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.     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Generalized Kinetic Equation for Far-from-Equilibrium Many-Body Systems

A kinetic equation for the single particle distribution function in an open many-body system, when in far away from equilibrium conditions is derived in the context of a Non-Equilibrium Thermo-Statistics of ample scope. It consists of a generalization of traditional kinetic equations in that no restrictions are imposed on the characteristics of the nonequilibrium thermodynamic state of the system. This kinetic equation do contain some contributions that become relevant in systems with a nonlinear kinetics when driven sufficiently far from equilibrium (certain complex systems). Moreover, the handling of the kinetic equation in a multiple-moment approach provides a generalized nonlinear higher-order thermo-hydrodynamics.     

Initial state dependence of nonlinear kinetic equations: The classical electron gas

The method of nonequilibrium cluster expansion is used to stydy the decay to equilibrium of a weakly coupled inhomogeneous electron gas prepared in a local equilibrium state at the initial time,t=0. A nonlinear kinetic equation describing the long time behavior of the one-particle distribution function is obtained. For consistency, initial correlations have to be taken into account. The resulting kinetic equation-differs from that obtained when the initial state of the system is assumed to be factorized in a product of one-particle functions. The question of to what extent correlations in the initial state play an essential role in determining the form of the kinetic equation at long times is discussed. To that end, the present calculations are compared with results obtained before for hard sphere gases and in general gases with strong short-range forces. A partial answer is proposed and some open questions are indicated.     

Dynamical model for nonlinear mirror modes near threshold

Using a reductive perturbative expansion of the Vlasov-Maxwell (VM) equations for magnetized plasmas, a pseudodifferential equation of gradient type is derived for the nonlinear dynamics of mirror modes near the instability threshold. This model, where kinetic effects arise at a linear level only, develops a finite-time singularity, indicating the existence of a subcritical bifurcation. A saturation mechanism based on the local variations of the ion Larmor radius, is then phenomenologically supplemented. In contrast with previous models where saturation is due to the cooling of a population of trapped particles, the resulting equation correctly reproduces results of numerical simulations of VM equations, such as the development of magnetic humps from an initial noise, and the existence of stable large-amplitude magnetic holes both below and slightly above threshold.     

Kinetic modeling of a slow flow CW CO2 laser

A kinetic model for analysis of the slow-flow CW-discharge CO₂ laser with diffusion cooling has been developed in which the gas temperature is obtained from energy balance equations. The method is based on the numerical solution of a set of nonlinear differential equations for vibrational kinetics. The numerical predictions from the model are compared with some experimental results and a good agreement is obtained.     

Nonlinear fir magneto-photoconductivity on quasi-bound Coulomb states of light holes in high purity p-Ge

The far-infrared magneto-photoconductivity due to optical transitions from the acceptor ground state into quasi-bound Coulomb states in p-Ge has been investigated at low temperatures as a function of intensity applying a high-power cw molecular laser. For intensities above about 1 mW/cm² the photoconductive signal shows a square root dependence on intensity, which is attributed to nonlinear free carrier capture in the low compensated material. The experimental results are analyzed in terms of a rate equation model yielding the kinetic parameters of the carrier generation- and recombination process involved.     

强迫耗散系统的有序结构和系统的发展(Ⅰ)，最小熵产生原理和有序结构

一个系统的发展总是由不可逆热力过程和非线性动力过程所驱动.将大气动力学方程组同考虑了动能变化的Gibbs关系结合起来构建的熵平衡方程，才能更好地描述大气系统的不可逆热力过程和非线性动力过程.至今非平衡态热力学仅利用Onsager线性唯象关系证明了最小熵产生原理.利用新建立的熵平衡方程和大气动力学方程的性质证明，最小熵产生原理在热力学线性区和非线性区都是普遍成立的.且当热量输送平衡、水汽输送平衡和动量输送平衡时，系统达到不可逆过程最弱的最小熵产生热力学状态.当系统又是动力平衡且无平流时，这种最小熵产生态就是 关键词： 非线性热力学 熵产生 最小熵产生原理 有序结构     

