Boltzmann-type Equation Research Articles

Lattice effects in the quasiparticle dynamics and kinetics

In this work we present a full selfconsistent set of nonlinear equations which unifies the nonlinear elasticity theory equations, the Boltzmannn transport theory and the Maxwell equations for quasiparticles with arbitrary dispersion laws in nonstationarily deformed crystals with arbitrary (but linear) constitutive relations. Transformations to replace the Galilean ones are obtained, the quasiparticle mechanics in a Hamiltonian form is deduced, and a Boltzmann-type transport equation (valid in the whole Brillouin zone) is derived. The theory may be applied to metals, semiconductors, quantum crystals, low-dimensional structures etc.

Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas: the multiparticle system.

We study the stationary nonequilibrium states of N-point particles moving under the influence of an electric field E among fixed obstacles (disk) in a two-dimensional torus. The total kinetic energy of the system is kept constant through a Gaussian thermostat that produces a velocity dependent mean field interaction between the particles. The current and the particle distribution functions are obtained numerically and compared for small /E/ with analytic solutions of a Boltzmann-type equation obtained by treating the collisions with the obstacles as random independent scatterings. The agreement is surprisingly good for both small and large N. The latter system in turn agrees with a self-consistent one-particle evolution expected to hold in the N-->infinity limit.

Nonlocal Transport Phenomena and Various Structure Formations in Plasmas. Nonlocal Transport in Laser Implosion and Supernova Explosion.

Nonlocal transport becomes important in a variety of situations in physics. We briefly review under what conditions it appears to be important in modeling energy transport by neutral gas, neutrons, charged particles, photons, and neutrinos. In the cases of these last three, the cross-sections strongly depend upon particle energies. In such cases, nonlocal transport becomes important even when each mean free path is much shorter than the length of the change of its energy density. We mainly review the research that has been done on nonlocal electron energy transport in relation to laser-produced plasmas and the way in which precise calculation of neutrino transport changes the properties of a shock wave, which is used as a driver to explode a massive star such as a supernova, by solving Boltzmann-type equations for neutrinos and anti-neutrinos.

Multi-fluid modelling of laminar polydisperse spray flames: origin, assumptions and comparison of sectional and sampling methods

A first attempt at deriving a fully Eulerian model for polydisperse evaporating sprays was developed by Tambour et al with the so-called sectional approach. However, the complete derivation of the sectional ‘multi-fluid’ conservation equations from the Boltzmann-type spray equation was never provided, neither was the set of underlying assumptions nor the comparison with the classical Lagrangian model: the sampling method. In this paper, we clarify the set of assumptions necessary in order to derive the multi-fluid sectional model from the spray equation at the ‘kinetic level’ and provide the derivation of the whole set of conservation equations describing the dispersed liquid phase. Whereas the previous derivation is conducted in any space dimension, we restrict ourselves to one-dimensional stationary flows where the droplets do not turn back and derive a Eulerian sampling model which is equivalent in this context to the usual Lagrangian particle approach. We then identify some situations, even within this restrictive framework, where the sectional approach fails to reproduce the coupling of the vaporization and dynamics of the spray, the sampling method then being required. In the domain of applicability of the sectional approach, the two methods are then compared numerically in the configuration of counterflow spray diffusion flames. The two methods, if refined enough, give quite similar results, except for some small differences, the origin of which is identified. It is proved that the sampling method is more precise even if it generates oscillations due to the intrinsic representation of a continuous function by Dirac delta functions. We thus provide a comprehensive analysis of the sectional approach from both the modelling and numerical points of view.

Hydrodynamic Equation for the Breakdown of the Quantum Hall Effect in a Uniform Current

The hydrodynamic equation for the spatial and temporal evolution of the electron temperature T_e in the breakdown of the quantum Hall effect at even-integer filling factors in a uniform current density j is derived from the Boltzmann-type equation, which takes into account electron-electron and electron-phonon scatterings. The derived equation has a drift term, which is proportional to j and to the first spatial derivative of T_e. Applied to the spatial evolution of T_e in a sample with an abrupt change of the width along the current direction, the equation gives a distinct dependence on the current direction as well as a critical relaxation, in agreement with the recent experiments.

America's plan for MOX

The one speed neutron transport equation with anisotropic scattering has been considered for the slab albedo problem. The scattering function is assumed to be a combination of forward—backward and with isotropic scattering (FBIS model). The transport equation is transformed into a Boltzmann-type equation, corresponding to isotropic single scattering and the resulting equation is solved by using the recently developed FN method. The albedo and transmission factor are calculated for a slab and various types of scattering parameters. Numerical results are given, accurate to five significant figures and compared with those already available in the literature. Finally, it is shown that the FN method gives more accurate results and easily applicable to this type of anisotropic scattering kernel.

