Anomalous transport and relaxation in classical one-dimensional models

Basile, Giada; Delfini, Luca; Lepri, Stefano; Livi, Roberto; Olla, Stefano; Politi, Antonio

doi:10.1140/epjst/e2007-00364-7

Cited by 40 publications

(59 citation statements)

References 49 publications

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“…One possibility is that the choice of a harmonic Hamiltonian makes the model special, and thus solvable, and at the same time makes it a non-generic case. Some simulations with a dynamics which is roughly similar to the above stochastic dynamics were recently done with FPU type anharmonic terms included in the Hamiltonian [79]. These are equilibrium simulations using the Green-Kubo formula.…”

Section: Exactly Solvable Modelmentioning

confidence: 99%

Heat transport in low-dimensional systems

Dhar

2008

Advances in Physics

922

1,114

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Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet nontrivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard particle and hard disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ, diverges with system size L, as κ ∼ L α . For one dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given. IntroductionIt is now about two hundred years since Fourier first proposed the law of heat conduction that goes by his name. Consider a macroscopic system subjected to different temperatures at its boundaries. One assumes that it is possible to have a * Corresponding author. Email: dabhi@rri.res.in November 19, 2008 19:56 Advances in Physics reva 2 coarse-grained description with a clear separation between microscopic and macroscopic scales. If this is achieved, it is then possible to define, at any spatial point x in the system and at time t, a local temperature field T (x, t) which varies slowly both in space and time (compared to microscopic scales). One then expects heat currents to flow inside the system and Fourier argued that the local heat current density J(x, t) is given bywhere κ is the thermal conductivity of the system. If u(x, t) represents the local energy density then this satisfies the continuity equation ∂u/∂t + ∇.J = 0. Using the relation ∂u/∂T = c, where c is the specific heat per unit volume, leads to the heat diffusion equation:Thus, Fourier's law implies diffusive transfer of energy. In terms of a microscopic picture, this can be understood in terms of the motion of the heat carriers, i.e. , molecules, electrons, lattice vibrations(phonons), etc., which suffer random collisions and hence move diffusively. Fourier's law is a phenomological law and has been enormously succesful in providing an accurate description of heat transport phenomena as observed in experimental systems. However there is no rigorous derivation of this law starting from a microscopic Hamiltonian description and this basic question has motivated a large number of studies on heat conduction in model systems. One important and somewhat surprising conclusion that emerges from these studies is that Fourier's law is probably not valid in one and two dimensional systems, except when the ...

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Section: Exactly Solvable Modelmentioning

confidence: 99%

Heat transport in low-dimensional systems

Dhar

2008

Advances in Physics

922

1,114

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“…Low-dimensional [i.e., one-(1D) and two-dimensional (2D)] systems where total size, energy, and momentum are the only conserved quantities, typically exhibit anomalous relaxation and transport properties [1][2][3][4]. Under these conditions the standard hydrodynamic description fails, because the transport coefficients are ill defined in the thermodynamic limit.…”

Section: Introductionmentioning

confidence: 99%

Anomalous dynamical scaling in anharmonic chains and plasma models with multiparticle collisions

et al. 2015

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We study the anomalous dynamical scaling of equilibrium correlations in one-dimensional systems. Two different models are compared: the Fermi-Pasta-Ulam chain with cubic and quartic nonlinearity and a gas of point particles interacting stochastically through multiparticle collision dynamics. For both models-that admit three conservation laws-by means of detailed numerical simulations we verify the predictions of nonlinear fluctuating hydrodynamics for the structure factors of density and energy fluctuations at equilibrium. Despite this, violations of the expected scaling in the currents correlation are found in some regimes, hindering the observation of the asymptotic scaling predicted by the theory. In the case of the gas model this crossover is clearly demonstrated upon changing the coupling constant.

