Showing papers on "Boltzmann constant published in 2004"

Fluctuations and Irreversibility: An Experimental Demonstration of a Second-Law-Like Theorem Using a Colloidal Particle Held in an Optical Trap

Abstract: The puzzle of how time-reversible microscopic equations of mechanics lead to the time-irreversible macroscopic equations of thermodynamics has been a paradox since the days of Boltzmann. Boltzmann simply sidestepped this enigma by stating "as soon as one looks at bodies of such small dimension that they contain only very few molecules, the validity of this theorem [the second law of thermodynamics and its description of irreversibility] must cease." Today we can state that the transient fluctuation theorem (TFT) of Evans and Searles is a generalized, second-law-like theorem that bridges the microscopic and macroscopic domains and links the time-reversible and irreversible descriptions. We apply this theorem to a colloidal particle in an optical trap. For the first time, we demonstrate the TFT in an experiment and show quantitative agreement with Langevin dynamics.

Boltzmann test of Slonczewski's theory of spin-transfer torque

Abstract: We use a matrix Boltzmann equation formalism to test the accuracy of Slonczewski's theory of spin-transfer torque in thin-film heterostructures where a nonmagnetic spacer layer separates two noncollinear ferromagnetic layers connected to nonmagnetic leads. When applicable, the model predictions for the torque as a function of the angle between the two ferromagnets agree extremely well with the torques computed from a Boltzmann equation calculation. We focus on asymmetric structures (where the two ferromagnets and two leads are not identical) where the agreement pertains to an analytic formula for the torque derived by us using Slonczewski's theory. In almost all cases, we can predict the correct value of the model parameters directly from the geometric and transport properties of the multilayer. For some asymmetric geometries, we predict a mode of stable precession that does not occur for the symmetric case studied by Slonczewski.

Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions

Abstract: We study high-energy asymptotics of the steady velocity distributions for model kinetic equations describing various regimes in dilute granular flows. The main results obtained are integral estimates of solutions of the Boltzmann equation for inelastic hard spheres, which imply that steady velocity distributions behave in a certain sense as C exp(−r∣v∣ s ), for ∣v∣ large. The values of s, which we call the orders of tails, range from s = 1 to s = 2, depending on the model of external forcing. To obtain these results we establish precise estimates for exponential moments of solutions, using a sharpened version of the Povzner-type inequalities.

Individual and average behavior of particles in a protoplanetary nebula

Abstract: We study the interaction between gas and particles in a protoplanetary disk, using both analytical and numerical approaches. We first present analytical expressions for the trajectories of individual particles undergoing gas drag in the disk, in the asymptotic cases of very small particles (Epstein regime) and very large particles (Stokes regime). Using a Boltzmann averaging method, we obtain an analytic expression for the evolution of the average density, velocity, and dispersion of the particles as a function of distance above the midplane of the disk. Using successive moments of the Boltzmann equation, we derive the equivalent fluid equations for the average motion of the particles; these are intrinsically different in the Epstein and Stokes regimes. A simple closure of the moment equations is proposed in both regimes. These fluid equations provide much better prospects for the study of more complex problems related to protoplanetary accretion disks, since for general initial size and phase-space distributions the evolution of the average behavior of the particles can be evaluated numerically with much less computational time than that required for the numerical integration of the orbits of all individual particles. In a companion paper, for instance, we use them for the analysis of a shearing instability induced by the sedimentation of the particles. In the present work we test the adequacy of the fluid formulation against a set of idealized numerical experiments. In the Epstein regime, we study an idealized uniform initial distribution of small particles. We obtain a set of analytic solutions for the fluid equations, which are found to be in good agreement with those obtained from numerical integration of the orbits of many particles. We also verify that any initial velocity dispersion is quickly damped out by the surrounding gas on the short stopping timescale, which provides closure and justifies the description of the particles as a fluid with a linear drag force and negligible pressure. In the Stokes regime, as the large particles oscillate across the midplane with declining amplitude, their velocity dispersion remains comparable to their average speed. Their sedimentation is analogous to the cooling of a pressure-supported fluid. We propose an empirical closure scheme for the moment equations of the Stokes particles fluid and test it against idealized numerical experiments. In both cases, this method can eventually be applied to study the evolution of particle distributions in protostellar disks after additional effects such as collision, sublimation, and condensation are included.

