H-theorem for the (modified) nonlinear Enskog equation

We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that _t H 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=– k_B H becomes the exact equilibrium entropy of the hard-sphere fluid.     

LocalH-theorem for a kinetic variational theory

A localH-theorem is derived for a recently proposed extension of Enskog kinetic theory to a dense model fluid composed of particles with interactions extending beyond a hard core.On leave from: Katedra Fizyki, Uniwersytetu Szczecinskiego, 70-451 Szczecin, Poland.     

H-theorem for a linear kinetic theory

A strongH-theorem is proved for the approximate linear kinetic theory of Bawzdziewicz and Cichocki, obtained by truncating a transformed hierarchy of evolution equations. For an ith truncation we define an entropy functional that is strictly increasing in time, unless the ith reduced distribution function depends on position coordinates only. It also follows that the only stationary solution of the linear kinetic theory is the equilibrium solution. In addition, we show that the usual symmetry properties of equilibrium time correlation functions are preserved by the approximate kinetic theory under consideration.On leave of absence from Institute of Physics, Szczecin University, Wielkopolska 15, Szczecin, Poland.     

On the kinetic theory of a dense gas of rough spheres

The revised Enskog equation for a dense gas of rough spheres is considered. TheH theorem and the conservation equations are discussed.     

Nonexistence of H Theorem for Some Lattice Boltzmann Models

In this paper, we provide a set of sufficient conditions under which a lattice Boltzmann model does not admit an H theorem. By verifying the conditions, we prove that a number of existing lattice Boltzmann models does not admit an H theorem. These models include D2Q6, D2Q9 and D3Q15 athermal models, and D2Q16 and D3Q40 thermal (energy-conserving) models. The proof does not require the equilibria to be polynomials.     

A particle method with adjustable transport properties—the generalized consistent Boltzmann algorithm

The consistent Boltzmann algorithm (CBA) for dense, hard-sphere gases is generalized to obtain the van der Waals equation of state and the corresponding exact viscosity at all densities except at the highest temperatures. A general scheme for adjusting any transport coefficients to higher values is presented.     

On the Brownian motion of a massive sphere suspended in a hard-sphere fluid. I. Multiple-time-scale analysis and microscopic expression for the friction coefficient

The Fokker-Planck equation governing the evolution of the distribution function of a massive Brownian hard sphere suspended in a fluid of much lighter spheres is derived from the exact hierarchy of kinetic equations for the total system via a multiple-time-scale analysis akin to a uniform expansion in powers of the square root of the mass ratio. The derivation leads to an exact expression for the friction coefficient which naturally splits into an Enskog contribution and a dynamical correction. The latter, which accounts for correlated collisions events, reduces to the integral of a time-displaced correlation function of dynamical variables linked to the collisional transfer of momentum between the infinitively heavy (i.e., immobile) Brownian sphere and the fluid particles.     

Bulk and shear viscosities of a polydisperse hard-sphere fluid

The bulk and shear viscosity expressions for a multicomponent hard-sphere mixture in the Enskog transport theory are generalized to the polydisperse limit. The effect of polydispersity is expressed in terms of correction factors to the monodisperse fluid results. These correction factors have been evaluated for both a log-normal size distribution with the mass-size relation of a power-law form and a log-normal mass distribution with fixed particle size, which is the continuous limit of isotopes.     

Comment on symmetry properties of the linear Enskog kinetic operators

The one-particle average consistent with the structure of the revised Enskog theory is introduced. Symmetry properties of the linear kinetic operators reflecting those of theN-particle pseudo-Liouville operators are derived, implying a recently proved symmetry of kinetic expressions for equilibrium time correlation functions.     

Entropy Function Approach to the Lattice Boltzmann Method

In this paper, we present the construction of the Lattice Boltzmann method equipped with the H-theorem. Based on entropy functions whose local equilibria are suitable to recover the Navier–Stokes equations in the framework of the Lattice Boltzmann method, we derive a collision integral which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium. We discuss performance of this approach as compared to the standard realizations.     