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“…The characterization of steady-states is a widely investigated problem within non-equilibrium statistical mechanics [1], since it provides the basis for understanding a large variety of phenomena, including transport processes, pattern formation and the dynamics of living systems. In a nutshell, the simplest setup amounts to determining the currents that emerge as a result of the application of an external force, either across the system, as for electric currents, or at the boundaries, as in heat conduction [2][3][4]. Anyway, it is quite a nontrivial task to be accomplished, even when the departure from equilibrium is minimal and one can rely on the Green-Kubo formalism for establishing a connection between the microscopic and the hydrodynamic descriptions.…”

mentioning

confidence: 99%

“…In a nutshell, the simplest setup amounts to determining the currents that emerge as a result of the application of an external force, either across the system, as for electric currents, or at the boundaries, as in heat conduction [2][3][4]. Anyway, it is quite a nontrivial task to be accomplished, even when the departure from equilibrium is minimal and one can rely on the Green-Kubo formalism for establishing a connection between the microscopic and the hydrodynamic descriptions.…”

mentioning

confidence: 99%

Boundary-Induced Instabilities in Coupled Oscillators

et al. 2014

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A novel class of nonequilibrium phase-transitions at zero temperature is found in chains of nonlinear oscillators. For two paradigmatic systems, the Hamiltonian XY model and the discrete nonlinear Schrödinger equation, we find that the application of boundary forces induces two synchronized phases, separated by a non-trivial interfacial region where the kinetic temperature is finite. Dynamics in such supercritical state displays anomalous chaotic properties whereby some observables are non-extensive and transport is superdiffusive. At finite temperatures, the transition is smoothed, but the temperature profile is still non-monotonous. The characterization of steady-states is a widely investigated problem within non-equilibrium statistical mechanics [1], since it provides the basis for understanding a large variety of phenomena, including transport processes, pattern formation and the dynamics of living systems. In a nutshell, the simplest setup amounts to determining the currents that emerge as a result of the application of an external force, either across the system, as for electric currents, or at the boundaries, as in heat conduction [2][3][4]. Anyway, it is quite a nontrivial task to be accomplished, even when the departure from equilibrium is minimal and one can rely on the Green-Kubo formalism for establishing a connection between the microscopic and the hydrodynamic descriptions. For instance, this is testified by the discrepancy that still persists, after many years of careful studies, between the most advanced theories of heat conduction and some numerical simulations. The level of difficulty typically increases when one considers coupled transport [5-11] (i.e. when two or more currents coexist, such as heat and electric ones in thermo-electric effects) or, even worse, far-from-equilibrium. This is why most of the theoretical studies concentrate on stochastic models, where fluctuations can be easily controlled, although they lack a truly microscopic justification. This approach proved, nevertheless, very effective, since it has allowed discovering non-equilibrium transitions, such as those exhibited by TASEP-like models, that have been used to describe translation of proteins, or traffic flows [12].In this Letter we describe a novel class of boundaryinduced transitions for two models that are typically used as test beds for a wide range of physical phenomena: the so-called Hamiltonian XY (or rotor) model [13][14][15][16] sub- * Electronic address: stefano.lepri@isc.cnr.it ject to an applied mechanical torque and the Discrete NonLinear Schrödinger (DNLS) equation [17][18][19] under a gradient of the chemical potential. This type of qualitative change of the dynamics results from the joint effect of thermal and mechanical forces. It can be interpreted as a desynchronization phenomenon in a spatially-extended dynamical system, whereby mutual entrainment of oscillators' phases is abruptly destroyed. As a result of such unlocking, a regime characterized by phase-coexistence sets in where, although the chain...

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Anomalous transport and relaxation in classical one-dimensional models

Cited by 40 publications

References 49 publications

Heat transport in low-dimensional systems

Heat transport in low-dimensional systems

Anomalous dynamical scaling in anharmonic chains and plasma models with multiparticle collisions

Boundary-Induced Instabilities in Coupled Oscillators

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