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

The dynamical properties of a tracer or impurity particle immersed in a host gas of inelastic and rough hard spheres in the homogeneous cooling state is studied. Specifically, the breakdown of energy equipartition as characterized by the tracer/host ratios of translational and rotational temperatures is analyzed by exploring a wide spectrum of values of the control parameters of the system (masses, moments of inertia, sizes, and coefficients of restitution). Three complementary approaches are considered. On the theoretical side, the Boltzmann and Boltzmann-Lorentz equations (both assuming the molecular chaos ansatz) are solved by means of a multitemperature Maxwellian approximation for the velocity distribution functions. This allows us to obtain explicit analytical expressions for the temperature ratios. On the computational side, two different techniques are used. First, the kinetic equations are numerically solved by the direct simulation Monte Carlo (DSMC) method. Second, molecular dynamics simulations for dilute gases are performed. Comparison between theory and simulations shows a general good agreement. This means that (i) the impact of the molecular chaos ansatz on the temperature ratios is not significant (except at high inelasticities and/or big impurities) and (ii) the simple Maxwellian approximation yields quite reliable predictions.

Energy nonequipartition in gas mixtures of inelastic rough hard spheres: The tracer limit.

An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier--Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman--Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman--Enskog-like method and generalized (tensorial) transport coefficients are obtained.

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Hydrodynamics of inelastic Maxwell models

Non-Newtonian transport properties of an inertial suspension of inelastic rough hard spheres under simple shear flow are determined by the Boltzmann kinetic equation. The influence of the interstitial gas on rough hard spheres is modeled via a Fokker–Planck generalized equation for rotating spheres accounting for the coupling of both the translational and rotational degrees of freedom of grains with the background viscous gas. The generalized Fokker–Planck term is the sum of two ordinary Fokker–Planck differential operators in linear v and angular ω velocity space. As usual, each Fokker–Planck operator is constituted by a drag force term (proportional to v and/or ω) plus a stochastic Langevin term defined in terms of the background temperature Tex. The Boltzmann equation is solved by two different but complementary approaches: (i) by means of Grad’s moment method and (ii) by using a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to inelastic rough hard spheres. As in the case of smooth inelastic hard spheres, our results show that both the temperature and the non-Newtonian viscosity increase drastically with an increase in the shear rate (discontinuous shear thickening effect) while the fourth-degree velocity moments also exhibit an S-shape. In particular, while high levels of roughness may slightly attenuate the jump of the viscosity in comparison to the smooth case, the opposite happens for the rotational temperature. As an application of these results, a linear stability analysis of the steady simple shear flow solution is also carried out showing that there are regions of the parameter space where the steady solution becomes linearly unstable. The present work extends previous theoretical results (H. Hayakawa and S. Takada, “Kinetic theory of discontinuous rheological phase transition for a dilute inertial suspension,” Prog. Theor. Exp. Phys. 2019, 083J01 and R. G. Gonzalez and V. Garzo, “Simple shear flow in granular suspensions: Inelastic Maxwell models and BGK-type kinetic model,” J. Stat. Mech. 2019, 013206) to rough spheres.

Non-Newtonian rheology in inertial suspensions of inelastic rough hard spheres under simple shear flow

Expressions for the heat and momentum transport are obtained for a dilute gas in a steady state with both temperature and velocity gradients. The results are derived from a kinetic model recently proposed: the Liu model [Phys. Fluids A 2, 277 (1990)]. This model improves the well‐known Bhatnagar–Gross–Krook (BGK) model equation. At a hydrodynamic level, the solution is characterized by a constant pressure, a linear profile of the flow velocity with respect to a space variable scaled with the local collision frequency, and a parabolic profile of the temperature with respect to the same variable. The shear viscosity and thermal conductivity coefficients are explicitly obtained. They are nonlinear functions of the shear rate and their only dependence on the temperature gradient is through their zero shear‐rate values. In addition, the shear‐rate dependence of the viscometric functions is also analyzed. A comparison with previous results derived from the BGK equation is carried out.

Kinetic model for heat and momentum transport

In this paper we evaluate corrections to the Navier-Stokes constitutive equations induced by the action of a gravitational field $\mathbf{g}=\ensuremath{-}g\mathbf{z}\^{}$ in a gas subjected to a thermal gradient parallel to a field with no convection. The analysis is performed from an exact perturbation solution of the Boltzmann equation for Maxwell molecules through second order in the field. The reference state (zeroth-order approximation) corresponds to the exact solution of the Boltzmann equation in the pure planar Fourier flow, which holds for arbitrary values of the thermal gradient. The results show that the pressure tensor becomes anisotropic, so that the momentum flux along the field direction is enhanced: ${(P}_{z}z\ensuremath{-}p)/p\ensuremath{\approx}{14.4(p}^{\ensuremath{-}2}{\ensuremath{\eta}}^{2}g\ensuremath{\partial}\mathrm{ln}T/\ensuremath{\partial}{z)}^{2}$. In addition, the heat flux increases (decreases) with respect to its Navier-Stokes value when the gas is heated from above (below): ${q}_{z}{/q}_{z}^{\mathrm{NS}}\ensuremath{-}1\ensuremath{\approx}{20.7(p}^{\ensuremath{-}2}{\ensuremath{\eta}}^{2}g\ensuremath{\partial}\mathrm{ln}T/\ensuremath{\partial}z)$.

Vicente Garzó

Papers

Energy nonequipartition in gas mixtures of inelastic rough hard spheres: The tracer limit.

Hydrodynamics of inelastic Maxwell models

Non-Newtonian rheology in inertial suspensions of inelastic rough hard spheres under simple shear flow

Kinetic model for heat and momentum transport

Nonlinear heat transport in a dilute gas in the presence of gravitation