Showing papers on "Boltzmann constant published in 2002"

Kinetic Theory and Fluid Dynamics

Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Presence of Perfectly Conducting Channel in Metallic Carbon Nanotubes

Abstract: It is proved that a conducting channel transmitting perfectly without backscattering is present independent of energy in metallic carbon nanotubes for scatterers with potential range larger than the lattice constant. In the case that several traveling channels are present, the conductance decreases from the ideal value determined by the number of traveling modes to the single-channel value when the length exceeds the mean free path determined by a Boltzmann transport equation. Further, inelastic scattering makes the conductance decrease in proportion to the inverse of the length in qualitative agreement with the Boltzmann result.

Freeze-out problem in hydrokinetic approach to A + A collisions.

Abstract: A new method for evaluating spectra and correlations in the hydrodynamic approach is proposed. It is based on an analysis of the Boltzmann equations (BE) in terms of probabilities for constituent particles to escape from the interacting system. The conditions of applicability of the Cooper-Frye freeze-out prescription are considered within the method. The results are illustrated with a nonrelativistic exact solution of BE for an expanding spherical fireball as well as with approximate solutions for ellipsoidally expanding ones.

Entropy, energy, and proximity to criticality in global earthquake populations

Abstract: [1] We examine the question of proximity of the global earthquake population to the critical point characterised by the energy E and entropy S based on annual frequency data from the Harvard CMT catalogue. The results are compared with a theoretical model corresponding to a Boltzmann probability density distribution of the form p(E) ∝ E−B−1e−E/θ. The data are consistent with the model predictions for fluctuations in the characteristic energy θ at constant B value, of the form S∼B〈lnE〉. This approximation is valid for large θ, relative to the maximum possible event size, confirming that the Earth is perpetually in a near-critical state, reminiscent of self-organized criticality. However, the results also show fluctuations of ±10% in entropy that may be more consistent with the notion of intermittent criticality. A more precise definition of the two paradigms, and a similar analysis of numerical models, are both needed to distinguish between these competing models.

Gaussian disorder model for high carrier densities: Theoretical aspects and application to experiments

Abstract: It is shown that the precondition for the common Boltzmann approximation usually utilized in many hopping models for organic semiconductors is easily violated in most experiments. A vivid hopping model which takes the Fermi distribution into account is presented to describe the equilibrium transport in organic semiconductors also for higher carrier densities. The description is based on the Miller-Abraham model for hopping in a disordered material and utilizes the so-called ``transport energy concept.'' Deviations from the common low-density approximation are discussed. The formalism is applied to recently published transport data of doped organic semiconductors.

A new consistent discrete-velocity model for the Boltzmann equation

Abstract: A discrete velocity model (DVM) of the Boltzmann equation based on a suitable transformation (Carleman transform) of the velocity variables in the collision integral is introduced. The convergence of the discrete collision sums to the Boltzmann operator and convergence of solutions of the DVM to solutions of the Boltzmann equation are then proven in a three-dimensional velocity space. In the space-homogeneous case, a numerical example compares the solutions to the DVM with the exact solution of the Boltzmann equation

Effective Kinetic Theory for High Temperature Gauge Theories

Abstract: Quasiparticle dynamics in relativistic plasmas associated with hot, weakly-coupled gauge theories (such as QCD at asymptotically high temperature $T$) can be described by an effective kinetic theory, valid on sufficiently large time and distance scales. The appropriate Boltzmann equations depend on effective scattering rates for various types of collisions that can occur in the plasma. The resulting effective kinetic theory may be used to evaluate observables which are dominantly sensitive to the dynamics of typical ultrarelativistic excitations. This includes transport coefficients (viscosities and diffusion constants) and energy loss rates. We show how to formulate effective Boltzmann equations which will be adequate to compute such observables to leading order in the running coupling $g(T)$ of high-temperature gauge theories [and all orders in $1/\log g(T)^{-1}$]. As previously proposed in the literature, a leading-order treatment requires including both $2 2$ particle scattering processes as well as effective ``$1 2$'' collinear splitting processes in the Boltzmann equations. The latter account for nearly collinear bremsstrahlung and pair production/annihilation processes which take place in the presence of fluctuations in the background gauge field. Our effective kinetic theory is applicable not only to near-equilibrium systems (relevant for the calculation of transport coefficients), but also to highly non-equilibrium situations, provided some simple conditions on distribution functions are satisfied.

