A kinetic theory of spectral line shapes

The methods of kinetic theory are used to describe the radiation from an atom immersed in a gas of perturbing particles. It is shown that the line shape can be expressed in terms of a one-particle distribution function. The appropriate BBGKY hierarchy of equations is derived. This hierarchy is then truncated by assuming that only two-body collisions are important. The resulting equations are solved to obtain a non-Markovian kinetic equation which describes the combined effects of Doppler and pressure broadening. When the Markovian assumption is applied, a generalized linear Boltzmann equation is obtained which describes the line shape in the region where the impact limit is valid and which also describes the phenomenon of collisional narrowing.This research was supported in part by the Advanced Research Projects Agency of the Department of Defense, monitored by Army Research Office-Durham under Contract No. DA-31-124-ARO-D-139.     

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.     

Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.     

A generalized Boltzmann equation for nuclear dynamics

We derive a generalized Boltzmann equation from an extended time-dependent mean-field theory, which is self-consistent and incorporates two-body collisions due to the residual interaction. Obtained from a systematic reduction of a more general, but complicated, many-body theory, this kinetic equation retains many of the quantum effects of the many-body system. Due to proper treatment of quantum causality, its collision integral contains terms which are associated with the principal-value parts of propagators in a quantum-mechanically correct memory kernel. Thus, it properly generalizes the Uehling-Uhlenbeck equation and provides a quantum kinetic theory for nuclear dynamics in both low- and intermediate-energy regions. The new features in this Boltzmann equation are investigated in a nuclear-matter model with a simple effective interaction. We solve for the small-amplitude solutions corresponding to the response of the system to an arbitrary weak external field. The results are contrasted with the collisionless limit and Uehling-Uhlenbeck limit and conclusions are drawn about the dynamical effects of the two-body collisions on the quasiparticles.     

Derivation of the density functional theory from the cluster expansion

The density functional theory is derived from a cluster expansion by truncating the higher-order correlations in one and only one term in the kinetic energy. The formulation allows self-consistent calculation of the exchange correlation effect without imposing additional assumptions to generalize the local density approximation. The pair correlation is described as a two-body collision of bound-state electrons, and modifies the electron- electron interaction energy as well as the kinetic energy. The theory admits excited states, and has no self-interaction energy.     

Relativistic BBGKY Theory and Collision Integral of Generalized Boltzmann Equation in a Relativistic Magnetized Plasma

Usually, only Coulomb interactions between charged particles which are independent of time are considered in BBGKY theory of a nonrelativistic plasma. In relativistic case, the induced electromagnetic forces between charged particles which are dependent on time obviously should be considered. A Lorentz-covariant generalized n-time Liouville equation for classical plasma is established. A convenient form applicable to the laboratory frame of this equation is also given. The relativistic BBGKY hierarchy is developed in which both Coulomb and electromagnetic forces between particles are included. A method for solving the relativistic pair correlation equation is given in polarization approximation. A new formula for calculating collision integral in terms of discrete particle Green functions is given. A number of generalized Boltzmann equations for relativistic plasmas are derived.     

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.     

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.     

Nonlinear momentum relaxation of an impurity in a harmonic chain

A microscopic derivation of the generalized Langevin equation for arbitrary powers of the momentum of an impurity in a harmonic chain is presented. As a direct consequence of the Gaussian character of the conditional momentum distribution function, nonlinear momentum coupling effects are absent for this system and the Langevin equation takes on a particularly simple form. The kernels which characterize the decay of higher powers of the impurity momentum depend on the ratio of the masses of the impurity and bath particles, in contrast to the situation for the momentum Langevin equation for this system. The simplicity of the harmonic chain dynamics is exploited in order to investigate several features of the relaxation, such as the factorization approximation for time-dependent correlation functions and the decay of the kinetic energy autocorrelation function.     

