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

____ : ..... '"i-t.-

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

A mixture of dissipative hard grains generically exhibits a breakdown of kinetic energy equipartition. The undriven and thus freely cooling binary problem, in the tracer limit where the density of one species becomes minute, may exhibit an extreme form of this breakdown, with the minority species carrying a finite fraction of the total kinetic energy of the system. We investigate the fingerprint of this non-equilibrium phase transition, akin to an ordering process, on transport properties. The analysis, performed by solving the Boltzmann kinetic equation from a combination of analytical and Monte Carlo techniques, hints at the possible failure of hydrodynamics in the ordered region. As a relevant byproduct of the study, the behaviour of the second- and fourth-degree velocity moments is also worked out.

/pdf/anomalous-transport-of-impurities-in-inelastic-maxwell-gases-38e7e778p8.pdf

Anomalous transport of impurities in inelastic Maxwell gases

A recent model for monodisperse granular suspensions is used to analyze transport properties in spatially inhomogeneous states close to the simple (or uniform) shear flow The kinetic equation is based on the inelastic Boltzmann (for low-density gases) with the presence of a viscous drag force that models the influence of the interstitial gas phase on the dynamics of grains A normal solution is obtained via a Chapman-Enskog-like expansion around a (local) shear flow distribution which retains all the hydrodynamic orders in the shear rate To first order in the expansion, the transport coefficients characterizing momentum and heat transport around shear flow are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by using a kinetic model of the Boltzmann equation To simplify the analysis, the steady-state conditions when viscous heating is compensated by the cooling terms arising from viscous friction and collisional dissipation are considered to get the explicit forms of the set of generalized transport coefficients The shear-rate dependence of some of the transport coefficients of the set is illustrated for several values of the coefficient of restitution

Shear-rate-dependent transport coefficients in granular suspensions.

The Boltzmann kinetic equation for low-density granular suspensions under simple shear flow is considered to determine the velocity moments through the fourth degree. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. Two independent but complementary approaches are followed to achieve exact results. First, to keep the structure of the Boltzmann collision operator, the so-called inelastic Maxwell models (IMM) are considered. In this model, since the collision rate is independent of the relative velocity of the two colliding particles, the forms of the collisional moments can be obtained without the knowledge of the velocity distribution function. As a complement of the previous effort, a BGK-type kinetic model adapted to granular gases is solved to get the velocity moments of the velocity distribution function. The analytical predictions of the rheological properties (which are \emph{exactly} obtained in terms of the coefficient of restitution $\alpha$ and the reduced shear rate $a^*$) show in general an excellent agreement with event-driven simulations performed for inelastic hard spheres. In particular, both theoretical approaches show clearly that the temperature and non-Newtonian viscosity exhibit an $S$ shape in a plane of stress-strain rate (discontinuous shear thickening effect). With respect to the fourth-degree velocity moments, we find that while those moments have unphysical values for IMM in a certain region of the parameter space of the system, they are well defined functions of both $\alpha$ and $a^*$ in the case of the BGK kinetic model. The explicit shear-rate dependence of the fourth-degree moments beyond this critical region is also obtained and compared against available computer simulations.

Simple shear flow in granular suspensions: inelastic Maxwell models and BGK-type kinetic model

Comment on the paper J. Solsvik and E. Manger, "Kinetic theory models for granular mixtures with unequal granular temperature: Hydrodynamic velocity," Phys. Fluids \textbf{33}, 043321 (2021).

Comment on "Kinetic theory models for granular mixtures with unequal granular temperature: Hydrodynamic velocity" [Phys. Fluids 33, 043321 (2021)]

The Mpemba effect is a phenomenon in which an initially hotter sample cools sooner. In this paper, we show the emergence of a anomalous Mpemba-like effect in a molecular binary mixture immersed in a viscous gas. Namely, a crossover in the temperature evolution when at least one of the samples presents non-monotonic relaxation. The influence of the bath on the dynamics of the particles is modeled via a viscous drag force plus a stochastic Langevin-like term. Each component of the mixture interchanges energy with the bath depending on the mechanical properties of its particles. This discrimination causes the coupling between the time evolution of temperature with that of the partial temperatures of each component. The anomalous Mpemba effect---and its inverse and mixed counterparts---stems from this coupling. In order to obtain analytical results, the velocity distribution functions of each component are approximated by considering multitemperature Maxwellian distributions. The theoretical results derived from the Enskog kinetic theory show an excellent agreement with direct simulation Monte Carlo (DMSC) data.

Vicente Garzó

Papers

Anomalous transport of impurities in inelastic Maxwell gases

Shear-rate-dependent transport coefficients in granular suspensions.

Simple shear flow in granular suspensions: inelastic Maxwell models and BGK-type kinetic model

Comment on "Kinetic theory models for granular mixtures with unequal granular temperature: Hydrodynamic velocity" [Phys. Fluids 33, 043321 (2021)]

Anomalous Mpemba effect in binary molecular suspensions