The isothermal local resistivity and its connection to different theories

The purpose of the present paper is to discuss the theory of the isothermal local resistivity in the sense of linear response. Different methods, as the Langevin equation, the non-equilibrium density operator technique and the linear response theory of conduction, have been related with each other to clear up different ambiguities in the literature.The first two sections are devoted to introduce the hydrodynamic and linear response equations for the electron gas in a medium of scattering mechanisms (phonons, impurities, etc.). The inversion of the conductivity formula into the isothermal local resistivity is performed with help of a generalized Langevin equation in the isothermal limit (lim_{q → 0} lim_{ω → 0}A_qω). This result agrees with that of the non-equilibrium operator technique. Then the many-variable projection technique of Mori is used to establish the relations between microscopic theory of electrical conduction and the hydrodynamic equations. The relaxation matrix formulation of Fermi-liquid in a metal can describe sound wave propagation in the Fermi-liquid which corresponds to charge density waves. Further, the relation between the isothermal local resistivity and Köhler's variation principle is established for electron-phonon system on a general way, which allows one to make contact with the Boltzmann equation.In the one-electron approximation the isothermal local resistivity is discussed in terms of phase shifts of non-overlapping scatterers. The result is valid for a dense system of resonant scatterers.     

Relaxation and collective motion in superconductors: a two-fluid description

A simple, unified discussion of branch imbalance and gap relaxation in superconductors is presented. Both phenomena are treated within the framework of the ordinary Boltzmann equation, supplemented by the BCS gap equation. We show that the physics of the process commonly referred to as quasiparticle branch imbalance relaxation may be understood simply if one introduces a two-fluid model for the charge in the superconductor, and regards the process as one in which charge associated with the normal component is converted into charge associated with the superfluid. We derive in detail the exact solutions of the Boltzmann equation, which are valid near T_c, and allow for the effects of anisotropy. We discuss the comparison between relaxation rates measured in the superfluid and those obtained from normal state measurements and calculations. We then derive a set of two-fluid hydrodynamic equations based on the two-fluid model for the charge, and find that the current of charge associated with the normal component is not in general equal to the usual normal current. On the basis of these equations we derive expressions for the characteristic quasiparticle diffusion length near phase slip centers, and for the frequency of the recently observed collective mode. We compare our result with those of both microscopic and phenomenological calculations.     

A Model of Heat Conduction

In this paper, we first define a deterministic particle model for heat conduction. It consists of a chain of N identical subsystems, each of which contains a scatterer and with particles moving among these scatterers. Based on this model, we then derive heuristically, in the limit of N → ∞ and decreasing scattering cross-section, a Boltzmann equation for this limiting system. This derivation is obtained by a closure argument based on memory loss between collisions. We then prove that the Boltzmann equation has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

The dynamical-quantization approach to open quantum systems

The dynamical-quantization approach to open quantum systems does consist in quantizing the Brownian motion starting directly from its stochastic dynamics under the framework of both Langevin and Fokker–Planck equations, without alluding to any model Hamiltonian. On the ground of this non-Hamiltonian quantization method, we can derive a non-Markovian Caldeira–Leggett quantum master equation as well as a non-Markovian quantum Smoluchowski equation. The former is solved for the case of a quantum Brownian particle in a gravitational field whilst the latter for a harmonic oscillator. In both physical situations, we come up with the existence of a non-equilibrium thermal quantum force and investigate its classical limit at high temperatures as well as its quantum limit at zero temperature. Further, as a physical application of our quantum Smoluchowski equation, we take up the tunneling phenomenon of a non-inertial quantum Brownian particle over a potential barrier. Lastly, we wish to point out, corroborating conclusions reached in our previous paper [A. O. Bolivar, Ann. Phys. 326 (2011) 1354], that the theoretical predictions in the present article uphold the view that our non-Hamiltonian quantum mechanics is able to capture novel features inherent in quantum Brownian motion, thereby overcoming shortcomings underlying the Caldeira–Leggett Hamiltonian model.     

q-Oscillator Algebra And d-Orthogonal Polynomials

In this paper we express the matrix coefficients of the Fock representation of a q-oscillator algebra in terms of the d-orthogonal Al-Salam Carlitz polynomials. Also, we derive a generating functions, recurrence relations and q-difference equations for these d-orthogonal polynomials.     