On the approach to kinetic theory due to Gross

A classical kinetic theory introduced by Gross is explored in further detail. The theory consists of a sequence of approximations to the Liouville distribution function, with each approximation leading to a truncation of the BBGKY hierarchy at successively higher order. We formulate the truncation scheme at general order in terms of a set of time-dependent equilibrium correlation functions. It has the correct symmetries and, as is implied by the work of Gross with the first two approximations, is such that the interparticle potential appears only implicitly via static equilibrium correlation functions. We arrange the theory as a sequence of linear kinetic equations for the phase-space density correlation function, and solve for the collision kernels which result in each order. The collision kernel of the second approximation, which involves only binary dynamics, is shown to be a mean-field generalization of the known low-density kernel. The third approximation gives a similar generalization of the triple-collision kernel. The nth approximation also reproduces the frequency moments of S(kω) through order ω²ⁿ. More generally, the approximations are shown to give a continued-fraction expansion of the collision kernel, with the levels governed by the dynamics of successively larger numbers of particles. This is a renormalized kinetic theory in the sense that the potential is eliminated and clusters of particles are never isolated.     

Study of Kinetic Equation for Non-Ideal Gases Using Lattice Boltzmann Method: Interparticle Forces from BBGKY Hierarchy

The first equation of the BBGKY hierarchy is closed with a truncation scheme which conserves mass, momentum and total energy. Resulting kinetic equation is solved numerically using the lattice Boltzmann method over an isothermal \(D2Q9\) lattice. The inter particle forces are derived using a density gradient approximation to the collision term of the BBGKY hierarchy and the resulting non-ideal equation of state is calculated. In this paper, we show analytically that the time rate of change of Boltzmann’s H-function i.e., \(\frac{dH \left( t \right) }{dt}\) is zero for this truncation scheme, implying reversibility in time. Therefore, a BGK term is added to the collision integral to describe the macroscopic irreversibility. Both the local and global \(H\) -functions are calculated numerically for significant variations in the density. We also have simulated a switching experiment over this system and obtained the non-equilibrium work distributions.     

Nonstationary and Quasi-stationary Treatment of Distribution Anisotropy in the Temporal Relaxation of an Energetic Electron Group

A relaxation study of an electron group in collision dominated weakly ionized plasmas has been performed. The study is based on the two-term approximation of the Legendre polynomial expansion of the electron velocity distribution in the nonstationary Boltzmann equation. To overcome the limitation of the conventional quasi-stationary treatment of the distribution anisotropy, a very efficient solution approach of the nonstationary kinetic equation in two-term approximation has been developed which allows for a strict nonstationary treatment of the distribution anisotropy. By using this approach the temporal evolution of the isotropic and anisotropic distribution of the electrons has been investigated for a model plasma, which involves typical features of an inert gas plasma. A comparison of the results with corresponding ones obtained by applying the conventional approach under various parameter conditions clearly indicates a pronounced falsification of the real relaxation course by the latter approach.     

Not to Normal Order—Notes on the Kinetic Limit for Weakly Interacting Quantum Fluids

The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrödinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation 〈a(t)^* a(0)〉.     

Integral equations in the theory of crystals

The hierarchy of Bogolyubov (BBGKY) equations is considered. On the basis of an approximation which follows from the expression for the zeroth approximation of the ternary distribution function a system of integral equations is obtained that describes both homogeneous and periodic structures.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 82–85, February, 1981.     

Nucleus-nucleus collisions: Intrinsic motion with external correlations

A consistent treatment of the relative and intrinsic motion which goes beyond the mean-field approach allows to include the fluctuations of the time-dependent mean field for the intrinsic as well as for the relative motion. Starting with the v. Neumann equation for the total density matrix, we derive a modified equation for the intrinsic many-body density matrix. This equation is used to obtain the quantum kinetic equations for the one-body density matrix and the two-body correlator. In the time-dependent single-particle basis, the occupation numbers change in time due to a collision term originating from residual two-body interactions which account for equilibration, and due to the fluctuations of the external mean field. The relations to TDHF with collision term are discussed. Special attention is paid to the conditions for a diabatic evolution of the single-particle states and to finite size effects which play an important role to make two-body collisions operative in finite nuclei.     

A stochastic master equation representation of statistical mechanics

A functional master equation for a probability density of the velocity distribution function is presented. This constitutes a representation of statistical mechanics which is equivalent to the BBGKY hierarchy but makes alternative approximation schemes available.     

