Cédric Bernardin

Momentum conserving model with anomalous thermal conductivity in low dimensional systems

Anomalous large thermal conductivity has been observed numerically and experimentally in one- and two-dimensional systems. There is an open debate about the role of conservation of momentum. We introduce a model whose thermal conductivity diverges in dimensions 1 and 2 if momentum is conserved, while it remains finite in dimension d > or = 3. We consider a system of harmonic oscillators perturbed by a nonlinear stochastic dynamics conserving momentum and energy. We compute explicitly the time correlation function of the energy current C(J)(t), and we find that it behaves, for large time, like t(-d/2) in the unpinned cases, and like t(-d/2-1) when an on-site harmonic potential is present. This result clarifies the role of conservation of momentum in the anomalous thermal conductivity in low dimensions.

Thermal Conductivity for a Momentum Conservative Model

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 oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t−d/2 in the unpinned case and like t−d/2–1 if an on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.

Transport Properties of a Chain of Anharmonic Oscillators with random flip of velocities

We consider the stationary states of a chain of n anharmonic coupled oscillators, whose deterministic Hamiltonian dynamics is perturbed by random independent sign change of the velocities (a random mechanism that conserve energy). The extremities are coupled to thermostats at different temperature Tℓ and Tr and subject to constant forces τℓ and τr. If the forces differ τℓ≠τr the center of mass of the system will move of a speed Vs inducing a tension gradient inside the system. Our aim is to see the influence of the tension gradient on the thermal conductivity. We investigate the entropy production properties of the stationary states, and we prove the existence of the Onsager matrix defined by Green-Kubo formulas (linear response). We also prove some explicit bounds on the thermal conductivity, depending on the temperature.

Harmonic Systems with Bulk Noises

We consider a harmonic chain in contact with thermal reservoirs at different temperatures and subject to bulk noises of different types: velocity flips or self-consistent reservoirs. While both systems have the same covariances in the non-equilibrium stationary state (NESS) the measures are very different. We study hydrodynamical scaling, large deviations, fluctuations, and long range correlations in both systems. Some of our results extend to higher dimensions.

Anomalous diffusion for a class of systems with two conserved quantities

We introduce a class of one-dimensional deterministic models of energy–volume conserving interfaces. Numerical simulations show that these dynamics are genuinely super-diffusive. We then modify the dynamics by adding a conservative stochastic noise so that it becomes ergodic. A system of conservation laws are derived as hydrodynamic limits of the modified dynamics. Numerical evidence shows that these models are still super-diffusive. This is proven rigorously for harmonic potentials.

3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise

We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.

Anomalous Fluctuations for a Perturbed Hamiltonian System with Exponential Interactions

A one-dimensional Hamiltonian system with exponential interactions perturbed by a conservative noise is considered. It is proved that energy superdiffuses and upper and lower bounds describing this anomalous diffusion are obtained.

Regularity of the diffusion coefficient for lattice gas reversible under Bernoulli measures

We prove the smoothness of a diffusion coefficient with respect to the density of particles for a non-gradient type model. This fact gives a complete proof of the hydrodynamic equation for lattice gas reversible under Bernoulli measures.

Stationary nonequilibrium properties for a heat conduction model

We consider a stochastic heat conduction model for solids composed of N interacting atoms. The system is in contact with two heat baths at different temperatures Tl and Tr. The bulk dynamics conserves two quantities: the energy and the deformation between atoms. If Tl not equal to Tr, a heat flux occurs in the system. For large N, the system adopts a linear temperature profile between Tl and Tr. We establish the hydrodynamic limit for the two conserved quantities. We introduce the fluctuation field of the energy and of the deformation in the nonequilibrium steady state. As N goes to infinity, we show that this field converges to a Gaussian field and we compute the limiting covariance matrix. The main contribution of the paper is the study of large deviations for the temperature profile in the nonequilibrium stationary state. A variational formula for the rate function is derived following the recent macroscopic fluctuation theory of Bertini [J. Stat. Phys. 107, 635 (2002); Math. Phys., Anal. Geom. 6, 231 (2003); J. Stat. Phys. 121, 843 (2005)].

Thermal Conductivity for a Noisy Disordered Harmonic Chain

We consider a d-dimensional 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 Green-Kubo formula. These bounds indicate a positive finite conductivity. We prove also that the infinite volume homogenized Green-Kubo formula converges.