DS10 - Défi de tous les savoirs

Energy Diffusion in Noisy Hamiltonian Systems – EDNHS

Energy Diffusion in Noisy Hamiltonian Systems

The derivation of macroscopic laws from microscopic models has a long history and spans several fields of mathematics and physics. For one dimensional conservative asymmetric systems, anomalous diffusion of the conserved quantities is expected and the main goal is to understand its nature. Since our understanding of this question is too difficult for realistic physical systems, the goal of the project is to make progress for Hamiltonian systems with bulk noise.

What is the nature of the macroscopic superdiffusion and how is it influenced by the microscopic details of the dynamics ?

For these noisy Hamiltonian systems the challenging problems (and the tasks of the present project) are:<br /><br />1. To derive Euler equations (hyperbolic system of conservation laws) beyond the shocks;<br />2. To derive the heat equation in the diffusive case;<br />3. To understand the nature of the anomalous diffusion (and its relations with the Kardar-Parisi-Zhang universality class);<br />4. To understand the influence of the disorder on the energy diffusion properties (Anderson’s or Many Body localization theory);<br />5. To get some informations on the diffusion in purely deterministic chains<br />from the study of the noisy Hamiltonian dynamics.

The goals of the present proposal can be identified as follows: Providing rigorous results in a field where numerics are still controversial and theoretical explanations are too speculative; Developing methods coming from the probabilistic community of stochastic interacting particle systems to systems closer to deterministic systems.

The main achievement has been the rigorous proof of the heuristic predictions by H. Spohn about the nature of the superdiffusion for a chain of harmonic oscillators perturbed by a conservative noise. These results have been extended in a weak anharmonic regime confirming some expected universality.

The next goal is to extend the last results in the context a strong anharmonicity and precize the nature on the superdiffusion on more solid grounds than numerics and heuristic discussions.

[BGJ1] C. Bernardin, P. Gonçalves, M. Jara. 3/4-fractional
superdiffusion in a system of harmonic oscillators perturbed by a
conservative noise. Arch. Ration. Mech. Anal. 220 (2016), no. 2,
505–542.
[BGJ2] C. Bernardin, P. Gonçalves, M. Jara. Weakly harmonic
oscillators perturbed by a conserving noise, to appear in Ann.
Appl. Probab., (2017).
[BGJSS] C. Bernardin, P. Gonçalves, M. Jara, M. Sasada, M.
Simon. From normal diffusion to superdiffusion of energy in the
evanescent flip noise limit. J. Stat. Phys. 159 (2015), no. 6,
1327–1368.
[BGJS1] C. Bernardin, P. Gonçalves, M. Jara, M., Simon.
Interpolation process between standard diffusion and fractional
diffusion, to appear in Ann. Inst. H. Poincaré Probab. Statist.,
(2017).
[BGJS2] C. Bernardin, P. Gonçalves, M. Jara, M., Simon.
Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field
Model: Universality Persistence, online at HAL and submited,
(2017).
[BGS] C. Bernardin, P. Gonçalves, S. Sethuraman. Occupation
times of long-range exclusion and connections to KPZ class
exponents. Probab. Theory Related Fields 166 (2016), no. 1-2,
365–428.
[BHLLO] C. Bernardin, F. Huveneers, J.L. Lebowitz, C.
Liverani, S. Olla. Green-Kubo formula for weakly coupled system
with dynamical noise. Communications in Mathematical Physics,
Springer Verlag, 2015, 334 (3), pp.1377-1412.
[BO] C. Bernardin, B. Oviedo Jimenez. Fractional Fick's Law for
the Boundary Driven Exclusion Process with Long Jumps, ALEA
Lat. Am. J. Probab. Math. Stat. 14 (2017), no. 1, 473–501.
[DDHS] W. De Roeck, A. Dhar, F. Huveneers, M. Schuetz. Step
density profiles in localized chains. J. Stat. Phys. 167 (2017), no. 5, 1143–1163.
[GJS] P. Gonçalves, M. Jara, M. Simon. Second Order
Boltzmann–Gibbs Principle for Polynomial Functions and
Applications. J. Stat. Phys. 166 (2017), no. 1, 90–113.

The understanding of the deep mechanisms responsible of normal or anomalous energy diffusion in chains of coupled oscillators is one of the most important questions of non-equilibrium statistical mechanics, in the mathematical and theoretical physics literature. A phenomenological description has been proposed (see e.g. [Derrida-Dhar-Saito]) and some theoretical predictions are available ([Spohn]) but mathematical proofs are highly challenging. During the last few years, to attack the problem in a rigorous way, it has been proposed to replace the purely deterministic chains by hybrid models: a conservative stochastic noise (in energy and in some cases in momentum) is superposed to the Hamiltonian dynamics. It turns out that, as proved by Basile, Bernardin and Olla in the harmonic case, these noisy Hamiltonian systems behave qualitatively similarly to the deterministic systems. The energy diffusion properties satisfied by the hybrid models are due to a subtle interplay between the Hamiltonian part and the stochastic part of the dynamics and are not reproduced by purely stochastic dynamics.

Even if the study of hybrid systems is simpler than the study of purely deterministic chains, the former are the source of many important challenging problems that have not yet been solved, in particular in the anharmonic case. We plan to work on these problems for the models introduced in [Basile-Bernardin-Olla] and also for simplified perturbed Hamiltonian systems considered recently by Bernardin and Stoltz. These stochastically perturbed systems will be studied thanks to the techniques introduced by Varadhan and Yau in the middle of the nineties and currently investigated in the context of scaling limits for stochastic interacting particle systems. The development of these techniques is a very active field of research nowadays and the hybrid models give rise to several new interesting difficulties.

Shortly, we are interested in: hydrodynamic limits (given by hyperbolic system of conservation laws) in the Euler time scale after the shocks; behavior of the thermal conductivity in the weak coupling (small interaction) limit and when the noise intensity goes to zero; effect of the disorder (e.g. random masses) on the transport properties of the noisy system (interplay between the noise and the Anderson's type localization phenomenon); study of the anomalous energy fluctuations.

To achieve these advances we will combine complementary skills and techniques from four mathematicians specialized and well trained in scaling limits of interacting particles systems, two mathematical physicists specialized in non-equilibrium statistical mechanics and one physicist. A PhD student, who will defend her thesis on June 17, 2014, will complete the team. The project being at the interface of physics and mathematics the exchange of ideas from the two communities is for us a necessity.

Project coordination

Cédric Bernardin (UNIVERSITE NICE SOPHIA ANTIPOLIS/Laboratoire Dieudonné)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

UNS UNIVERSITE NICE SOPHIA ANTIPOLIS/Laboratoire Dieudonné

Help of the ANR 104,000 euros
Beginning and duration of the scientific project: September 2014 - 48 Months

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