This paper studies heat flow in a simple microscopic model: a chain of N coupled oscillators that are fixed in place and driven at the two ends by separate heat baths. The baths are modeled by Langevin thermostats, which add friction and random noise to the boundary momenta. When the two bath temperatures differ, the system settles into a non-equilibrium steady state (NESS) that is not given by an explicit formula. The authors derive new inequalities that describe how concentrated that steady state is and how fast arbitrary initial states relax to it.

The main mathematical objects are logarithmic Sobolev inequalities (LSIs). An LSI bounds the relative entropy of a probability density (a measure of disorder) by a kind of ‘‘derivative energy’’ called Fisher information, which measures how rapidly the density changes. The paper proves two complementary LSIs for the boundary-driven chain in a weakly anharmonic regime (that is, the chain is a small perturbation of a harmonic, purely spring-like, chain). First, they obtain a full-gradient LSI whose constant does not grow with N. In plain terms, this gives uniform control on fluctuations in the steady state no matter how long the chain is.

Second, for the special case of a homogeneous pinned chain (each oscillator has the same on-site pinning and nearest-neighbor coupling) and under an extra quantitative regularity assumption, they prove a boundary space-time LSI. That inequality links entropy decay to dissipation coming only from the two boundary momenta. Using that bound they show that relative entropy decays to zero on the same O(N^3) time scale known for the harmonic chain. In other words, the chain relaxes to its steady state in a time proportional to N^3, and this scaling survives the small anharmonic perturbation under the stated assumptions.

The proofs use two main ideas. To get the dimension-free full-gradient LSI the authors extract a finite-dimensional Gaussian component from the boundary Brownian noise and transfer Gross’s Gaussian LSI to the chain by estimating the linearized propagator. For the boundary space-time LSI they exploit controllability of the harmonic reference chain and compare transition kernels by a finite-dimensional change of variables. The estimates are uniform over bounded positive temperatures and do not require the two baths to be nearly equal in temperature.