Open Access
2020 Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain
Lu Xu
Electron. J. Probab. 25: 1-40 (2020). DOI: 10.1214/20-EJP488


We consider a chain of $n$ coupled oscillators placed on a one-dimensional lattice with periodic boundary conditions. The interaction between particles is determined by a weakly anharmonic potential $V_{n} = r^{2}/2 + \sigma _{n}U(r)$, where $U$ has bounded second derivative and $\sigma _{n}$ vanishes as $n \to \infty $. The dynamics is perturbed by noises acting only on the positions, such that the total momentum and length are the only conserved quantities. With relative entropy technique, we prove for dynamics out of equilibrium that, if $\sigma _{n}$ decays sufficiently fast, the fluctuation field of the conserved quantities converges in law to a linear $p$-system in the hyperbolic space-time scaling limit. The transition speed is spatially homogeneous due to the vanishing anharmonicity. We also present a quantitative bound for the speed of convergence to the corresponding hydrodynamic limit.


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Lu Xu. "Hyperbolic scaling limit of non-equilibrium fluctuations for a weakly anharmonic chain." Electron. J. Probab. 25 1 - 40, 2020.


Received: 26 November 2019; Accepted: 27 June 2020; Published: 2020
First available in Project Euclid: 18 July 2020

zbMATH: 07252716
MathSciNet: MR4125789
Digital Object Identifier: 10.1214/20-EJP488

Primary: 60K35 , 82C05 , 82C22

Keywords: Boltzmann–Gibbs principle , hyperbolic scaling limit , non-equilibrium fluctuation , Relative entropy

Vol.25 • 2020
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