Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing
Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing
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DOI：
10.1039/c8cp01863d
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发表时间：
2018-08-28
影响因子：
3.3
通讯作者：
Akimoto, Takuma
中科院分区：
文献类型：
--
作者：
Hou, Ru;Cherstvy, Andrey G.;Akimoto, Takuma
我们研究更新过程的幂律等待时间分布（WTDs）和非零漂移通过计算分析和计算机模拟他们的合奏和时间平均传播特性。对于类似于1/t（1+alpha）的WTD psi（t），考虑标度指数alpha的所有可能值。我们处理连续时间随机游动（CTRW）0 < alpha < 1的平均等待时间发散，并研究1 < alpha 2的普通和平衡CTRW的过程的行为< 2 and alpha >。我们证明，在漂移的存在下，α < 1的CTRW是老化和非遍历的意义上的非等价的合奏和时间平均位移特性的滞后时间的限制，远远短于轨迹长度。在系综和时间平均等价的意义下，1 <α < 2的CTRW过程在平衡态下是遍历的，在一般情况下是非遍历的。最后，α> 2的CTRW更新过程--无论是平衡的还是普通的--总是遍历的。然而，对于1 <α 2的情况< 2 and alpha >，扩散过程的方差取决于初始系综。对于α> 1的有偏CTRW，我们还研究了遍历性破缺参数的行为。此外，我们证明，有偏见的CTRW的爱因斯坦关系是有效的合奏和时间平均位移的水平上，在整个范围内的WTD指数α。
We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent alpha are considered for the WTD psi(t) similar to 1/t(1+alpha). We treat continuous-time random walks (CTRWs) with 0 < alpha < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < alpha < 2 and alpha > 2. We demonstrate that in the presence of a drift CTRWs with alpha < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < alpha < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with alpha > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < alpha < 2 and alpha > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with alpha > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent alpha.