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

The transport coefficients of a granular binary mixture driven by a stochastic bath with friction are determined from the inelastic Boltzmann kinetic equation. A normal solution is obtained via the Chapman-Enskog method for states near homogeneous steady states. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. They are given in terms of the solutions of a set of coupled linear integral equations. As in the monocomponent case, since the collisional cooling cannot be compensated for locally by the heat produced by the external driving, the reference distributions (zeroth-order approximations) ${f}_{i}^{(0)}$ ($i=1,2$) for each species depend on time through their dependence on the pressure and the temperature. Explicit forms for the diffusion transport coefficients and the shear viscosity coefficient are obtained by assuming the steady-state conditions and by considering the leading terms in a Sonine polynomial expansion. A comparison with previous results obtained for granular Brownian motion and by using a (local) stochastic thermostat is also carried out. The present work extends previous theoretical results derived for monocomponent dense gases [Garz\'o, Chamorro, and Vega Reyes, Phys. Rev. E 87, 032201 (2013)] to granular mixtures at low density.

Transport coefficients for driven granular mixtures at low density.

The Navier-Stokes transport coefficients of a granular dense fluid of smooth inelastic hard disks or spheres are explicitly determined by solving the inelastic Enskog equation by means of Grad's moment method. The transport coefficients are explicitly determined as functions of the (constant) coefficient of restitution and the solid volume fraction. In addition, the cooling rate is also calculated to first order in the spatial gradients. The calculations are performed for an arbitrary number of dimensions. The results are not limited to small dissipation and are expected to apply at moderate densities. It is found that the expressions of the Navier-Stokes transport coefficients and the cooling rate agree with those previously obtained from the Chapman-Enskog method by using the leading terms in a Sonine polynomial expansion. This shows the equivalence between both methods for granular fluids in the Navier-Stokes approximation. A comparison with previous results derived from Grad's moment method for inelastic disks and spheres is also carried out.

Grad's moment method for a granular fluid at moderate densities. Navier-Stokes transport coefficients

The homogeneous state of a granular flow of smooth inelastic hard spheres or disks described by the Enskog–Boltzmann kinetic equation is analyzed. The granular gas is fluidized by the presence of a random force and a drag force. The combined action of both forces, which act homogeneously on the granular gas, tries to mimic the interaction of the set of particles with a surrounding fluid. The first stochastic force thermalizes the system, providing for the necessary energy recovery to keep the system in its gas state at all times, whereas the second force allows us to mimic the action of the surrounding fluid viscosity. After a transient regime, the gas reaches a steady state characterized by a scaled distribution function that depends not only on the dimensionless velocity c ≡ v/v0 (v0 being the thermal velocity) but also on the dimensionless driving force parameters. The dependence of and its first relevant velocity moments a2 and a3 (which measure non-Gaussian properties of ) on both the coefficient of restitution α and the driven parameters is widely investigated by means of the direct simulation Monte Carlo method. In addition, approximate forms for a2 and a3 are also derived from an expansion of in Sonine polynomials. The theoretical expressions of the above Sonine coefficients agree well with simulation data, even for quite small values of α. Moreover, the third order expansion of the distribution function makes a significant improvement in accuracy for larger velocities and inelasticities. Results also show that the non-Gaussian corrections to the distribution function are smaller than those observed for undriven granular gases.

Homogeneous steady states in a granular fluid driven by a stochastic bath with friction

Enskog theory for polydisperse granular mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture

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

Transport coefficients for driven granular mixtures at low density.

Grad's moment method for a granular fluid at moderate densities. Navier-Stokes transport coefficients

Homogeneous steady states in a granular fluid driven by a stochastic bath with friction

Enskog theory for polydisperse granular mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture

Nonlinear Transport in Inelastic Maxwell Mixtures Under Simple Shear Flow