Abstract:We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostat 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 for the stationary state, in the limit as N → ∞, the Fourier's law and … Show more
“…Related works in systems defined by dynamical local rules include [4,5,6,7,17]. For stochastic models, the collection of results is much larger, and we mention only some that are closer to this paper in spirit: [1,2,8,9,10,16,18,19,23,24,26,29].…”
Abstract. We consider systems of moving particles in 1-dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the near-integrable dynamics, transport properties are found to be more normal than expected.
“…Related works in systems defined by dynamical local rules include [4,5,6,7,17]. For stochastic models, the collection of results is much larger, and we mention only some that are closer to this paper in spirit: [1,2,8,9,10,16,18,19,23,24,26,29].…”
Abstract. We consider systems of moving particles in 1-dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the near-integrable dynamics, transport properties are found to be more normal than expected.
“…21,1 They later appeared as the high-energy limit of a chain with deterministic dynamics 9 and were also used as a stochastic perturbation ͑mimicking nonlinearities͒ of an oscillator chain. 3,8 In the present paper we construct for our energy diffusion model a dual process that expresses the evolution of the K-point correlation functions of the kinetic energies in terms of a process of K interacting random walkers. We also give a closed expression for the stationary covariance ͗x i 2 ; x j 2 ͘ which confirms the presence of long-range correlations in the nonequilibrium stationary state ͓as already found before in the SEP Ref.…”
We study a model of heat conduction with stochastic diffusion of energy. We obtain a dual particle process which describes the evolution of all the correlation functions. An exact expression for the covariance of the energy exhibits long-range correlations in the presence of a current. We discuss the formal connection of this model with the simple symmetric exclusion process.
“…Later, purely stochastic models, where energy is assumed to diffuse between neighboring boxes (the oscillators), have been considered [4,5]. More recently, systems of harmonic oscillators exchanging energy with ''conservative'' noise have been proven to admit a unique stationary state with a constant heat flux and a linear temperature profile [6]. Admittedly, the leap from such class of models to even the simplest deterministic, nonlinear ones is still a challenge for the theory [7].…”
We provide an explicit representation of the nonequilibrium invariant measure for a chain of harmonic oscillators with conservative noise in the presence of stationary heat flow. By first determining the covariance matrix, we are able to express the measure as the product of Gaussian distributions aligned along some collective modes that are spatially localized with power-law tails. Numerical studies show that such a representation applies also to a purely deterministic model, the quartic Fermi-Pasta-Ulam chain.
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