Abstract
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 $V_n = r^2/2 + \sigma_nU(r)$, 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.
Highlights
One of the central topics in statistical physics is to derive macroscopic equations in scaling limits of microscopic dynamics
For Hamiltonian lattice field, Euler equations can be formally obtained in the limit, under a generic assumption of local equilibrium
Proper noises can provide the dynamics with enough ergodicity, in the sense that the only conserved quantities are those evolving with the macroscopic equations [13]
Summary
One of the central topics in statistical physics is to derive macroscopic equations in scaling limits of microscopic dynamics. For Hamiltonian dynamics with noises conserving volume, momentum and energy, Euler equations are obtained under the hyperbolic space-time scale [23, 4]. They are proved by relative entropy technique and restricted to the smooth regime of Euler equations. Theorem 2.4, shows that non-equilibrium fluctuations evolve following a linear p-system with spatially homogeneous sound speed, provided that U has bounded second order derivative and σn decays fast enough This is the first rigorous result obtained for non-equilibrium fluctuations for a Hamiltonian dynamics presents some level of nonlinearity. Let Cα([0, T ]; H−k) be the subset of C([0, T ]; H−k), consisting of Holder continuous trajectories with order α > 0
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
