N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

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Fluctuation-dissipation: Response theory in statistical physics

Granular materials are ubiquitous in our daily lives. While they have been the subject of intensive engineering research for centuries, in the last two decades granular matter has attracted significant attention from physicists. Yet despite major efforts by many groups, the theoretical description of granular systems remains largely a plethora of different, often contradictory concepts and approaches. Various theoretical models have emerged for describing the onset of collective behavior and pattern formation in granular matter. This review surveys a number of situations in which nontrivial patterns emerge in granular systems, elucidates important distinctions between these phenomena and similar ones occurring in continuum fluids, and describes general principles and models of pattern formation in complex systems that have been successfully applied to granular systems.

/pdf/patterns-and-collective-behavior-in-granular-media-2t8b6sr2ja.pdf

Patterns and collective behavior in granular media: Theoretical concepts

▪ Abstract The recent avalanche of research activity in the field of granular matter has yielded much progress. The use of state-of-the-art (and other) computational and experimental methods has led to the discovery of new states and patterns and enabled detailed tests of theories and models. The application of statistical mechanical methods and phenomenology has contributed to the understanding of the microscopic a nd macroscopic properties of granular systems. Some previously open problems seem to be solved. Fluidized granular systems (rapid granular flows), recently referred to as granular gases, are often modeled by hydrodynamic equations of motion, some of which are based on systematic expansions applied to the pertinent Boltzmann equation. The undeniable success of granular hydrodynamics is somewhat surprising in view of the lack of scale separation in these systems and the neglect of certain correlations in most derivations of the hydrodynamic equations. Microstructures have been recognized as key ...

Rapid granular flows

(Dated: February 2, 2008)The rheological properties for dilute and moderately dense granular binary mixtures of smooth,inelastic hard disks/spheres under uniform shear ﬂow in steady state conditions are reported. Theresults are based on the Enskog kinetic theory, numerically solved by a dense gas extension of theDirect Simulation Monte Carlo method for dilute gases. These results are confronted to the onesalso obtained by performing molecular dynamics simulations with very good agreement for the lowerdensities and higher coeﬃcients of restitution. It appears that the range of densities for which theEnskog equation applies decreases with increasing dissipation.I. INTRODUCTION

http://arxiv.org/pdf/cond-mat/0411548.pdf

Rheology of granular mixtures under uniform shear flow: Enskog kinetic theory versus molecular dynamics simulations

The kinetic granular temperatures of a binary granular mixture in simple shear flow are determined from the Boltzmann kinetic theory by using a Sonine polynomial expansion. The results show that the temperature ratio is clearly different from unity (as may be expected since the system is out of equilibrium) and strongly depends on the restitution coefficients as well as on the parameters of the mixture. The approximate analytical calculations are compared with those obtained from Monte Carlo simulations of the Boltzmann equation showing an excellent agreement over the range of parameters investigated. Finally, the influence of the temperature differences on the rheological properties is also discussed.

Energy nonequipartition in a sheared granular mixture

Transport properties of granular gases driven by a stochastic bath with friction are determined. This type of thermostats attempt to mimic the effect of the interstitial fluid surrounding the solid particles. As a first step, the steady homogeneous state is analyzed. As with undriven granular gases, it can be seen that the kinetic equation admits a scaled solution where dependence on granular temperature is encoded not only through the scaled velocity \(\mathbf {c}=\mathbf {v}/\upsilon _{\text {th}}\) (\(\upsilon _{\text {th}}\) being the thermal velocity) but also through the (scaled) driven parameters. The Boltzmann kinetic equation is solved then by means of the Chapman–Enskog method around the homogeneous driven state. Momentum and heat fluxes are determined to first-order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. The relevant transport coefficients are identified and compared against computer simulations. Similarly to undriven systems, the theory compares quite well with simulations for conditions of practical interest. Finally, thermal diffusion segregation for driven granular mixtures is also analyzed.

Transport Properties for Driven Granular Gases

The present chapter provides a concise introduction to the kinetic theory of granular gases (namely, gases of hard spheres with inelastic collisions) at low and moderate densities. We briefly review first the dynamics of binary collisions for some of the models most widely used in the literature, and then we outline heuristically the derivation of the Boltzmann and Enskog kinetic equations for monocomponent granular gases. A connection with hydrodynamics is established where the corresponding macroscopic balance equations for the densities of mass, momentum and energy are exactly derived from the above kinetic equations with expressions for the momentum and heat fluxes and the cooling rate as functionals of the one-particle velocity distribution function. These kinetic equations are then extended to the interesting case of multicomponent granular mixtures. The complexity of the kernel of the Enskog-Boltzmann collision operator, however, prevents the possibility of obtaining exact results, and for this reason there is often a preference for kinetic models that are mathematically simpler than the original equations but capture their most relevant physical properties. Thus, the chapter ends with the construction of some kinetic models proposed in the literature of granular gases based on the popular BGK model for ordinary (elastic) gases.

Kinetic Theory of Inelastic Hard Spheres

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures Ti (measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows.

https://www.mdpi.com/1099-4300/24/6/826/pdf?version=1655203292

Vicente Garzó

Papers

Rheology of granular mixtures under uniform shear flow: Enskog kinetic theory versus molecular dynamics simulations

Energy nonequipartition in a sheared granular mixture

Transport Properties for Driven Granular Gases

Kinetic Theory of Inelastic Hard Spheres

Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties—A Review