Coherent state semiclassical initial value representation for the Boltzmann operator in thermal correlation functions

Abstract: A semiclassical methodology for evaluating the Boltzmann operator entering semiclassical approximations for finite temperature correlation functions is described. Specifically, Miller’s imaginary time semiclassical approach is applied to the Herman–Kluk coherent state initial value representation (IVR) for the time evolution operator in order to obtain a coherent state IVR for the Boltzmann operator. The phase-space representation gives rise to exponentially decaying factors for the coordinates and momenta of the real time trajectories employed in the dynamical part of the calculation. A Monte Carlo procedure is developed for evaluating dynamical observables, in which the absolute value of the entire exponential part of the integrand serves as the sampling function. Numerical tests presented show that the methodology is accurate as well as stable over the temperature range relevant to chemical applications.

Tracer diffusion in granular shear flows.

Abstract: Tracer diffusion in a granular gas in simple shear flow is analyzed. The analysis is made from a perturbation solution of the Boltzmann kinetic equation through first order in the gradient of the mole fraction of tracer particles. The reference state (zeroth-order approximation) corresponds to a Sonine solution of the Boltzmann equation, which holds for arbitrary values of the restitution coefficients. Due to the anisotropy induced in the system by the shear flow, the mass flux defines a diffusion tensor D(ij) instead of a scalar diffusion coefficient. The elements of this tensor are given in terms of the restitution coefficients and mass and size ratios. The dependence of the diffusion tensor on the parameters of the problem is illustrated in the three-dimensional case. The results show that the influence of dissipation on the elements D(ij) is in general quite important, even for moderate values of the restitution coefficients. In the case of self-diffusion (mechanically equivalent particles), the trends observed in recent molecular-dynamics simulations are similar to those obtained here from the Boltzmann kinetic theory.

A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Abstract: Using the main ideas of Tanaka, the measure-solution {Pt}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {Pt}t of solutions {Ptl}t of approximating Boltzmann equations with cutoff is proved, Then, a result of Graham-Meleard is used and allows us to approximate {Ptl}t with the empirical measure {µtl,n}t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

Kinetical foundations of non-conventional statistics

Abstract: After considering the kinetical interaction principle (KIP) introduced by Kaniadakis (Physica A 296 (2001) 405), we study in the Boltzmann picture, the evolution equation and the H-theorem for non-extensive systems. The q -kinetics and the κ -kinetics are studied in detail starting from the most general non-linear Boltzmann equation compatible with the KIP.

Semiclassical initial value representation for the Boltzmann operator in thermal rate constants

Abstract: The thermal rate constant for a chemical reaction, k(T), can be expressed as the long time limit of the flux-side correlation Cfs(t)=tr[e−βĤ/2Fe−βĤ/2eiĤt/ℏĥe−iĤt/ℏ]. Previous work has focused on semiclassical (SC) approximations [implemented via an initial value representation (IVR)] for the time evolution operators exp(±iĤt/ℏ) in the correlation function, and this paper shows how an SC-IVR can also be used to approximate the Boltzmann operators exp(−βĤ/2). Test calculations show that over a wide temperature range little error is introduced in the rate constant by this SC approximation for the Boltzmann operator.

Numerical methods for the semiconductor boltzmann equation based on spherical harmonics expansions and entropy discretizations

Abstract: We present a class of numerical methods for the semiconductor Boltzmann Poisson problem in the case of spherical band energies. The methods are based on spherical harmonics expansions in the wave vector and difference discretizations in space–time. The resulting class of approximate solutions dissipate a certain type of entropy.

A WENO-Solver for the 1D Non-Stationary Boltzmann–Poisson System for Semiconductor Devices

Abstract: In this work we present preliminary results of a high order WENO scheme applied to a new formulation of the Boltzmann equation (BTE) describing electron transport in semiconductor devices with a spherical coordinate system for the phase velocity space. The problem is two dimensional in the phase velocity space and one dimensional in the physical space, plus the time variable driving to steady states. The new formulation avoids the singularity due to the spherical coordinate system.

