Abstract:We study the hyperbolic scaling limit for a chain of N coupled anharmonic oscillators. The chain is open and with the following adiabatic boundary conditions: it is attached to a wall on the left and there is a force (tension) τ acting on the right. In order to provide the system of the good ergodic properties, we perturb the Hamiltonian dynamics with random local exchanges of velocities between the particles, so that momentum and energy are locally conserved. We prove that in the macroscopic limit the distrib… Show more
“…We follow the lines of the proof given in [4], Section 3.3 and inspired from [16]. A sketch of the proof for the one-block estimate is given in Appendix B.…”
We study the diffusive scaling limit for a chain of N coupled oscillators. In order to provide the system with good ergodic properties, we perturb the Hamiltonian dynamics with random flips of velocities, so that the energy is locally conserved. We derive the hydrodynamic equations by estimating the relative entropy with respect to the local equilibrium state, modified by a correction term.Acknowledgements. I thank Cédric Bernardin and Stefano Olla for giving me this problem, and for the useful discussions and suggestions on this work.
“…We follow the lines of the proof given in [4], Section 3.3 and inspired from [16]. A sketch of the proof for the one-block estimate is given in Appendix B.…”
We study the diffusive scaling limit for a chain of N coupled oscillators. In order to provide the system with good ergodic properties, we perturb the Hamiltonian dynamics with random flips of velocities, so that the energy is locally conserved. We derive the hydrodynamic equations by estimating the relative entropy with respect to the local equilibrium state, modified by a correction term.Acknowledgements. I thank Cédric Bernardin and Stefano Olla for giving me this problem, and for the useful discussions and suggestions on this work.
“…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.
“…This requires proving that the macroscopic system has some very strong ergodic properties, e.g. that the only time invariant measures locally absolutely continuous w.r.t Lebesgue measure are, for infinitely extended spatially uniform systems, of the Gibbs type 30,70,63 . This has only been proven so far for systems evolving via stochastic dynamics, e.g.…”
We present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier's Law, J(r) = −κ∇T (r); J the heat flux, T the temperature and κ, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with κ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for κ, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.
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