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.

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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

We study in this work a steady shearing laminar flow with null heat flux (usually called "uniform shear flow") in a gas-solid suspension at low density. The solid particles are modeled as a gas of smooth hard spheres with inelastic collisions while the influence of the surrounding interstitial fluid on the dynamics of grains is modeled by means of a volume drag force, in the context of a rheological model for suspensions. The model is solved by means of three different but complementary routes, two of them being theoretical (Grad's moment method applied to the corresponding Boltzmann equation and an exact solution of a kinetic model adapted to granular suspensions) and the other being computational (Monte Carlo simulations of the Boltzmann equation). Unlike in previous studies on granular sheared suspensions, the collisional moment associated with the momentum transfer is determined in Grad's solution by including all the quadratic terms in the stress tensor. This theoretical enhancement allows for the detection and evaluation of the normal stress differences in the plane normal to the laminar flow. In addition, the exact solution of the kinetic model gives the explicit form of the velocity moments of the velocity distribution function. Comparison between our theoretical and numerical results shows in general a good agreement for the non-Newtonian rheological properties, the kurtosis (fourth velocity moment of the distribution function), and the velocity distribution of the kinetic model for quite strong inelasticity and not too large values of the (scaled) friction coefficient characterizing the viscous drag force. This shows the accuracy of our analytical results that allows us to describe in detail the flow dynamics of the granular sheared suspension.

Non-Newtonian hydrodynamics for a dilute granular suspension under uniform shear flow.

Diffusion induced by a thermal gradient in a granular binary mixture is analyzed in the context of the (inelastic) Enskog equation. Although the Enskog equation neglects velocity correlations among particles which are about to collide, it retains spatial correlations arising from volume exclusion effects and thus it is expected to apply to moderate densities. In the steady state with gradients only along a given direction, a segregation criterion is obtained from the thermal diffusion factor $\Lambda$ measuring the amount of segregation parallel to the thermal gradient. As expected, the sign of the factor $\Lambda$ provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the parameters of the mixture (masses, sizes, concentration, solid volume fraction, and coefficients of restitution). The form of the phase diagrams for the BNE/RBNE transition is illustrated in detail for several systems, with special emphasis on the significant role played by the inelasticity of collisions. In particular, an effect already found in dilute gases (segregation in a binary mixture of identical masses and sizes {\em but} different coefficients of restitution) is extended to dense systems. A comparison with recent computer simulation results shows a good qualitative agreement at the level of the thermal diffusion factor. The present analysis generalizes to arbitrary concentration previous theoretical results derived in the tracer limit case.

Thermal diffusion segregation in granular binary mixtures described by the Enskog equation

The Ikenberry-Truesdell exact solution to the Boltzmann equation for Maxwell molecules is revisited. This solution refers to a state characterized by a linear profile of the velocity flow and spatially uniform density and temperature. The solution is extended to include explicit expressions for the fourth-degree moments. It is shown that if the shear rate is larger than a certain critical value, the fourth-degree moments do not reach stationary values, even when the temperature is kept constant. The explicit shear-rate dependence of the moments below this critical value are obtained.

Exact moment solution of the Boltzmann equation for uniform shear flow

A method is devised to measure the first-order Chapman–Enskog (CE) velocity distribution function associated with the heat flux in a dilute granular gas. The method is based on the application of a homogeneous, anisotropic velocity-dependent external force which produces heat flux in the absence of gradients. The form of the force is found under the condition that, in the linear response regime, the deviation of the velocity distribution function from that of the homogeneous cooling state obeys the same linear integral equation as the one derived from the conventional CE expansion. The direct simulation Monte Carlo method is used to solve the corresponding Boltzmann equation and measure the dependence of the (modified) thermal conductivity on the coefficient of normal restitution α . Comparison with previous simulation data obtained from the Green–Kubo relations [J.J. Brey, M.J. Ruiz-Montero, P. Mayner, M.I. Garcia de Soria, J. Phys.: Condens. Matter 17 (2005) S2489] shows an excellent agreement, both methods consistently showing that the first Sonine approximation dramatically overestimates the thermal conductivity for high inelasticity ( α ≲ 0.7 ). Since our method is tied to the Boltzmann equation, the results indicate that the failure of the first Sonine approximation is not due to velocity correlation effects absent in the Boltzmann framework. This is further confirmed by an analysis of the first-order CE velocity distribution function and its three first Sonine coefficients obtained from the simulations.

https://arxiv.org/pdf/cond-mat/0604078.pdf

First-order Chapman–Enskog velocity distribution function in a granular gas

The Boltzmann equation for inelastic Maxwell models is used to analyze nonlinear transport in a granular binary mixture in the steady simple shear flow. Two different transport processes are studied. First, the rheological properties (shear and normal stresses) are obtained by solving exactly the velocity moment equations. Second, the diffusion tensor of impurities immersed in a sheared inelastic Maxwell gas is explicitly determined from a perturbation solution through first order in the concentration gradient. The corresponding reference state of this expansion corresponds to the solution derived in the (pure) shear flow problem. All these transport coefficients are given in terms of the restitution coefficients and the parameters of the mixture (ratios of masses, concentration, and sizes). The results are compared with those obtained analytically for inelastic hard spheres in the first Sonine approximation and by means of Monte Carlo simulations. The comparison between the results obtained for both interaction models shows a good agreement over a wide range values of the parameter space.

Vicente Garzó

Papers

Non-Newtonian hydrodynamics for a dilute granular suspension under uniform shear flow.

Thermal diffusion segregation in granular binary mixtures described by the Enskog equation

Exact moment solution of the Boltzmann equation for uniform shear flow

First-order Chapman–Enskog velocity distribution function in a granular gas

Nonlinear transport in inelastic Maxwell mixtures under simple shear flow