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Understanding the rich spatial and temporal structures in nonequilibrium thermal environments is a major subject of statistical mechanics. Because universal laws, based on an ensemble of systems, are mute on an individual system, exploring nonequilibrium statistical mechanics and the ensuing universality in individual systems has long been of fundamental interest. Here, by adopting the wave description of microscopic motion, and combining the recently developed eigenchannel theory and the mathematical tool of the concentration of measure, we show that in a single complex medium, a universal spatial structure - the diffusive steady state - emerges from an overwhelming number of scattering eigenstates of the wave equation. Our findings suggest a new principle, dubbed the wave thermalization, namely, a propagating wave undergoing complex scattering processes can simulate nonequilibrium thermal environments, and exhibit macroscopic nonequilibrium phenomena.
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Granular fluids consist of collections of activated mesoscopic or macroscopic particles (e.g., powders or grains) whose flows often appear similar to those of normal fluids. To explore the qualitative and quantitative description of these flows an idealized model for such fluids, a system of smooth inelastic hard spheres, is considered. The single feature distinguishing granular and normal fluids being explored in this way is the inelasticity of collisions. The dominant differences observed in real granular fluids are indeed captured by this feature. Following a brief introductory description of real granular fluids and motivation for the idealized model, the elements of nonequilibrium statistical mechanics are recalled (observables, states, and their dynamics). Peculiarities of the hard sphere interactions are developed in detail. The exact microscopic balance equations for the number, energy, and momentum densities are derived and their averages described as the origin for a possible macroscopic continuum mechanics description. This formally exact analysis leads to closed, macroscopic hydrodynamic equations through the notion of a normal state. This concept is introduced and the Navier-Stokes constitutive equations are derived, with associated Green-Kubo expressions for the transport coefficients. A parallel description of granular gases is described in the context of kinetic theory, and the Boltzmann limit is identified critically. The construction of the normal solution to the kinetic equation is outlined, and Navier-Stokes order hydrodynamic equations are re-derived for a low density granular gas.
The local equilibrium approach previously developed by the Authors [J. Mabillard and P. Gaspard, J. Stat. Mech. (2020) 103203] for matter with broken symmetries is applied to crystalline solids. The macroscopic hydrodynamics of crystals and their local thermodynamic and transport properties are deduced from the microscopic Hamiltonian dynamics. In particular, the Green-Kubo formulas are obtained for all the transport coefficients. The eight hydrodynamic modes and their dispersion relation are studied for general and cubic crystals. In the same twenty crystallographic classes as those compatible with piezoelectricity, cross effects coupling transport between linear momentum and heat or crystalline order are shown to split the degeneracy of damping rates for modes propagating in opposite generic directions.
We derive a class of mesoscopic virial equations governing energy partition between conjugate position and momentum variables of individual degrees of freedom. They are shown to apply to a wide range of nonequilibrium steady states with stochastic (Langevin) and deterministic (Nose--Hoover) dynamics, and to extend to collective modes for models of heat-conducting lattices. A generalised macroscopic virial theorem ensues upon summation over all degrees of freedom. This theorem allows for the derivation of nonequilibrium state equations that involve dissipative heat flows on the same footing with state variables, as exemplified for inertial Brownian motion with solid friction and overdamped active Brownian particles subject to inhomogeneous pressure.
We review the field of the glass transition, glassy dynamics and aging from a statistical mechanics perspective. We give a brief introduction to the subject and explain the main phenomenology encountered in glassy systems, with a particular emphasis on spatially heterogeneous dynamics. We review the main theoretical approaches currently available to account for these glassy phenomena, including recent developments regarding mean-field theory of liquids and glasses, novel computational tools, and connections to the jamming transition. Finally, the physics of aging and off-equilibrium dynamics exhibited by glassy materials is discussed.
The majority game, modelling a system of heterogeneous agents trying to behave in a similar way, is introduced and studied using methods of statistical mechanics. The stationary states of the game are given by the (local) minima of a particular Hopfield like hamiltonian. On the basis of a replica symmetric calculations, we draw the phase diagram, which contains the analog of a retrieval phase. The number of metastable states is estimated using the annealed approximation. The results are confronted with extensive numerical simulations.
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