Entropy Inequalities for Evaporation Condensation Problem in Rarefied Gas Dynamics

The present paper is devoted mainly to the half space problem for stationary Boltzmann-type equations. Using only conservation laws and the Boltzmann H-theorem we derive an inequality for unknown constant fluxes of mass, energy, and momentum. This inequality is expressed in terms of three parameters (pressure p, temperature T and the Mach number M) of the asymptotic Maxwellian at infinity. Geometrically the inequality describes a “physical” domain with positive entropy production in the 3-d space of the parameters. The domain appears to be qualitatively different for evaporation and condensation problems. We show that for given M, the curve p=p(M), T=T(M) of maximal entropy production practically coincides with the experimental evaporation curve obtained by Sone et al. on the basis of numerical solutions of BGK equation. Similar consideration for the condensation problem is also in qualitative agreement with known numerical results.

Steady states of a boltzmann equation for driven granular media

We study a three-dimensional model for driven granular media in which particles interact inelastically while they follow Brownian dynamics in between collisions. A steady Boltzmann-type kinetic equation associated with a pseudo-Maxwellian model is analyzed. Homogeneous steady states are found by a small inelasticity expansion. These states are given by a Maxwellian distribution corrected by the second Sonine polynomial up to third order in the expansion. The resulting correction is a quartic polynomial in velocity space. This result agrees qualitatively with the molecular dynamics simulation in C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Phys. Rev. E 60, 4340 (1999).

Kadanoff–Baym equations and non-Markovian Boltzmann equation in generalized T-matrix approximation

A recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized T-matrix approximation for the self-energies. It is shown that the T-matrix obtains additional contributions arising from initial correlations. Using these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a far-reaching generalization of Boltzmann-type kinetic equations: It self-consistently includes memory effects (retardation, off-shell T-matrices) as well as many-particle effects (damping, in-medium T-matrices) and spin-statistics effects (Pauli-blocking).

Interference effects in the scattering of identical atoms in a laser radiation field

The properties of the collision integral in a quantum Boltzmann-type kinetic equation are studied under the conditions of spatially nonuniform distributions of colliding particles interacting with an external electromagnetic field. The components of the nonlinear resonances and the velocity distribution of the excited atoms, which are due to polarization transitions, are determined on the basis of the Kazantsev collision integral.

Variation of the albedo and the transmission factor with forward and backward scattering in neutron transport theory- the F N method

The one speed neutron transport equation with anisotropic scattering has been considered for the slab albedo problem. The scattering function is assumed to be a combination of forward—backward and with isotropic scattering (FBIS model). The transport equation is transformed into a Boltzmann-type equation, corresponding to isotropic single scattering and the resulting equation is solved by using the recently developed F N method. The albedo and transmission factor are calculated for a slab and various types of scattering parameters. Numerical results are given, accurate to five significant figures and compared with those already available in the literature. Finally, it is shown that the F N method gives more accurate results and easily applicable to this type of anisotropic scattering kernel.

Kinetic model of a collisional admixture in dusty gas and its application to calculating flow past bodies

Using the methods of statistical physics, the basic kinetic equation describing the dynamics of a polydisperse admixture of solid particles in a dilute dusty-gas flow is derived. Particle rotation, inelastic collisions, and interaction with the carrier gas are taken into account. The basic kinetic equation is used to obtain a Boltzmann-type equation for the one-particle distribution function, for which the boundary conditions for the problem of dusty-gas flow past a body are formulated. On the basis of the kinetic model developed, using direct statistical modeling, the flow patterns and the fields of the dispersed-phase macroparameters in a uniform crosswise dusty-gas flow past a cylinder are obtained for various free-stream particle sizes and concentrations.

On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds

We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails: in particular, Boltzmann-type equations with (smoothed) soft potentials. We compensate the lack of uniform-in-time estimates by the use of precise logarithmic Sobolev-type inequalities, and the assumption that the initial datum decays rapidly at large velocities. Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which compactness arguments apparently do not apply (even mere convergence to equilibrium was an open problem for soft potentials).

Multiradii modeling of counterflow spray diffusion flames

A multiradii spray flame model is derived in a kinetic framework in which the droplets' distribution function is taken as a sum of delta products. The droplet class equations are obtained by taking moments of the Boltzmann-type spray equation. The gas-phase equations are coupled to the droplets equations through various integral sources terms. In the counterflow geometry, using a self-similar assumption, multiradii counterflow spray flame models are then obtained as exact solutions of multidimensional small Mach number governing equations. The structure of multiradii counterflow spray flames is investigated numerically. To solve the discretized system of equation, a fully coupled adaptive computation was conducted between the gas and liquid phase. The equations are discretized by using a finite difference, and highly optimized libraries were used in order to evaluate thermochemistry and transport properties. We first investigate an n -heptane counterflow monodisperse spray flame studied experimentally by Chen and Gomez. We then investigate a multiradii methanol spray flame studied experimentally by Gomez and coworkers. The multiradii approach is used in order to describe accurately the multiple size behavior, which is found to be strongly radius-dependent. Good agreement is found in both cases up to experimental uncertainties in the measured injection velocities and class number densities.