A new hierarchy system on the basis of a “Master” Boltzmann equation for microscopic density

It is shown that Boltzmann's equation written in terms of microscopic density (namely the unaveraged Boltzmann function) has a wider range of validity as well as finer resolvability for fluctuations than the conventional Boltzmann equation governing Boltzmann's function. In fact the new Boltzmann equation for ideal gases has implications as a microscopically exact continuity equation like Klimontovich's equation for plasmas, and can be derived without invoking any statistical concepts, e.g., distribution functions, or molecular chaos. The Boltzmann equation in the older formalism is obtained by averaging this equation only under a restricted condition of the molecular chaos. The new Boltzmann equation is seen to contain information comparable with Liouville's equation, and serves as a master kinetic equation. A new hierarchy system is formulated in a certain parallelism to the BBGKY hierarchy. They are shown to yield an identical one-particle equation. The difference between the two hierarchy systems first appears in the two-particle equation. The difference is twofold. First, the present formalism includes thermal fluctuations that are missing in the BBGKY formalism. Second, the former allows us to formulate multi-time correlations as well, whereas the latter is restricted to simultaneous correlation. These two features are favorably utilized in deriving the Landau-Lifshitz fluctuation law in a most straightforward manner. Also, equations describing the nonequilibrium interaction between thermal and fluid-dynamical fluctuations are derived.     

Dissipative dynamics for hard spheres

The dynamics for a system of hard spheres with dissipative collisions is described at the levels of statistical mechanics, kinetic theory, and simulation. The Liouville operator(s) and associated binary scattering operators are defined as the generators for time evolution in phase space. The BBGKY hierarchy for reduced distribution functions is given, and an approximate kinetic equation is obtained that extends the revised Enskog theory to dissipative dynamics. A Monte Carlo simulation method to solve this equation is described, extending the Bird method to the dense, dissipative hard-sphere system. A practical kinetic model for theoretical analysis of this equation also is proposed. As an illustration of these results, the kinetic theory and the Monte Carlo simulations are applied to the homogeneous cooling state of rapid granular flow.     

Relativistic quantum kinetic analysis of a pion-nucleon system

A relativistic plasma of nucleons interacting through pions via the usual isospin-invariant Yukawa coupling is analyzed in the framework of the covariant Wigner function technique. The method is manifestly covariant and the temperature effects are considered. The relativistic quantum BBGKY hierarchy for the pion-nucleon system is derived. By generalizing the Bogolioubov analysis of the classical BBGKY hierarchy a non-perturbative renormalizable method is elaborated which allows the solution of the kinetic problem in form of power series of two cluster parameters which measure the importance of correlations. In the lowest order of the cluster expansion (Hartree approximation or zero-order approximation) the quasi-nucleon Fock space is introduced, the fermion Wigner function in the thermodynamic equilibrium is obtained and the vacuum effects are renormalized. In this approximation the plasma behaves as a perfect Fermi gas of nucleons and antinucleons, but there exists an abnormal configuration with a uniform pion condensate which is unstable. In the next approximation (quadratic in the small parameters) the quasi-pion dispersion relation is obtained and the vacuum polarization tensor is renormalized. The quasi-pion rest-mass spectra (“plasma frequency”) and the effective-coupling behaviour as functions of the thermodynamic state are given. By estimating the size of the cluster parameters the self-consistency of the approximation scheme is proved. The quasi-pion Fock space is introduced and the quasi-pion equilibrium Wigner function is obtained. From these results the problem of the higher-order corrections to the Hartree thermodynamics is outlined.     

Comparison of monte carlo and boltzmann two-term calculations of time-dependent velocity distribution functions of electrons in an electric field in a gas

Summary An extended comparison has been made between Boltzmann two-term and rigorous Monte Carlo calculations of time-dependent velocity distribution functions of electrons in a d.c. electric field in a gas to assess the limits of the conventional approximations. It is shown that under various conditions the two-term approximation is unable to represent the velocity distribution correctly even if the conventional quasi-stationary approximation of the velocity distribution anisotropy, usually adopted in this kind of calculations, is abandoned. The conditions which permit to use the two-term theory are discussed.