Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise
Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise
复制标题
带噪声的一维聚焦非线性薛定谔方程的指数弛豫到平衡
DOI：
10.1007/s00220-015-2511-9
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发表时间：
2014
影响因子：
2.4
通讯作者：
J. Lebowitz
中科院分区：
文献类型：
--
作者：
E. Carlen;J. Froehlich;J. Lebowitz
我们构造了一个哈密顿系统的广义巨正则和正则吉布斯测度，该哈密顿系统用一个定义在圆上的复标量场来描述，并满足一个非线性薛定谔方程，其聚焦非线性度为p < 6。这些吉布斯措施的关键属性，特别是没有“相变”和规律性的字段样本，建立。然后，我们研究了这个系统的时间演化的随机噪声项，模仿耦合系统的热浴在一些固定的温度的影响扰动的哈密顿演化。对于场的傅里叶模式，噪声是Ornstein-Uhlenbeck型的，随着模式的频率趋于∞，噪声的强度衰减到零。我们证明了指数的方法的系统的状态的巨正则吉布斯措施的温度和“化学势”确定的随机噪声项。
We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schrödinger equation with a focusing nonlinearity of order p < 6. Key properties of these Gibbs measures, in particular absence of “phase transitions” and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein–Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to ∞. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and “chemical potential” determined by the stochastic noise term.