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24
Fourier’s law for a microscopic model of heat conduction
 J. Stat. Phys
, 2005
"... Abstract. We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor osc ..."
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Cited by 26 (11 self)
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Abstract. We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N → ∞, satisfies Fourier’s law and the linear profile for the energy average. 1.
THERMAL CONDUCTIVITY FOR A MOMENTUM CONSERVING MODEL
, 2006
"... Abstract. We present here complete mathematical proofs of the results annouced in condmat/0509688. We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of harmonic oscillators perturbed by a stochastic dynamics c ..."
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Cited by 15 (13 self)
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Abstract. We present here complete mathematical proofs of the results annouced in condmat/0509688. We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of harmonic oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute the finitesize thermal conductivity via GreenKubo formula. In the limit as the size N of the system goes to infinity, conductivity diverges like N in dimension 1 and like ln N in dimension 2. Conductivity remains finite if d ≥ 3 or if a pinning (on site potential) is present. 1.
Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats.
, 802
"... We discuss the DonskerVaradhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the DonskerVaradhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, ..."
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Cited by 6 (0 self)
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We discuss the DonskerVaradhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the DonskerVaradhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary state as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling. 1 Introduction. Attempts to study large systems out of equilibrium through fluctuation theory has received a lot of attention in recent years [2, 3, 4, 5, 16, 18, 19, 20]. In a recent series of papers [27, 8, 28, 29, 30], it has been understood that in random systems driven out of equilibrium, the theory of large deviations provides naturally a variational
Heat conduction and entropy production in anharmonic crystals with selfconsistent stochastic reservoirs
, 2008
"... Abstract. We investigate a class of anharmonic crystals in d dimensions, d ≥ 1, coupled to both external and internal heat baths of the OrnsteinUhlenbeck type. The external heat baths, applied at the boundaries in the 1direction, are at specified, unequal, temperatures Tl and Tr. The temperatures ..."
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Cited by 6 (2 self)
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Abstract. We investigate a class of anharmonic crystals in d dimensions, d ≥ 1, coupled to both external and internal heat baths of the OrnsteinUhlenbeck type. The external heat baths, applied at the boundaries in the 1direction, are at specified, unequal, temperatures Tl and Tr. The temperatures of the internal baths are determined in a selfconsistent way by the requirement that there be no net energy exchange with the system in the nonequilibrium stationary state (NESS). We prove the existence of such a stationary selfconsistent profile of temperatures for a finite system and show it minimizes the entropy production to leading order in (Tl − Tr). In the NESS the heat conductivity κ is defined as the heat flux per unit area divided by the length of the system and (Tl − Tr). In the limit when the temperatures of the external reservoirs goes to the same temperature T, κ(T) is given by the GreenKubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This κ(T) remains bounded as the size of the system goes to infinity. We also show that the corresponding infinite system GreenKubo formula yields a finite result. Stronger results are obtained under the assumption that the selfconsistent profile remains bounded. 1.
Thermal Conductivity for a Noisy Disordered Harmonic Chain
, 808
"... Abstract. We consider a ddimensional disordered harmonic chain (DHC) perturbed by an energy conservative noise. We obtain uniform in the volume upper and lower bounds for the thermal conductivity defined through the GreenKubo formula. These bounds indicate a positive finite conductivity. We prove ..."
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Cited by 5 (4 self)
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Abstract. We consider a ddimensional disordered harmonic chain (DHC) perturbed by an energy conservative noise. We obtain uniform in the volume upper and lower bounds for the thermal conductivity defined through the GreenKubo formula. These bounds indicate a positive finite conductivity. We prove also that the infinite volume homogenized GreenKubo formula converges. 1.
Fourier’s law and maximum path information
, 2004
"... By using a path information defined for the measure of the uncertainty of instable dynamics, a theoretical derivation of Fourier’s law of heat conduction is given on the basis of maximum information method associated with the principle of least action. PACS numbers: 05.45.a (Nonlinear dynamics); 66 ..."
