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

Energy and momentum transport in a strongly sheared dilute gas is analyzed in the context of the nonlinear Boltzmann equation. Thermal transport is generated in the system by the action of a nonconservative external force which tries to mimick the effects of a temperature gradient. By performing a perturbation expansion in powers of the field strength, we obtain expressions for the field susceptibility tensor and the shear viscosity coefficient up to second order in the force. They are highly nonlinear functions of the shear rate. It is shown that the choice of the heat field used in computer simulations yields a field susceptibility tensor different from the thermal conductivity tensor. In order to get equivalent results, a new shear‐rate dependent force is suggested. The calculations presented here extend previous results derived from the Bhatnagar–Gross–Krook approximation [J. Chem. Phys. 101, 1423 (1994)].

Thermal transport generated by an external force in a sheared dilute gas

The transport coefficients of a dilute classical gas in the presence of a nonconservative force proportional to the particle velocity are determined from the Boltzmann equation. First, when this force is the only external action on the state of the system, the Boltzmann equation admits a Maxwellian solution f0(v,t) with a time-dependent temperature. Then, the Boltzmann equation is solved by means of the Chapman-Enskog expansion around the local version of the distribution f0 to obtain the relevant transport coefficients of the system: the shear viscosity η, the thermal conductivity κ, and a new transport coefficientµ (which is also present in granular gases) relating the heat flux with the density gradient. While the three coefficients areexactly determined from the Boltzmann equation for Maxwell molecules, expressions for them are obtained for non-Maxwell molecules by considering the BGK kinetic model of the Boltzmann equation. Finally, a stability analysis of the linear hydrodynamic equations with respect to the time-dependent equilibrium state is performed, showing that the onset of instability is associated with the transversal shear mode that could be unstable for wave numbers smaller than a certain critical wave number.

Influence of a velocity-dependent force on linear transport in low-density gases. Stability analysis

Steady simple shear flow of a low-density binary mixture of inelastic smooth hard spheres is studied in the context of the Boltzmann equation. This equation is solved by using two different and complementary approaches: a Sonine polynomial expansion and the direct simulation Monte Carlo method. The dependence of the shear and normal stresses as well as of the steady granular temperature on both the dissipation and the parameters of the mixture (ratios of masses, concentration, and sizes) is analyzed. In contrast to previous studies, the theory predicts and the simulation confirms that the partial temperatures of each species are different, even in the weak dissipation limit. In addition, the simulation shows that the theory reproduces fairly well the values of the shear stress and the phenomenon of normal stress differences. On the other hand, here we are mainly interested in analyzing transport in the homogeneous shear flow so that, the possible formation of particle clusters is ignored in our description.

Rheological properties in a low-density granular mixture

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.

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High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

This corrects the article DOI: 10.1103/PhysRevE.96.052901.

Vicente Garzó

Papers

Thermal transport generated by an external force in a sheared dilute gas

Influence of a velocity-dependent force on linear transport in low-density gases. Stability analysis

Rheological properties in a low-density granular mixture

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

Erratum: Energy nonequipartition in gas mixtures of inelastic rough hard spheres: The tracer limit [Phys. Rev. E 96, 052901 (2017)].