S-matrix theory and the density matrix formalism

A general equation governing the time development of the diagonal part of the density matrix is proposed for weakly interacting systems possessing no off-diagonal long range order. This equation, which involves generalized Møller operators of the type employed in S-matrix theory, is solved for two cases and leads to the generalized Pauli and Boltzmann equations.     

Coarsely grained stochastic Boltzmann equation and its moment equations

The stochastic Boltzmann equation is coarsely grained. The coarsely grained stochastic (CGS) Boltzmann equation has fluctuating terms in its collision term. On the basis of the CGS Boltzmann equation, reduced Grad’s 26 moment equations are derived. Coarsely grained moment equations obtained from the CGS Boltzmann equation show that fluctuating terms remain as nonvanishing terms owing to the nonlinearity in the collision term of the CGS Boltzmann equation. The Navier-Stokes-Fourier law obtained using the CGS Boltzmann equation indicates that the pressure deviator and heat flux include fluctuations of their one-order higher moments.     

Transport equations for chiral fermions to order ? and electroweak baryogenesis: Part I

This is the first in a series of two papers. In this first part, we use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations, accurate to order ?, for massive chiral fermions, scalar particles, and for the corresponding CP-conjugate states. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. We provide a quantitative criterion in which case the reduction to the diagonal kinetic equations in the mass eigenbasis is justified, leading to a quasiparticle picture even in the case of mixing scalar or fermionic particles. Within the approximations we make, it is possible to first study the Boltzmann equations without the collision term. In a second paper [Ann. Phys. xxx (2004) xxx] we discuss the collision terms and reduce the Boltzmann equations to fluid equations.     

The Boltzmann equation from quantum field theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff–Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff–Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.     

Investigation of Stability and Hydrodynamics of Different Lattice Boltzmann Models

Stability and hydrodynamic behaviors of different lattice Boltzmann models including the lattice Boltzmann equation (LBE), the differential lattice Boltzmann equation (DLBE), the interpolation-supplemented lattice Boltzmann method (ISLBM) and the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) are studied in detail. Our work is based on the von Neumann linearized stability analysis under a uniform flow condition. The local stability and hydrodynamic (dissipation) behaviors are studied by solving the evolution operator of the linearized lattice Boltzmann equations numerically. Our investigation shows that the LBE schemes with interpolations, such as DLBE, ISLBM and TLLBM, improve the numerical stability by increasing hyper-viscosities at large wave numbers (small scales). It was found that these interpolated LBE schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper-viscosities.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

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.     

A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases

We propose an extension of the Fokker–Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier–Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar–Gross–Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman–Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.     

Spin-dependent balance equations in spintronics

It is commonly known that the hydrodynamic equations can be derived from the Boltzmann equation. In this paper,we derive similar spin-dependent balance equations based on the spinor Boltzmann equation. Besides the usual charge current, heat current, and pressure tensor, we also explore the characteristic spin accumulation and spin current as well as the spin-dependent pressure tensor and heat current in spintronics. The numerical results of these physical quantities are demonstrated using an example of spin-polarized transport through a mesoscopic ferromagnet.     

New Discrete Model Boltzmann Equations for Arbitrary Partitions of the Velocity Space

Modified discrete Boltzmann equations for arbitrary partitions of the velocity space are established. The new equations can be derived from the continuous Boltzmann equation and are a generalization of previous discrete-velocity models. They preserve mass, momentum, and energy, and an H-theorem holds. The new model equations are tested by comparing their solutions with the analytical ones of the continuous Boltzmann equation for the Krook–Wu and the very hard particle models.     

Hilbert Expansion from the Boltzmann Equation to Relativistic Fluids

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellians. The Maxwellians are constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations.     

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.     

From Reactive Boltzmann Equations to Reaction–Diffusion Systems

We consider the reactive Boltzmann equations for a mixture of different species of molecules, including a fixed background. We propose a scaling in which the collisions involving this background are predominant, while the inelastic (reactive) binary collisions are very rare. We show that, at the formal level, the solutions of the Boltzmann equations converge toward the solutions of a reaction-diffusion system. The coefficients of this system can be expressed in terms of the cross sections of the Boltzmann kernels. We discuss various possible physical settings (gases having internal energy, presence of a boundary, etc.), and present one rigorous mathematical proof in a simplified situation (for which the existence of strong solutions to the Boltzmann equation is known).     

Lattice Kinetic Formulation for Ferrofluids

The lattice Boltzmann approach is used to solve continuum equations describing colloids of ferromagnetic particles (ferrofluids) in a regime, where the particle spins are in equilibrium with magnetic torques. This limit of rapid spin adjustment yields a symmetric total stress tensor that is essential for a kinetic formulation based on the Boltzmann equation. The magnetisation equation is solved using a vector-valued distribution function analogous to the earlier treatment (J. Comput. Phys. 179, 95) of the induction equation in magnetohydrodynamics, but the details are rather more complex because the magnetisation equation is not in conservation form except in a weakly magnetised limit.     

Illustrations of a dynamical theory of the ether

The Schrödinger and Klein-Gordon equations for free, structureless particles are derived classically from two different continuum approximations to a Boltzmann equation for the trace component of a mixture. The majority component is designated as the ether. Deviations from these continuum approximations (rarefied ether) yield deviations from the Schrödinger and Klein-Gordon equations which are shown explicitly.     

Classical Theory of Hot-Electron Transport in Electric and Magnetic Fields

Balance equation approach to the hot-electron transport in electric and magnetic fields is reformulated.The balance equations are re-derived from the Boltzmann equation. A new expression for the distribution function isreported in the present paper. It is homogeneous steady solution of the Boltzmann equation in constant relaxation timeapproximation. It holds when ωocτ < i or ωc < Te. As an example, the mobility of 2D electron gas in the GaAs-AlGaAsheterojunction is computed as a function of electric field and magnetic field.     

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.     

Applicability of kinetic approaches to heavy-ion reactions

The Boltzmann (or Vlasov-Uehling-Uhlenbeck) equation for heavy-ion collisions is derived on the basis of the Dyson equation. Special attention is given to correctly choosing a statistical ensemble. This makes it possible to determine physically small parameters in the gradient expansion; to establish the range of applicability of kinetic approaches to heavy-ion reactions; to write, in a simple form, generalized Boltzmann equations consistently taking into account all corrections for medium effects; and to analyze the role of these corrections.     

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.