Derivation of non-Markovian transport equations for trapped cold atoms in nonequilibrium thermal field theory

The non-Markovian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of nonequilibrium thermal field theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markovian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markovian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability.     

Longitudinal relaxation spectra of the transverse Ising model close to Tc

The correlative effects of the nature of the interaction and of the method of calculation on the shape of the longitudinal relaxation function (LRF) for the transverse Ising model are analysed. The LRF is calculated in two ways: (i) its continued fraction representation within the three pole approximation (TPA); and (ii) the resolution of kinetic equations derived for the correlation functions beyond the random phase approximation (RPA). The effects of the nature of the interaction on the LRF spectral characteristics are investigated using an interaction made of three variable contributions: uniaxial dipolar, isotropic infinite range and anisotropic nearest-neighbour interactions. Contrary to the TPA, the kinetic-equation-method (KEM) leads to LRF's exhibiting a three peak structure for every q-value except q = 0 (q = 0_⊥ if the interaction is of dipolar nature) whatever the interaction. The approximations underlying both methods are specified and discussed. Comments on recent neutron scattering experiments on Li Tb_pY_1-pF₄ by Youngblood et al. are made.     

Entropy Function Approach to the Lattice Boltzmann Method

In this paper, we present the construction of the Lattice Boltzmann method equipped with the H-theorem. Based on entropy functions whose local equilibria are suitable to recover the Navier–Stokes equations in the framework of the Lattice Boltzmann method, we derive a collision integral which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium. We discuss performance of this approach as compared to the standard realizations.     

Free <Emphasis Type="Italic">q</Emphasis>-deformed relativistic wave equations by representation theory

In a representation theoretic approach a free q-relativistic wave equation must have the property that the space of solutions is an irreducible representation of the q-Poincaré algebra. It is shown how this requirement uniquely determines the q-wave equations. As examples, the q-Dirac equation (including q-gamma matrices which satisfy a q-Clifford algebra), the q-Weyl equations, and the q-Maxwell equations are computed explicitly.Received: 8 May 2002, Revised: 14 July 2003, Published online: 29 August 2003     

BGK and Fokker-Planck Models of the Boltzmann Equation for Gases with Discrete Levels of Vibrational Energy

We propose two models of the Boltzmann equation (BGK and Fokker-Planck models) for rarefied flows of diatomic gases in vibrational non-equilibrium. These models take into account the discrete repartition of vibration energy modes, which is required for high temperature flows, like for atmospheric re-entry problems. We prove that these models satisfy conservation and entropy properties (H-theorem), and we derive their corresponding compressible Navier–Stokes asymptotics.

     

Investigation of Noise-Induced Escape Rate: A Quantum Mechanical Approach

A quantum system coupled to a heat-bath in non-equilibrium environment is considered to study the problem of noise-induced escape rate from a metastable state in the moderate to strong friction limit (Kramers’ regime). It is known that starting from an initial coherent state representation of bath oscillators, one can derive a c-number generalized quantum Langevin equation where the quantum correction terms appear as a coupled infinite set of hierarchy of equations. For practical purpose, one should truncate these equations after a certain order. In our present development, we calculate the quantum correction terms in a closed analytical form based on a systematic perturbation technique and then derive the lowest order quantum correction factor exactly in the case of an Ohmic dissipative bath. Finally, to demonstrate its applicability, the effective equation of motions has been used to study the barrier crossing dynamics which incorporates the quantum correction factors.     