Two-time temperature Green’s functions in kinetic theory and molecular hydrodynamics: II. Equations for pair-interaction systems

We consider two approaches to the calculation of correlation functions for a system of particles with direct pair interaction. The first is based on a chain of equations that determines a Boltzmann-type kinetic equation; the second is based on a chain of molecular hydrodynamic equations. We demonstrate that the two approaches are equivalent in the sense that they completely describe the system under consideration. We discuss the advantages of the approach based on the molecular hydrodynamic equations.

Conductivities of a quantum Hall system with a model disorder potential: Influence of electron-phonon interaction on plateau widths

We consider noninteracting electrons in a quasi-two-dimensional strip (extended in the $x$ direction) in the presence of a strong perpendicular magnetic field and a constant electric field in the $y$ direction, and subject to a model disorder potential $V(x,y)$ (``toy model'') which allows us to obtain the exact solutions of the time-dependent Schr\"odinger equation in very good approximation. Further, the electrons are coupled to a heat bath (accoustic phonons) at temperature $T$ that modifies the coherent Schr\"odinger time evolution induced by the electric field ${E}_{y}.$ After elimination of macroscopically unobservable fluctuations by averaging over suitable short time intervals, the time evolution that is relevant for the macroscopic current density can be described by a Boltzmann-type equation, which is solved numerically. The steady-state solution allows us to calculate ${\ensuremath{\sigma}}_{\mathrm{yy}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ as a function of any physical parameter of the system. Here we give results of ${\ensuremath{\sigma}}_{\mathrm{yy}}$ and of ${\ensuremath{\sigma}}_{\mathrm{xy}}$ for all filling factors and as a function of temperature. Quantized Hall plateaus are obtained at the correct values with high precision. Between two plateaus, the Shubnikov--de Haas peak spreads out and its maximum decreases with increasing temperature in qualitative accordance with typical quantum Hall samples. Further, we obtain the remarkable result that for temperatures below 5 K the plateaus of ${\ensuremath{\sigma}}_{\mathrm{xy}}$ become sensibly larger than those of ${\ensuremath{\sigma}}_{\mathrm{yy}}.$ Our analysis shows that this effect results from electron-phonon interaction. (Such a phenomenon has been known experimentally for a long time, but it has seemed unexplained so far.) Our method of calculation could, in principle, be extended to more complex disorder potentials.

A generalized collision mechanism for stochastic particle schemes approximating Boltzmann-type equations

A generalized collision mechanism for Boltzmann-type stochastic particle schemes is developed. This mechanism is based on the idea of random weight transfer originating from random discrete velocity models. The problem of applying the new degrees of freedom for the purpose of variance reduction is studied. Some results of numerical experiments are given.

Boltzmann-type equations for elementary reversible cellular automata

A Boltzmann-type equation is introduced as an approximation for calculating the time evolution of the probability distribution in elementary reversible cellular automata. A number of properties are discussed from the approximation. Applications to heat conduction problem are exhibited. Simulation results suggest a connection between conserved quantities and diffusive or ballistic behavior.

Molecular dissection of gating in the ClC-2 chloride channel.

The ClC-2 chloride channel is probably involved in the regulation of cell volume and of neuronal excitability. Site-directed mutagenesis was used to understand ClC-2 activation in response to cell swelling, hyperpolarization and acidic extracellular pH. Similar to equivalent mutations in ClC-0, neutralizing Lys566 at the end of the transmembrane domains results in outward rectification and a shift in voltage dependence, but leaves the basic gating mechanism, including swelling activation, intact. In contrast, mutations in the cytoplasmic loop between transmembrane domains D7 and D8 abolish all three modes of activation by constitutively opening the channel without changing its pore properties. These effects resemble those observed with deletions of an amino-terminal inactivation domain, and suggest that it may act as its receptor. Such a 'ball-and-chain' type mechanism may act as a final pathway in the activation of ClC-2 elicited by several stimuli.

Description of accelerator and cosmic ray data in terms of relativistic hadron thermodynamics

Evidences for thermodynamical description of relativistic hadrons are presented. Relativistic hadron distribution does not obey any known statistics, though a prevailing term leads to Stephan — Boltzmann type equation of state. A common hadron state asymptotic regime is compatible with S-matrix models describing the low energy accelerator data with Pomeron exchange phenomenology. The gross features of giant showers can be explained in the frame of relativistic hadron thermodynamics. It cannot be achieved by S-matrix models with additional production of mini-jets.