Micro-canonical statistical mechanics of some non-extensive systems

Abstract: Non-extensive systems are not allowed to go to the thermodynamic limit. Therefore statistical mechanics has to be reformulated without invoking the thermodynamical limit. That is, we have to go back to pre-Gibbsian times. It is shown that Boltzmann's mechanical definition of entropy S as a function of the conserved “extensive” variables energy E , particle number N , etc., allows the description of even the most sophisticated cases of phase transitions unambiguously for “small” systems like nuclei, atomic clusters, and self-gravitating astrophysical systems. The rich topology of the curvature of S ( E , N ) shows the whole “zoo” of transitions: transitions of first order including the surface tension at phase separation, continuous transitions, critical and multi-critical points. The transitions are the “catastrophes” of the Laplace transform from the “extensive” to the “intensive” variables. Moreover, this classification of phase transitions is much more natural than the Yang–Lee criterion.

Collision group and renormalization of the Boltzmann collision integral.

Abstract: On the basis of a recently discovered collision group [V. L. Saveliev, in Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. Gallis, AIP Conf. Proc. No. 585 (AIP, Melville, NY, 2001), p. 101], the Boltzmann collision integral is exactly rewritten in two parts. The first part describes the scattering of particles with small angles. In this part the infinity due to the infinite cross sections is extracted from the Boltzmann collision integral. Moreover, the Boltzmann collision integral is represented as a divergence of the flow in velocity space. Owing to this, the role of collisions in the kinetic equation can be interpreted in terms of the nonlocal friction force that depends on the distribution function.

Fluid Dynamic Limits of the Boltzmann Equation

Abstract: In 1866, James Clerk Maxwell (1831–1879) developed a fundamental theoretical basis for the kinetics theory of gases. Maxwell’s theory is based on the idea of Daniel Bernoulli (1738), which gave birth to the kinetic theory of gases, that gases are formed of electric molecules rushing hither and thither at high speeds, colliding and rebounding according to the laws of elementary mechanics (see, Cercignani, Illner, and Pulvirenti 1994, pp. 8–12). In fact, Maxwell developed, first, a theory of transport processes and gave a heuristic derivation of the velocity distribution function that bears his name. Next, he developed a much more accurate model (Maxwell 1867), based on transfer equations, in fact, a model, according to which the molecules interact with a force inversely proportional to the fifth power of the distance between them (now commonly called Maxwellian molecules). With these transfer equations, Maxwell came very close to an evolution equation for the distribution, but this step (1872) must be credited to Ludwig Boltzmann (1844–1906). The equation under consideration is usually called the Boltzmann equation.

Thermo-statistics or Topology of the Microcanonical Entropy Surface

Abstract: Boltzmann’s principle S(E, N, V ...) = lnW(E, N, V ...) allows the interpretation of Statistical Mechanics of a closed system as Pseudo-Riemannian geometry in the space of the conserved parameters E, N, V ... (the conserved mechanical parameters in the language of Ruppeiner [1]) without invoking the thermodynamic limit. The topology is controlled by the curvature of S(E, N, V ...). The most interesting region is the region of (wrong) positive maximum curvature, the region of phase-separation. This is demonstrated among others for the equilibrium of a typical non-extensive system, a self-gravitating and rotating cloud in a spherical container at various energies and angular-momenta. A rich variety of realistic configurations, as single stars, multistar systems, rings and finally gas, are obtained as equilibrium microcanonical phases. The global phase diagram, the topology of the curvature, as function of energy and angular-momentum is presented. No exotic form of thermodynamics like Tsallis [2,3] non-extensive one is necessary. It is further shown that a finite (even mesoscopic) system approaches equilibrium with a change of its entropy ΔS ≥ 0 (Second Law) even when its Poincare recurrence time is not large.

Multiterm spherical tensor representation of Boltzmann's equation for a nonhydrodynamic weakly ionized plasma.