A criterion for the existence of spin waves in polarized gases in a wide temperature range

We derive a kinetic equation for a polarized paramagnetic gas that is exact in the Boltzmann (binary) approximation. The equation is written in a compact form and applies to both Fermi and Bose gases in a wide temperature range as long as the Boltzmann approximation remains applicable. The derived equation is used to analyze the conditions for the propagation of spin waves in polarized Fermi and Bose gases. We deduce a universal criterion for the propagation of weakly damped spin waves in a wide temperature range. The criterion is reduced to the condition that the real parts of the particle zero-angle scattering amplitudes (or T matrices) be much larger than their imaginary parts. We derive dispersion equations for spin waves at high and low gas temperatures and show that spin waves can propagate in both these limiting cases.     

A Bhatnagar-Gross-Krook kinetic approach to fast reactive mixtures: Relaxation problems

The paper deals with a consistent BGK-type approximation for the Boltzmann-like equations which govern the evolution of a gas undergoing bimolecular chemical reactions. In particular, model equations, specifically devised for physical situations in which chemical relaxation is as fast as mechanical relaxation, are discussed in comparison to previous models. This BGK approach preserves the main features of the reactive Boltzmann equations, including law of mass action and H-theorem. Numerical results illustrating the effects of the several varying parameters on the relaxation to equilibrium are presented and commented on.     

Moment Systems Derived from Relativistic Kinetic Equations

In this paper, we are interested in the derivation of macroscopic equations from kinetic ones using a moment method in a relativistic framework. More precisely, we establish the general form of moments that are compatible with the Lorentz invariance and derive a hierarchy of relativistic moment systems from a Boltzmann kinetic equation. The proof is based on the representation theory of Lie algebras. We then extend this derivation to the classical case and general families of moments that obey the Galilean invariance are also constructed. It is remarkable that the set of formal classical limits of the so-obtained relativistic moment systems is not identical to the set of classical moments quoted in Ref. 21 and one could use a new physically relevant criterion to derive suitable moment systems in the classical case. Finally, the ultra-relativistic limit is considered.     

Kinetic theory of long-range interacting systems with angle–action variables and collective effects

We develop a kinetic theory of systems with long-range interactions taking collective effects and spatial inhomogeneity into account. Starting from the Klimontovich equation and using a quasilinear approximation, we derive a Lenard–Balescu-type kinetic equation written in angle–action variables. We confirm the result obtained by Heyvaerts [Heyvaerts, Mon. Not. R. Astron. Soc. 407, 355 (2010)] who started from the Liouville equation and used the BBGKY hierarchy truncated at the level of the two-body distribution function (i.e., neglecting three-body correlations). When collective effects are ignored, we recover the Landau-type kinetic equation obtained in our previous papers [P.H. Chavanis, Physica A 377, 469 (2007); J. Stat. Mech., P05019 (2010)]. We also consider the relaxation of a test particle in a bath of field particles. Its stochastic motion is described by a Fokker–Planck equation written in angle–action variables. We determine the diffusion tensor and the friction force by explicitly calculating the first and second order moments of the increment of action of the test particle from its equations of motion, taking collective effects into account. This generalizes the expressions obtained in our previous works. We discuss the scaling with N$N$ of the relaxation time for the system as a whole and for a test particle in a bath.     

Das Fluktuationsspektrum von Elektronen in einem stationären Nichtgleichgewichtszustand

The Boltzmann equation is used to calculate the time correlation function and the fluctuation spectrum for electrons obeying classical statistics. The stationary joint distribution for one electron to be initially ink ₀=k(0) and finally ink=k(t) is given by the product of the conditional probability and the stationary distribution. These quantities can be found from the Boltzmann equation if there exists, for any initial distribution, a unique solution which satisfies the Markov equation and tends to a stationary solution for large times under stationary conditions. It is proved that these conditions hold for linear collision operators and in the relaxation approximation. General operator expressions for the fluctuation spectrum and the differential conductivity in a stationary electric field are given, which can be evaluated within the usual approximation schemes known for the stationary, nonequilibrium solutions of the Boltzmann equation. In equilibrium they reproduce the classical fluctuation dissipation theorem. In a nonequilibrium state they define a noise temperature depending on the field. In the relaxation approximation and for polynomial band structure the exact solution can be found. For parabolic and biparabolic spherical bands the result is discussed explicitly.