We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $$N^{-3}$$ N - 3 . For the purely harmonic chain the order of the rate is in $$ [N^{-3},N^{-1}]$$ [ N - 3 , N - 1 ] . This shows that, in this set up, the spectral gap decays at most polynomially with N .

中文翻译：

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一个弱非谐振荡器链收敛到非平衡稳态的定量速率

我们研究了 N 个弱非谐经典振荡器的一维链，其末端耦合到不同温度的热浴。每个振荡器都受到钉扎电位的影响，并且还与其最近的邻居相互作用。在我们的设置中，两个势能都是同质的，并且是谐波扰动的有界（有 N 个依赖边界）扰动。我们展示了 Bakry-Emery 理论的广义版本如何适用于这种受 Baudoin 启发的亚椭圆生成器（J Funct Anal 273(7):2275-2291, 2017）。通过这一点，我们证明了在 Wasserstein-Kantorovich 距离和相对熵与定量率中对非平衡稳态的指数收敛。我们通过求解李雅普诺夫型矩阵方程来估计速率常数，我们得到齐次链的指数速率，订单大于 $$N^{-3}$$ N - 3 。对于纯谐波链，速率的顺序为 $$ [N^{-3},N^{-1}]$$ [ N - 3 , N - 1 ] 。这表明，在此设置中，频谱间隙最多以 N 多项式衰减。