Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNonequilibrium statistical mechanics of crystals
115
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
For a given thermodynamic system, and a given choice of coarse-grained state variables, the knowledge of a force-flux constitutive law is the basis for any nonequilibrium modeling. In the first paper of this series we established how, by a generalization of the classical fluctuation-dissipation theorem (FDT), the structure of a constitutive law is directly related to the distribution of the fluctuations of the state variables. When these fluctuations can be expressed in terms of diffusion processes, one may use Green-Kubo-type coarse-graining schemes to find the constitutive laws. In this paper we propose a coarse-graining method that is valid when the fluctuations are described by means of general Markov processes, which include diffusions as a special case. We prove the success of the method by numerically computing the constitutive law for a simple chemical reaction $A rightleftarrows B$. Furthermore, we show that one cannot find a consistent constitutive law by any Green-Kubo-like scheme.
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.
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.
Understanding the fluctuations by which phenomenological evolution equations with thermodynamic structure can be enhanced is the key to a general framework of nonequilibrium statistical mechanics. These fluctuations provide an idealized representation of microscopic details. We consider fluctuation-enhanced equations associated with Markov processes and elaborate the general recipes for evaluating dynamic material properties, which characterize force-flux constitutive laws, by statistical mechanics. Markov processes with continuous trajectories are conveniently characterized by stochastic differential equations and lead to Green-Kubo-type formulas for dynamic material properties. Markov processes with discontinuous jumps include transitions over energy barriers with the rates calculated by Kramers. We describe a unified approach to Markovian fluctuations and demonstrate how the appropriate type of fluctuations (continuous versus discontinuous) is reflected in the mathematical structure of the phenomenological equations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