Autocorrelation functions in the generalized Boltzmann-Enskog model

The purpose of this paper is to study the asymptotic properties of time autocorrelation functions for the generalized nonlinear Boltzmann-Enskog model, which contains a long-range component of the interaction between the particles. On the basis of the analysis of non-linear features of the Boltzmann-Enskog kinetic equation, the role of nonlinear effects is directly revealed at the approach to an equilibrium state. It is shown that autocorrelation functions have power asymptotics t ^?3/2, and the effects that are related to the inclusion of the long-range component lead to a change in the coefficient at t ^?3/2. These results establish a closed expression for the determination of coefficients in the asymptotic expansion of the autocorrelation functions of rate and thermal diffusion.     

Uncondensed atoms in the regime of velocity-selective coherent population trapping

We consider the model of a Bose condensate in the regime of velocity-selective coherent population trapping. As a result of interaction between particles, some fraction of atoms is outside the condensate, remaining in the coherent trapping state. These atoms are involved in brief events of intense interaction with external resonant electromagnetic fields. Intense induced and spontaneous transitions are accompanied by the exchange of momenta between atoms and radiation, which is manifested as migration of atoms in the velocity space. The rate of such migration is calculated. A nonlinear kinetic equation for the many-particle statistical operator for uncondensed atoms is derived under the assumption that correlations of atoms with different momenta are insignificant. The structure of its steady-state solution leads to certain conclusions about the above-mentioned migration pattern taking the Bose statistics into consideration. With allowance for statistical effects, we derive nonlinear integral equations for frequencies controlling the migration. The results of numerical solution of these equations are represented in the weak interatomic interaction approximation.     

Propagating fronts,chaos and multistability in a cell replication model

Numerical solutions to a model equation that describes cell population dynamics are presented and analyzed. A distinctive feature of the model equation (a hyperbolic partial differential equation) is the presence of delayed arguments in the time (t) and maturation (x) variables due to the nonzero length of the cell cycle. This transport like equation balances a linear convection with a nonlinear, nonlocal, and delayed reaction term. The linear convection term acts to impress the value of u(t,x=0) on the entire population while the death term acts to drive the population to extinction. The rich phenomenology of solution behaviour presented here arises from the nonlinear, nonlocal birth term. The existence of this kinetic nonlinearity accounts for the existence and propagation of soliton-like or front solutions, while the increasing effect of nonlocality and temporal delays acts to produce a fine periodic structure on the trailing part of the front. This nonlinear, nonlocal, and delayed kinetic term is also shown to be responsible for the existence of a Hopf bifurcation and subsequent period doublings to apparent "chaos" along the characteristics of this hyperbolic partial differential equation. In the time maturation plane, the combined effects of nonlinearity, nonlocality, and delays leads to solution behaviour exhibiting spatial chaos for certain parameter values. Although analytic results are not available for the system we have studied, consistency and validation of the numerical results was achieved by using different numerical methods. A general conclusion of this work, of interest for the understanding of any biological system modeled by a hyperbolic delayed partial differential equation, is that increasing the spatio-temporal delays will often lead to spatial complexity and irregular wave propagation. (c) 1996 American Institute of Physics.     

Brownian particles with long- and short-range interactions

We develop the kinetic theory of Brownian particles with long- and short-range interactions. Since the particles are in contact with a thermal bath fixing the temperature T, they are described by the canonical ensemble. We consider both overdamped and inertial models. In the overdamped limit, the evolution of the spatial density is governed by the generalized mean field Smoluchowski equation including a mean field potential due to long-range interactions and a generically nonlinear barotropic pressure due to short-range interactions. This equation describes various physical systems such as self-gravitating Brownian particles (Smoluchowski-Poisson system), bacterial populations experiencing chemotaxis (Keller-Segel model) and colloidal particles with capillary interactions. We also take into account the inertia of the particles and derive corresponding kinetic and hydrodynamic equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations. For each model, we provide the corresponding form of free energy and establish the H-theorem and the virial theorem. Finally, we show that the same hydrodynamic equations are obtained in the context of nonlinear mean field Fokker-Planck equations associated with generalized thermodynamics. However, in that case, the nonlinear pressure is due to the bias in the transition probabilities from one state to the other leading to non-Boltzmannian distributions while in the former case the distribution is Boltzmannian but the nonlinear pressure arises from the two-body correlation function induced by the short-range potential of interaction. As a whole, our paper develops connections between the topics of long-range interactions, short-range interactions, nonlinear mean field Fokker-Planck equations and generalized thermodynamics. It also justifies from a kinetic theory based on microscopic processes, the basic equations that were introduced phenomenologically to describe self-gravitating Brownian particles, chemotaxis and colloidal suspensions with attractive interactions.     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.     

