The Green–Kubo formula for general Markov processes with a continuous time parameter

Abstract

For general Markov processes, the Green-Kubo formula is shown to be valid under a mild condition. A class of stochastic evolution equations on a separable Hilbert space and three typical infinite systems of locally interacting diffusions on Z d (irreversible in most cases) are shown to satisfy the Green-Kubo formula, and the Einstein relations for these stochastic evolution equations are shown explicitly as a corollary.

Stochastic Processes and Applications

“…Our approach to the Green-Kubo formula in Sect. 9.3, and in particular, Result 9.1, is based on [107,244]. See also [106].…”

Linear response in the nonequilibrium zero range process

“…which is the modification to the Green-Kubo relation [11] , for all times t > 0, for the boundary driven zero range process. Observe that it is the correlation between current J ℓ and dynamical activity I ℓ that governs the correction.…”

“…As a consequence, using (15) results in the linear response exactly of the same form (11) as in equilibrium, because (with V = N in ( 13)),…”

“…The first term is the Kubo expression (11). The second term will be useful whenever we know more about the time-reversed process.…”

Non-equilibrium stochastic dynamics

“…文献 [35] 证明了这个结果在可逆马氏链 (平衡态) 中对一个实观测量 ϕ 成立. 文献 [36] 对一般有 限状态马氏链 (它的向前速度 Dϕ(ξ t ) 与向后速度 D * ϕ(ξ t ) (倒逆过程的向前速度) 可能不相同) 给出 了 Green-Kubo 公式最一般的形式 (2.6), 事实上它也适用于扩散过程和一般马氏过程 [37] . 进而, 我们还可以考虑连续参数平稳马氏过程 (包括马氏链和扩散过程等) 的某个实值观测量 ϕ 随系统变化时的功率谱 (它的自相关函数的 Fourier 变换), 发现某些实值观测量的功率谱密度具有非 零点处的谱峰可以作为系统处于非平衡态的标识, 因为可逆平稳马氏过程的任意观测量 ϕ 过程的功 率谱总是在 [0, ∞) 上单调下降的 (参见文献 [35,36,38]).…”