1,833 research outputs found
Hyperbolicity of exact hydrodynamics for three-dimensional linearized Grad's equations
No full textWe extend to the three-dimensional situation a recent proof of hyperbolicity of the exact (to all orders in Knudsen number) linear hydrodynamics obtained from Grad's moment system [M. Colangeli, I. Karlin, M. Kröger, Phys. Rev. E 75, 051204 (2007)]. Proof of an H theorem is also presented
Microscopic models for uphill diffusion
No full textWe study a system of particles which jump on the sites of the interval [1, L] of Z. The density at the boundaries is kept fixed to simulate the action of mass reservoirs. The evolution depends on two parameters lambda' >= 0 and lambda '' >= 0 which are the strength of an external potential and respectively of an attractive potential among the particles. When lambda' = lambda '' = 0 the system behaves diffusively and the density profile of the final stationary state is linear, Fick's law is satisfied. In this paper we show that when lambda' > 0 and lambda '' = 0 the system models the diffusion of carbon in the presence of silicon as in the Darken experiment: the final state of the system is in qualitative agreement with the experimental one and uphill diffusion is present at the weld. Finally if lambda' = 0 and lambda '' > 0 is suitably large, the system simulates a vapor-liquid phase transition and we have a surprising phenomenon, as studied in Colangeli et al (2016 Phys. Lett. A 380 1710-3) and Colangeli et al (2017 J. Stat. Phys. 167 1081-111). Namely when the densities in the reservoirs correspond respectively to metastable vapor and metastable liquid we find a final stationary current which goes uphill from the reservoir with smaller density (vapor) to that with larger density (liquid). Our results are mainly numerical, we have theoretical explanations yet we miss a complete mathematical proof
Equilibrium, fluctuation relations and transport for irreversible deterministic dynamics
No full textIn a recent paper [M. Colangeli et al., J. Stat. Mech. P04021, (2011)] it was argued that the Fluctuation Relation for the phase space contraction rate A could suitably be extended to non-reversible dissipative systems. We strengthen here those arguments, providing analytical and numerical evidence based on the properties of a simple irreversible nonequilibrium baker model. We also consider the problem of response, showing that the transport coefficients are not affected by the irreversibility of the microscopic dynamics. In addition, we prove that a form of detailed balance, hence of equilibrium, holds in the space of relevant variables, despite the irreversibility of the phase space dynamics. This corroborates the idea that the same stochastic description, which arises from a projection onto a subspace of relevant coordinates, is compatible with quite different underlying deterministic dynamics. In other words, the details of the microscopic dynamics are largely irrelevant, for what concerns properties such as those concerning the Fluctuation Relations, the equilibrium behavior and the response to perturbations. (C) 2011 Elsevier B.V. All rights reserved
From kinetic models to hydrodynamics: some novel results
No full textFrom Kinetic Models to Hydrodynamics serves as an introduction to the asymptotic methods necessary to obtain hydrodynamic equations from a fundamental description using kinetic theory models and the Boltzmann equation. The work is a survey of an active research area, which aims to bridge time and length scales from the particle-like description inherent in Boltzmann equation theory to a fully established “continuum” approach typical of macroscopic laws of physics.The author sheds light on a new method—using invariant manifolds—which addresses a functional equation for the nonequilibrium single-particle distribution function. This method allows one to find exact and thermodynamically consistent expressions for: hydrodynamic modes; transport coefficient expressions for hydrodynamic modes; and transport coefficients of a fluid beyond the traditional hydrodynamic limit. The invariant manifold method paves the way to establish a needed bridge between Boltzmann equation theory and a particle-based theory of hydrodynamics. Finally, the author explores the ambitious and longstanding task of obtaining hydrodynamic constitutive equations from their kinetic counterparts. The work is intended for specialists in kinetic theory—or more generally statistical mechanics—and will provide a bridge between a physical and mathematical approach to solve real-world problems.
Nonequilibrium response from the dissipative Liouville equation
No full textThe problem of response of nonequilibrium systems is currently under intense investigation. We propose a general method of solution of the Liouville equation for thermostatted particle systems subjected to external forces which retains only the slow degrees of freedom, by projecting out the majority of fast variables. Response formulae, extending the Green-Kubo relations to dissipative dynamics, are provided, and comparison with numerical data is presented
Small Scale Hydrodynamics
No full textThe purpose of this work is to offer a brief survey of some of the mathematical methods useful to bridge different levels of description, i.e. from the set-up of classical kinetic theory of gases to hydrodynamics. Most of the standard mathematical techniques, in this field, stem from the seminal Chapman–Enskog expansion, which constitutes an important success in kinetic theory, as it made possible to formally derive the Navier-Stokes hydrodynamics from the Boltzmann equation. Yet, almost a century of effort to extend the hydrodynamic description beyond the Navier-Stokes-Fourier approximation failed even in the case of small deviations around the equilibrium, due to the onset of instabilities which also cause the violation of the H-Theorem. A different route, in kinetic theory, is represented by the recent Invariant Manifold method. The latter technique allows one to restore stability in the hydrodynamic equations, which remain thus applicable even at short length scales, under the hypothesis of validity of the Local Equilibrium condition
Emergence of stationary uphill currents in 2D Ising models: the role of reservoirs and boundary conditions
No full textWe investigate the dynamics of a 2D Ising model on a square lattice with conservative Kawasaki dynamics in the bulk, coupled with two external reservoirs that pull the dynamics out of equilibrium. Two different mechanisms for the action of the reservoirs are considered. In the first, called ISF, the condition of local equilibrium between reservoir and the lattice is not satisfied. The second mechanism, called detailed balance (DB), implements a DB condition, thus satisfying the local equilibrium property. We provide numerical evidence that, for a suitable choice of the temperature (i.e. below the critical temperature of the equilibrium 2D Ising model) and the reservoir magnetizations, in the long time limit the ISF model undergoes a ferromagnetic phase transition and gives rise to stationary uphill currents, namely positive spins diffuse from the reservoir with lower magnetization to the reservoir with higher magnetization. The same phenomenon does not occur for DB dynamics with properly chosen boundary conditions. Our analysis extends the results reported in Colangeli et al. [Phys. Rev. E 97, 030103(R) (2018)], shedding also light on the effect of temperature and the role of different boundary conditions for this model. These issues may be relevant in a variety of situations (e.g. mesoscopic systems) in which the violation of the local equilibrium condition produces unexpected phenomena that seem to contradict the standard laws of transport
Reduced Markovian Descriptions of Brownian Dynamics: Toward an Exact Theory
No full textWe outline a reduction scheme for a class of Brownian dynamics which leads to
meaningful corrections to the Smoluchowski equation in the overdamped regime.
The mobility coefficient of the reduced dynamics is obtained by exploiting the
Dynamic Invariance principle, whereas the diffusion coefficient fulfils the
Fluctuation-Dissipation theorem. Explicit calculations are carried out in the
case of a harmonically bound particle. A quantitative pointwise representation
of the reduction error is also provided and connections to both the Maximum
Entropy method and the linear response theory are highlighted. Our study paves
the way to the development of reduction procedures applicable to a wider class
of diffusion processes
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