Log-Sobolev inequalities for boundary-driven anharmonic chains
arXiv:2607.13953
The paper proves uniform logarithmic Sobolev inequalities for a weakly anharmonic chain of oscillators driven at its ends by Langevin thermostats at different temperatures, showing entropy decay on a relaxation time scaling like N³.
Abstract
We study the non-equilibrium steady state of a weakly anharmonic chain of oscillators driven at its boundary by Langevin thermostats at unequal temperatures. Under a perturbative weak-anharmonicity condition, we prove a full-gradient logarithmic Sobolev inequality whose constant is independent of the chain length . For homogeneous pinned chains, an additional quantitative regularity assumption yields a boundary space-time logarithmic Sobolev inequality and relative-entropy decay on the same relaxation time scale as the harmonic chain. The proof extracts a finite-dimensional Gaussian component from the boundary noise and compares conditional terminal-state laws by a change of variables. The estimates are uniform over bounded positive temperatures and require no near-equilibrium assumption on their difference.
29 pages