations is solved for the electron and phonon distribution functions, and the kinetic coefficients of a semiconductor are calculated in the linear approximation in the degeneracy parameter. The influence of normal scattering of quasiparticles on the electrical conductivity, thermopower, and heat conductivity of a degenerate semiconductor is analyzed. Redistribution of the phonon momentum in N processes within each branch of the vibrational spectrum, as well as among different branches, is taken into account.     

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.     

Elastic scattering of acoustic phonons on point mass defects

The problem of eigenvalues of the collision operator for a gas of acoustic phonons scattered by point mass defects of small concentration embedded in transversely isotropic media is considered. For this purpose the properties of solution of the Boltzmann-Peierls kinetic equation for spatially homogeneous states are studied. An analytic formula for the Laplace transform of the distribution function is obtained. The singularities of this Laplace transform and the initial distribution function determine the dependence of this distribution function on time. For several hexagonal materials characteristics of the singularity set are calculated. Usually the singularity set consists of a continuous part and four discrete points. However, there exist elastic hexagonal materials (⁴He, Cd, Ta, Zn) for which some of discrete points are absent. For some materials (e.g. Zr) the continuous part is very narrow.     

Non-Markovian Quantum Kinetics and Conservation Laws

A link between memory effects in quantum kinetic equations and nonequilibrium correlations associated with the energy conservation is investigated. In order that the energy be conserved by an approximate collision integral, the one-particle distribution function and the mean interaction energy are treated as independent nonequilibrium state parameters. The density operator method is used to derive a kinetic equation in second-order non-Markovian Born approximation and an evolution equation for the nonequilibrium quasi-temperature which is thermodynamically conjugated to the mean interaction energy. The kinetic equation contains a correlation contribution which exactly cancels the collision term in thermal equilibrium and ensures the energy conservation in nonequilibrium states. Explicit expressions for the entropy production in the non-Markovian regime and the time-dependent correlation energy are obtained.     

Influence of the energy distribution function on thermodesorption

A kinetic model of the desorption process based on the energy distribution function was set up. The influence of known distribution functions, such as those of Dirac, Gauss, Cauchy. Weibull and others, on the kinetics of thermodesorption was investigated. In those cases the energy distribution function is a kinetic parameter. It was shown that the shape and position of the thermodesorption peaks depend on the type of energy distribution function. On the basis of the theoretical kinetic model the dependencies of differential desorption heat upon degree of coverage were calculated. It was shown that the differential desorption heat also depends on the energy distribution function.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.     

Investigation of a nonequilibrium electron subsystem in low-temperature microwave detectors

The electron energy distributions arising in small-size metal absorbers of microwave radiation detectors operating at ultralow temperatures have been calculated using the kinetic equation. It is shown that the electron distributions are nonequilibrium, significantly different from the Fermi distribution, and are determined by the ratio of the rates of electron-electron and electron-phonon relaxation and by the effect of the measuring element (superconductor-insulator-normal metal junction) on the absorber. The response of such a bolometer is calculated and compared to the experimental data.     

Discrete ellipsoidal statistical BGK model and Burnett equations

A new discrete Boltzmann model, the discrete ellipsoidal statistical Bhatnagar–Gross–Krook (ESBGK) model, is proposed to simulate nonequilibrium compressible flows. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in the Burnett level, two kinds of discrete velocity model are introduced and the relations between nonequilibrium quantities and the viscous stress and heat flux in the Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, those based on the Navier–Stokes or the Burnett equations.     

Exchange Fluctuations in a Nonequilibrium Lorentz Gas

It has been shown that a correlation mechanism that is based on the exchange interaction and destroys the relation between distribution functions and response (Price relation) occurs in a nonequilibrium Lorentz gas (particles interact only with the thermostat). The physical nature of this phenomenon is that the scattering of particles of the gas in the same state on a single particle of the thermostat creates a flux of correlated pairs, which depends on the form of a nonequilibrium distribution function, making impossible the existence of a universal relation between distribution functions and response.     

Transport Theory and Collective Modes. I. The Case of Moderately Dense Gases

The complex spectral representation of the Liouville operator introduced by Prigogine and others is applied to moderately dense gases interacting through hard-core potentials in arbitrary d-dimensional spaces. Kinetic equations near equilibrium are constructed in each subspace as introduced in the spectral decomposition for collective, renormalized reduced distribution functions. Our renormalization is a nonequilibrium effect, as the renormalization effect disappears at equilibrium. It is remarkable that our renormalized functions strictly obey well-defined Markovian kinetic equations for all d, even though the ordinary distribution functions obey nonMarkovian equations with memory effects. One can now define transport coefficients associated to the collective modes for all dimensional systems including d = 2. Our formulation hence provides a microscopic meaning of the macroscopic transport theory. Moreover, this gives an answer to the long-standing question whether or not transport equations exist in two-dimensional systems. The non-Markovian effects for the ordinary distribution function, such as the long-time tails for arbitrary n-mode coupling, are estimated by superposition of the Markovian evolutions of the dressed distribution functions.