Generalized Boltzmann Physical Kinetics

Abstract: Preface Historical introduction and the problem formulation Chapter 1. Generalized Boltzmann Equation Chapter 2. Theory of generalized hydrodynamic equations Chapter 3. Strict theory of turbulence and some applications of the generalized hydrodynamic theory Chapter 4. Physics of a weakly ionized gas Chapter 5. Kinetic coefficients in the theory of the generalized kinetic equations Chapter 6. Some applications of the generalized Boltzmann physical kinetics Chapter 7. Numerical simulation of vortex gas flow using the generalized Euler equations Chapter 8. Generalized Boltzmann physical kinetics in physics of plasma and liquids Appendix 1. Derivation of energy equation for invariant E_alpha = (m_alpha V_alpha^2)/2 + epsilon_alpha Appendix 2. Three-diagonal method of Gauss elimination technique for the differential third order equation Appendix 3. Some integral calculations in the generalized Navier-Stokes approximation Appendix 4. Three-diagonal method of Gauss elimination technique for the differential second order equation Appendix 5. Characteristic scales in plasma physics Appendix 6. Dispersion relations in the generalized Boltzmann kinetic theory neglecting the integral collision term References Subject index

Macroscopic limits of the Boltzmann equation: a review

Abstract: This document is concerned with the modeling of particle systems via kinetic equations. First, the hierarchy of available models for particle systems is reviewed, from particle dynamics to fluid models through kinetic equations. In particular the derivation of the gas dynamics Boltzmann equation is recalled and a few companion models are discussed. Then, the basic properties of kinetic models and particularly of the Boltzmann collision operator are reviewed. The core of this work is the derivation of macroscopic models (as e.g., the Euler or Navier-Stokes equations) from the Boltzmann equation by means of the Hilbert and ChapmanEnskog methods. This matter is first discussed in the context of the BGK equation, which is a simpler model than the full Boltzmann equation. The extension to the Boltzmann equation is summarized at the end of this discussion. Finally, a certain number of current research directions are reviewed. Our goal is to give a synthetic description of this subject, so as to allow the reader to acquire a rapid knowledge of the basic aspects of kinetic theory. The reader is referred to the bibliography for more details on the various items which are reviewed here.

Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski

Abstract: We propose a formal extension of thermodynamics and kinetic theories to a larger class of entropy functionals. Kinetic equations associated to Boltzmann, Fermi, Bose and Tsallis entropies are recovered as a special case. This formalism first provides a unifying description of classical and quantum kinetic theories. On the other hand, a generalized thermodynamical framework is justified to describe complex systems exhibiting anomalous diffusion. Finally, a notion of generalized thermodynamics emerges in the context of the violent relaxation of collisionless stellar systems and two-dimensional vortices due to the existence of Casimir invariants and incomplete relaxation. A thermodynamical analogy can also be developed to analyze the non-linear dynamical stability of stationary solutions of the Vlasov and 2D Euler–Poisson systems. On general grounds, we suggest that generalized entropies arise due to the existence of “hidden constraints” that modify the form of entropy that we would naively expect. Generalized kinetic equations are therefore “effective” equations that are introduced heuristically to describe complex systems.

