Abstract We consider an infinite chain of coupled harmonic oscillators whose Hamiltonian dynamics is perturbed by a random exchange of momentum between particles such that total energy and momentum are conserved, modeling collision between atoms. This random exchange is rarefied in the limit, that corresponds to the hypothesis that in the macroscopic unit time only a finite number of collisions takes place (the Boltzmann-Grad limit). Furthermore, the system is in contact with a Langevin thermostat at temperature T through a single particle. We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function W ( t , y , k ) that evolves according to a linear kinetic equation, with the interface condition at y = 0 that corresponds to reflection, transmission and absorption of phonons caused by the presence of the thermostat. The paper extends the results of [15] , where a harmonic chain (with no inter-particle scattering) in contact with a Langevin thermostat has been considered.

中文翻译：

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带有朗之万点温控器的谐振子链的动力学极限

摘要 我们考虑了一个无限的耦合谐振子链，其哈密顿动力学受到粒子之间动量随机交换的扰动，使得总能量和动量守恒，模拟原子之间的碰撞。这种随机交换在极限中是稀少的，这对应于在宏观单位时间内只发生有限数量的碰撞的假设（玻尔兹曼-格拉德极限）。此外，该系统通过单个粒子与温度为 T 的朗之万恒温器接触。我们证明，在双曲线时空重新标度之后，描述空间频率域中声子能量密度的 Wigner 分布收敛到一个正能量密度函数 W ( t , y , k )，它根据线性动力学演化方程，y = 0 时的界面条件对应于由恒温器的存在引起的声子的反射、传输和吸收。该论文扩展了 [15] 的结果，其中考虑了与朗之万恒温器接触的谐波链（没有粒子间散射）。