AI 中文总结
研究边界驱动的弱非谐链非平衡稳态，在微扰弱非谐条件下证明全梯度对数 Sobolev 不等式，对均匀固定链在特定假设下得到相关不等式及相对熵衰减，通过提取高斯分量和变量变换证明，结果与温度无关且无需近平衡假设。
AI 中文摘要
我们研究了由不等温朗之万恒温器在边界驱动的\(N\)个振子的弱非谐链的非平衡稳态。在微扰弱非谐条件下，我们证明了一个全梯度对数 Sobolev 不等式，其常数与链长\(N\)无关。对于均匀固定链，一个额外的定量正则性假设产生了边界时空对数 Sobolev 不等式和在与谐链相同\(O(N^3)\)弛豫时间尺度上的相对熵衰减。证明从边界噪声中提取有限维高斯分量，并通过变量变换比较条件终端状态定律。估计在有界正温度上是均匀的，并且对它们的差异不需要近平衡假设。
英文摘要
We study the non-equilibrium steady state of a weakly anharmonic chain of $N$ 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 $N$. For homogeneous pinned chains, an additional quantitative regularity assumption yields a boundary space-time logarithmic Sobolev inequality and relative-entropy decay on the same $O(N^3)$ 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.
Comments29 pages