Effective temperature in nonequilibrium steady states of Langevin systems with a tilted periodic potential

Abstract: We theoretically study Langevin systems with a tilted periodic potential. It is known that the ratio Theta of the diffusion constant D to the differential mobility mu(d) is not equal to the temperature of the environment (multiplied by the Boltzmann constant), except in the linear response regime, where the fluctuation dissipation theorem holds. In order to elucidate the physical meaning of Theta far from equilibrium, we analyze a modulated system with a slowly varying potential. We derive a large scale description of the probability density for the modulated system by use of a perturbation method. The expressions we obtain show that Theta plays the role of the temperature in the large scale description of the system and that Theta can be determined directly in experiments, without measurements of the diffusion constant and the differential mobility. Hence the relation D= mu(d) Theta among the independent measurable quantities D, mu(d), and Theta can be interpreted as an extension of the Einstein relation.

Kinetic and Hydrodynamic Models of Nearly Elastic Granular Flows

Abstract: We introduce and discuss certain models of dilute granular systems of spheres with dissipative collisions and variable coefficient of restitution under the assumption of weak inelasticity. The dissipation is taken into account by introducing a correction to the Boltzmann collision operator in the form of a nonlinear friction type operator. Using this correction we obtain formally from the Boltzmann equation in a direct way a hydrodynamic description of a system of nearly elastic particles colliding with a variable coefficient of restitution. In one dimension of the velocity variable, the correction reduces to the nonlinear friction operator obtained in [24] as the quasi-elastic limit of a model Boltzmann equation for partially inelastic spheres. The large-time asymptotic of this one-dimensional model can be described in detail.

Diffusion approximation for the one dimensional Boltzmann-Poisson system

Abstract: The diffusion limit of the initial-boundary value problem for the Boltzmann-Poisson system is studied in one dimension. By carefully analyzing entropy production terms due to the boundary, $L^p$ estimates are established for the solution of the scaled Boltzmann equation (coupled to Poisson) with well prepared initial and boundary conditions. A hybrid Hilbert expansion taking advantage of the regularity of the limiting system allows to prove the convergence of the solution towards the solution of the Drift-Diffusion-Poisson system and to exhibit a convergence rate.

Model for computing superconfiguration temperatures in nonlocal-thermodynamic-equilibrium hot plasmas.

Abstract: A model is presented where the level-population densities in quasi-steady-state hot dense plasmas are described by means of large nonrelativistic superconfigurations (SC's), whose configuration populations follow a decreasing-exponential law versus energy (Boltzmann like) for a temperature depending on the SC. Two systems of linear equations are obtained. The first one yields the average-state population densities of the SC's. Using these results, the second system yields the SC temperatures. In this model, a very large number of atomic levels is accounted for in a simple way, thus yielding the configuration populations and, hence, the ionic distribution and average charge. It also yields accurate simulations of the spectra, which are of the essence for emissivity and absorption calculations. It opens a way to time-dependent calculations.

Micro- and nanoscale non-ideal gas Poiseuille flows in a consistent Boltzmann algorithm model

Abstract: The direct simulation Monte Carlo method in the consistent Boltzmann algorithm model has been developed and expanded for non-ideal gas predictions. The enhanced collision rate factor is determined by considering the excluded molecular volume and shadowing/screening effects based on the Enskog theory. The parameter for the attraction strength is also determined by comparison with the classical thermodynamics theory. Different pressure-driven gas Poiseuille flows in micro- and nanoscale channels are investigated. The van der Waals effect leads to a higher mass flow rate and different friction and heat transfer characteristics on the wall surface, compared to the results in the perfect gas model. The results also show that the van der Waals effect is dependent not only on the pressure but also on the channel size. A higher driving pressure or a smaller channel size will result in a larger van der Waals gas effect.

A New Thermodynamics from Nuclei to Stars

Abstract: Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with the given total energy. Due to Boltzmann's principle, eS=tr(δ(E-H)), its geometrical size is related to the entropy S(E,N,...). This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption, as are needed in conventional (canonical) thermo-statistics. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the fundamental definition of any classical equilibrium statistics. It can address nuclei and astrophysical objects as well. All kind of phase transitions can be distinguished sharply and uniquely for even small systems. It is further shown that the second law is a natural consequence of the statistical nature of thermodynamics which describes all systems with the same -- redundant -- set of few control parameters simultaneously. It has nothing to do with the thermodynamic limit. It even works in systems which are by far than any thermodynamic "limit".