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Cited by 2 (2 self)
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By using a path information defined for the measure of the uncertainty of instable dynamics, a theoretical derivation of Fourier’s law of heat conduction is given on the basis of maximum information method associated with the principle of least action. PACS numbers: 05.45.a (Nonlinear dynamics); 66.10.Cb (Diffusion); 05.60.Cd (Classical transport); 05.40.Jc (Brownian motion) 1
Heat transport in harmonic lattices
, 2009
"... We work out the nonequilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain ..."
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Cited by 2 (1 self)
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We work out the nonequilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the nonequilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to selfconsistent reservoirs and reproduce some exact results on this model, obtained recently by Bonetto, Lebowitz and Lukkarinen.
NONEQUILIBRIUM STATIONARY STATES OF HARMONIC CHAINS WITH BULK NOISES
"... Abstract. We consider a chain composed of N coupled harmonic oscillators in contact with heat baths at temperature Tℓ and Tr at sites 1 and N respectively. The oscillators are also subjected tononmomentum conserving bulkstochastic noises. These maketheheat conductivitysatisfy Fourier’s law. Here we ..."
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Cited by 1 (1 self)
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Abstract. We consider a chain composed of N coupled harmonic oscillators in contact with heat baths at temperature Tℓ and Tr at sites 1 and N respectively. The oscillators are also subjected tononmomentum conserving bulkstochastic noises. These maketheheat conductivitysatisfy Fourier’s law. Here we describe some new results about the hydrodynamical equations for typical macroscopic energy and displacement profiles, as well as their fluctuations and large deviations, in two simple models of this type. ensl00621060, version 2 26 Oct 2011 1.
ThermoElectric Transport Properties of a Chain of Quantum Dots with SelfConsistent Reservoirs
"... We introduce a model for charge and heat transport based on the LandauerBüttiker scattering approach. The system consists of a chain of N quantum dots, each of them being coupled to a particle reservoir. Additionally, the left and right ends of the chain are coupled to two particle reservoirs. All ..."
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We introduce a model for charge and heat transport based on the LandauerBüttiker scattering approach. The system consists of a chain of N quantum dots, each of them being coupled to a particle reservoir. Additionally, the left and right ends of the chain are coupled to two particle reservoirs. All these reservoirs are independent and can be described by any of the standard physical distributions: MaxwellBoltzmann, FermiDirac and BoseEinstein. In the linear response regime, and under some assumptions, we first describe the general transport properties of the system. Then we impose the selfconsistency condition, i.e. we fix the boundary values (TL, µL) and (TR, µR), and adjust the parameters (Ti, µi), for i = 1,..., N, so that the net electric and heat currents through all the intermediate reservoirs vanish. This leads to expressions for the temperature and chemical potential profiles along the system, which turn out to be independent of the distribution describing the reservoirs. We also determine the electric and heat currents flowing through the system and present some numerical results, using random matrix theory, showing that the statistical average currents are governed by Ohm and Fourier laws.
Diffusion laws and least action principle
, 2004
"... According to our recent theoretical result, an instable dynamical system, in order to follow the least action principle of mechanics, always maximizes its uncertainty of motion. In this work, we apply this methodology to study diffusion phenomena of particles in potential field. A straightforward de ..."
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According to our recent theoretical result, an instable dynamical system, in order to follow the least action principle of mechanics, always maximizes its uncertainty of motion. In this work, we apply this methodology to study diffusion phenomena of particles in potential field. A straightforward derivation of FokkerPlanck equation, Fick’s laws and Ohm’s law governing normal diffusion is given with the help of the stationary action distribution of probability, derived from maximization of a Shannon path information under the constraint associated with the average action over different phase paths due to dynamical uncertainty. PACS numbers: 05.45.a (Nonlinear dynamics); 66.10.Cb (Diffusion); 05.60.Cd (Classical transport); 05.40.Jc (Brownian motion) 1