On the Brownian motion of a massive sphere suspended in a hard-sphere fluid. II. Molecular dynamics estimates of the friction coefficient

The friction coefficient exerted by a hard-sphere fluid on an infinitely massive Brownian sphere is calculated for several size ratios , where and are the diameters of the Brownian and fluid spheres, respectively. The exact microscopic expression derived in part I of this work from kinetic theory is transformed and shown to be proportional to the time integral of the autocorrelation function of the momentum transferred from the fluid to the Brownian sphere during instantaneous collisions. Three different methods are described to extract the friction coefficient from molecular dynamics simulations carried out onfinite systems. The three independent methods lead to estimates of which agree within statisticalerrors (typically 5%). The results are compared to the predictions of Enskog theory and of the hydrodynamic Stokes law. The former breaks down as the size ratio and/or the packing fraction of the fluid increase. Somewhat surprisingly, Stokes' law is found to hold withstick boundary conditions, in the range 1/4.5 explored in the present simulations, with a hydrodynamic diameterd=. The analysis of the moleuclar dynamics data on the basis of Stokes' law withslip boundary conditions is less conclusive, although the right trend is found as / increases.     

Numerical analysis of a shock-wave solution of the Enskog equation obtained via a Monte Carlo method

In this paper a planar stationary shock-wave-like solution of the Enskog equation obtained via a Monte Carlo technique is studied; both the algorithm used to obtain the solution and the qualitative behavior of the macroscopic quantities are discussed in comparison with the corresponding solution of the Boltzmann equation.     

Exact solutions of the nonlinear Boltzmann equation

A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at ; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.Reference due to C. Cercignani.     

Critical dynamics in systems controlled by fractional kinetic equations

The field theory renormalization group is used for analyzing the fractional Langevin equation with the order of the temporal derivative 0<α<1$0<\alpha <1$, fractional Laplacian of the order σ$\sigma$, and Gaussian noise correlator. The case of non-linearity φ^m${\phi }^m$ with odd m≥3$m\ge 3$ is considered. It is proved that the model is multiplicatively renormalizable. Propagators were found in the momentum and coordinate representation, expressed in terms of Fox’s H functions.     

A scaling method for deriving kinetic equations from the BBGKY hierarchy

On the basis of the scale covariance of correlation functions under a coarsegraining in space and time, the Boltzmann equation for neutral gases, the Balescu-Lenard-Boltzmann-Landau equation for dilute plasmas, and linear equations for the variances of fluctuations are derived from the BBGKY hierarchy equations with no short-range correlations at the initial time. This is done by using Mori's scaling method in an extended form. Thus it is shown that the scale invariance of macroscopic features affords a useful principle in nonequilibrium statistical mechanics. It is also shown that there existtwo kinds of correlation functions, one describing the interlevel correlations of the kinetic level with its sublevels and the other representing the fluctuations in the kinetic level.Partially financed by the Scientific Research Fund of the Ministry of Education.     

H∞ consensus control of a class of second-order multi-agent systems without relative velocity measurement

This paper studies consensus control problems for a class of second-order multi-agent systems without relative velocity measurement. Some dynamic neighbour-based rules are adopted for the agents in the presence of external disturbances. A sufficient condition is derived to make all agents achieve consensus while satisfying desired H_∞ performance. Finally, numerical simulations are provided to show the effectiveness of our theoretical results.     

A lattice Bhatnagar--Gross--Krook model for a class of the generalized Burgers equations

A new lattice Bhatnagar--Gross--Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.     

Limit cycles for a class of second order differential equations

We study the limit cycles of a wide class of second order differential equations, which can be seen as a particular perturbation of the harmonic oscillator. In particular, by choosing adequately the perturbed function we show, using the averaging theory, that it is possible to obtain as many limit cycles as we want.     

The mechanism of the increase of the generalized dimension of a filtered chaotic time series

The determination of the attractor dimension from an experimental time series may be affected by the influence of filters which are incorporated into many measuring processes. While this is expected from the Kaplan-Yorke conjecture, we show that for one-dimensional maps a weak filter can induce a self-similarity which is responsible for the increase of the Hausdorff dimension. We are able to calculate the increase of the generalized dimensionD _q for the filtered time series of the logistic mapx _i +1=rx _i (1–x _i ) atr=4 analytically.