Entropic Lattice Boltzmann Simulation of the Flow Past Square Cylinder

Abstract: Minimal Boltzmann kinetic models, such as lattice Boltzmann, are often used as an alternative to the discretization of the Navier–Stokes equations for hydrodynamic simulations. Recently, it was argued that modeling sub-grid scale phenomena at the kinetic level might provide an efficient tool for large scale simulations. Indeed, a particular variant of this approach, known as the entropic lattice Boltzmann method (ELBM), has shown that an efficient coarse-grained simulation of decaying turbulence is possible using these approaches. The present work investigates the efficiency of the entropic lattice Boltzmann in describing flows of engineering interest. In order to do so, we have chosen the flow past a square cylinder, which is a simple model of such flows. We will show that ELBM can quantitatively capture the variation of vortex shedding frequency as a function of Reynolds number in the low as well as the high Reynolds number regime, without any need for explicit sub-grid scale modeling. This extends the previous studies for this set-up, where experimental behavior ranging from Re~O(10) to Re≤1000 was predicted by a single simulation algorithm.1–5

Charge transport in 1D silicon devices via Monte Carlo simulation and Boltzmann‐Poisson solver

Abstract: Time‐depending solutions to the Boltzmann‐Poisson system in one spatial dimension and three‐dimensional velocity space are obtained by using a recent finite difference numerical scheme. The collision operator of the Boltzmann equation models the scattering processes between electrons and phonons assumed in thermal equilibrium. The numerical solutions for bulk silicon and for a one‐dimensional n+‐n‐n+ silicon diode are compared with the Monte Carlo simulation. Further comparisons with the experimental data are shown.

Viscosity and thermalization

Abstract: I solve the relativistic Navier stokes equations assuming boost invariance and rotational symmetry. I compare the resulting numerical solutions for two limiting models of the shear viscosity. In the first model the shear viscosity is made proportional to the temperature. Thus, η ∝ T/σ0 where σ0 is some fixed cross section (perhaps σ0 ~ Λ−2QCD). This viscosity model is typical of the classical Boltzmann simulations of Gyulassy and Molnar. In the second model the shear viscosity is made proportional to T3. This model is typical of high temperature QCD. When the initial mean free path of the T3 model is four times larger than the T/σ0 model, the two models of viscosity produce the same radial flow. This result can be understood with simple scaling arguments. Thus, the large transport opacity needed in classical Boltzmann simulations is in part an artefact of the fixed scale σ0 in these models.

Thermodynamic stabilities of the generalized Boltzmann entropies

Abstract: We consider the thermodynamic stability conditions (TSC) on the Boltzmann entropies generalized by Tsallis’ q- and Kaniadakis’ κ-deformed logarithmic functions. It is shown that the corresponding TSCs are not necessarily equivalent to the concavity of the generalized Boltzmann entropies with respect to internal energy. Nevertheless, both the TSCs are equivalent to the positivity of standard heat capacity.

Does the Boltzmann principle need a dynamical correction

Abstract: In an attempt to derive thermodynamics from classical mechanics, an approximate expression for the equilibrium temperature of a finite system has been derived (M. Bianucci, R. Mannella, B. J. West and P. Grigolini, Phys. Rev. E 51: 3002 (1995)) which differs from the one that follows from the Boltzmann principle S = klnΩ(E) via the thermodynamic relation 1/T=∂S / ∂E by additional terms of “dynamical” character, which are argued to correct and generalize the Boltzmann principle for small systems (here Ω(E) is the area of the constant-energy surface). In the present work, the underlying definition of temperature in the Fokker–Planck formalism of Bianucci et al., is investigated and shown to coincide with an approximate form of the equipartition temperature. Its exact form, however, is strictly related to the “volume” entropy S = k ln Ф(E) via the thermodynamic relation above for systems of any number of degrees of freedom (Ф(E) is the phase space volume enclosed by the constant-energy surface). This observation explains and clarifies the numerical results of Bianucci et al., and shows that a dynamical correction for either the temperature or the entropy is unnecessary, at least within the class of systems considered by those authors. Explicit analytical and numerical results for a particle coupled to a small chain (N~10) of quartic oscillators are also provided to further illustrate these facts.