Temperature profile and heat flux inhibition in laser-driven plasmas

It is shown that the nonlinear heat flux calculated by means of the modified moment method applied to the Vlasov-Landau kinetic equation is adequate to account for the heat flux inhibition in laser-driven plasmas, if the nonlinear heat flux so obtained is used in the steady state energy balance equation to calculate the temperature profile self-consistently. Heat flux limit formulas are obtained for supersonic and subsonic fluid speeds. The flux limit factors have an upper bound of 0.14 in the case of the hydrogen plasma. This bound would be reduced, if the temperature at the sonic surface was less than the electron temperature at the critical density surface.     

Evolution of acoustic waves in microinhomogeneous media with quadratic elastic nonlinearity and relaxation

A nonlinear wave equation for the velocity “relaxator” is derived in the framework of the rheological model and the corresponding equation of state of a microinhomogeneous medium containing viscoelastic defects with quadratic nonlinear elasticity. The equation is qualitatively analyzed, and numerical solutions to it are presented for a stationary symmetric shock wave and the evolution of initially harmonic waves.     

Analytical and Numerical Studies of the One-Dimensional Spin Facilitated Kinetic Ising Model

The one-dimensional spin facilitated kinetic Ising model is studied analytically using the master equation and by simulations. The local state of the spins (corresponding to mobile and immobile cells) can change depending on the state of the neighbored spins, which reflects the high cooperativity inherent in glassy materials. The short-time behavior is analyzed using a Fock space representation for the master equation. The hierarchy of evolution equations for the averaged spin state and the time dependence of the spin autocorrelation function are calculated with different methods (mean-field theory, expansion in powers of the time, partial summation) and compared with numerical simulations. The long-time behavior can be obtained by mapping the one-dimensional spin facilitated kinetic Ising model onto a one-dimensional diffusion model containing birth and death processes. The resulting master equation is solved by van Kampen's size expansion, which leads to a Langevin equation with Gaussian noise. The predicted autocorrelation function and the global memory offer in the long-time limit a screened algebraic decay and a stretched exponential decay, respectively, consistent with numerical simulations.     

Nonlinear Dust Acoustic Waves in Strongly Coupled Dusty Plasmas with Positively Charged Dust Particles

The nonlinear propagation of dust acoustic waves is investigated in four-component plasmas consisting of positively charged dust grains, trapped ions, nonthermal electrons, and photoelectron due to ultraviolet irradiation.We use generalized viscoelastic hydrodynamic model for strongly coupled dust grain. In the weak nonlinearity limit, a modified Kadomstev–Petviashvili(KP) equation and a modified KP-Burger equation, which have a damping term coming from nonadiabatic charge variation, have been derived in the kinetic regime and hydrodynamic regime, respectively. With the increasing of UV photon flux, the hydrodynamic regime changes to kinetic regime. The approximate analytical line soliton and shock solutions are investigated in the kinetic regime and hydrodynamic regime, respectively.     

Effect of Defects on the Spatial Localization of Nonlinear Acoustic Waves

A self-consistent mathematical model that includes equations of elasticity theory and kinetic equations for the density of different types of point defects is reduced to a nonlinear equation of evolution that combines the familiar Korteweg–de Vries–Burgers and Klein–Gordon equations of wave dynamics. Exact analytical solutions for this equation are found and analyzed.