Abstract: The Boltzmann equation corresponding to a general ``multiterm'' representation of the phase space distribution function $f(\mathbf{r},\mathbf{c},t)$ for charged particles in a gas in an electric field was reformulated entirely in terms of spherical tensors ${f}_{m}^{l}$ some time ago, and numerous applications, including extension to time varying and crossed electric and magnetic fields, have followed. However, these applications have, by and large, been limited to the hydrodynamic conditions that prevail in swarm experiments and the full potential of the tensor formalism has thus never been realized. This paper resumes the discussion in the context of the more general nonhydrodynamic situation. Geometries for which a simple Legendre polynomial expansion suffices to represent f are discussed briefly, but the emphasis is upon cylindrical geometry, where such simplification does not arise. In particular, we consider an axisymmetric cylindrical column of weakly ionized plasma, and derive an infinite hierarchy of integrodifferential equations for the expansion coefficients of the phase space distribution function, valid for both electrons and ions, and for all types of binary interaction with neutral gas molecules.

Non-extensive Hamiltonian systems follow Boltzmann's principle not Tsallis statistics—phase transitions, Second Law of Thermodynamics

Abstract: Boltzmann's principle S(E,N,V)=k ln W(E,N,V) relates the entropy to the geometric area e S ( E , N , V ) of the manifold of constant energy in the N -body phase space. From the principle all thermodynamics and especially all phenomena of phase transitions and critical phenomena can be deduced. The topology of the curvature matrix C ( E , N ) (Hessian) of S ( E , N ) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Thus, C ( E , N ) describes all kinds of phase transitions with all their flavor. No assumptions of extensivity, concavity of S ( E ), additivity have to be invoked. Thus Boltzmann's principle and not Tsallis statistics describes the equilibrium properties as well the approach to equilibrium of extensive and non-extensive Hamiltonian systems. No thermodynamic limit must be invoked.

On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow

Abstract: On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow

GENERIC Projection-Operator Derivation of Boltzmanns Kinetic Equation

Abstract: The power of the atomistically founded GENERIC approach to nonequilibrium systems is illustrated by coarse graining from the full N-particle phase space to the level of single-particle distribution functions. For a dilute gas, the elements of GENERIC can be evaluated without any fundamental assumptions. Boltzmann’s famous kinetic equation is recovered.

Boltzmann-like modelling of a suspension

Abstract: This paper deals with the presentation of a kinetic model for a suspension of identical hard spheres. Considering that the collisions between particles are instantaneous, binary, inelastic and taking the diameter of the spheres into account, a Boltzmann equation for the dispersed phase is proposed. It allows one to obtain the conservation of mass and momentum as well as, for slightly inelastic collisions, an H-theorem which conveys the irreversibility of the evolution. The problem of the boundary conditions for the Boltzmann equation is then introduced. From an anisotropic law of rebound characterizing the inelastic and non-punctual impact of a particle to the wall, a parietal behavior for the first moments of the kinetic equation is deduced.

Transport equation and hard thermal loops in noncommutative Yang-Mills theory

Abstract: We show that the high temperature limit of the noncommutative thermal Yang-Mills theory can be directly obtained from the Boltzmann transport equation of classical particles As an illustration of the simplicity of the Boltzmann method, we evaluate the two- and the three-point gluon functions in the noncommutative $U(N)$ theory at high temperatures T These amplitudes are gauge invariant and satisfy simple Ward identities Using the constraint satisfied at order ${T}^{2}$ by the covariantly conserved current, we construct the hard thermal loop effective action of the noncommutative theory

Some new properties of the kinetic equation for the consistent boltzmann algorithm

Abstract: We study properties of the consistent Boltzmann algorithm for dense gases, using its limiting kinetic equation. First, we derive an H-theorem for this equation. Then, following the classical derivation by Chapman and Cowling, we find approximations to the equations of continuity, momentum, and energy. The first order correction terms with respect to the particle diameter turn out to be the same as for the Enskog equation. These results confirm previous derivations, based on the virial, of the corresponding equation of state.

Relaxation, the Boltzmann-Jeans Conjecture and Chaos

Abstract: Slow (logarithmic) relaxation from a highly excited state is studied in a Hamiltonian system with many degrees of freedom. The relaxation time is shown to increase as the exponential of the square root of the energy of excitation, in agreement with the Boltzmann-Jeans conjecture, while it is found to be inversely proportional to residual Kolmogorov-Sinai entropy, introduced in this Letter. The increase of the thermodynamic entropy through this relaxation process is found to be proportional to this quantity.