Boltzmann-Gibbs Entropy Versus Tsallis Entropy: Recent Contributions to Resolving the Argument of Einstein Concerning “Neither Herr Boltzmann nor Herr Planck has Given a Definition of W”?

Abstract: Classical statistical mechanics of macroscopic systems in equilibrium is based on Boltzmann's principle. Tsallis has proposed a generalization of Boltzmann-Gibbs statistics. Its relation to dynamics and nonextensivity of statistical systems are matters of intense investigation and debate. This essay review has been prepared at the occasion of awarding the 'Mexico Prize for Science and Technology 2003'to Professor Constantino Tsallis from the Brazilian Center for Research in Physics.

Stable Equilibrium Based on Lévy Statistics: A Linear Boltzmann Equation Approach

Abstract: To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m<

Burnett equations for the ellipsoidal statistical BGK model

Abstract: In order to discuss the agreement of the ellipsoidal statistical BGK (ES-BGK) model with the Boltzmann equation, Burnett equations are computed by means of the second-order Chapman-Enskog expansion of the ES-BGK model. It is found that the Burnett equations for the ES-BGK model with the correct Prandtl number are identical to the Burnett equations for the Boltzmann equation for Maxwell molecules (fifth-order power potentials). However, for other types of particle interaction, the Boltzmann Burnett equations cannot be reproduced from the ES-BGK model.

A Note on the Energy-Transport Limit of the Semiconductor Boltzmann Equation

Abstract: In this paper, we present a new scaling limit of the semiconductor Boltzmann equation which yields the so-called Energy-Transport model. This model consists of a set of continuity equations for the density and energy together with constitutive relations for the particle and energy fluxes. These fluxes are expressed in terms of gradients of the entropic variables, through a diffusivity matrix related to the Boltzmann collision operators. The present work is devoted to a scaling limit in which the only operator involved in the definition of the diffusivity is the elastic collision operator. Previous derivations required a two-step procedure, resorting to an intermediate model, the so-called Spherical Harmonics Expansion (or SHE) model. We shall present and review the relationship between all these models.

Convergence in higher mean of a random Schroedinger to a linear Boltzmann evolution

Abstract: We study the macroscopic scaling and weak coupling limit for a random Schroedinger equation on Z^3. We prove that the Wigner transforms of a large class of "macroscopic" solutions converge in r-th mean to solutions of a linear Boltzmann equation, for any finite value of r in R_+. This extends previous results where convergence in expectation was established.

Efficiency in the Generation of the Boltzmann−Gibbs Distribution by the Tsallis Dynamics Reweighting Method

Abstract: We numerically investigated deterministic methods for constructing the Boltzmann−Gibbs (BG) distributions used in molecular dynamics simulations. Tsallis dynamics (TD) is a deterministic dynamical ...

Electron and phonon relaxation in metal films perturbed by a femtosecond laser pulse

Abstract: A theoretical investigation of the electron and phonon time-dependent distributions in an Ag film subjected to a femtosecond laser pulse has been carried out. A system of two coupled time-dependent Boltzmann equations, describing electron and phonon dynamics, has been numerically solved. In the electron Boltzmann equation, electron–electron and electron–phonon collision integrals are considered together with a source term for laser perturbation. In the phonon Boltzmann equation, only electron–phonon collisions are considered, neglecting laser perturbation and phonon–phonon collisions. Screening of the interactions has been accounted for in both the electron–electron and the electron–phonon collisions. The results show the simultaneous electron and phonon time-dependent distributions from the initial non-equilibrium behaviour up to the establishment of a new final equilibrium condition.