Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions
Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions
复制标题
具有重尾分布的更新奖励过程的异常波动
DOI：
10.1103/physreve.106.034130
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发表时间：
2022
影响因子：
2.4
通讯作者：
Nemoto Takahiro
中科院分区：
文献类型：
--
作者：
Horii Hiroshi;Lefevere Raphael;Itami Masato;Nemoto Takahiro
对于具有幂律衰减等待时间分布的更新奖励过程，极大的概率被分配给渐近过程的非典型值。以前的工作表明，这种异常的标度导致在相应的大偏差函数的奇异性。为了进一步理解这个问题，我们在这篇文章中研究了几个更新奖励过程的方差标度：计数过程与两个不同的幂律衰减的等待时间分布和Knudsen气体（热传导模型）。通过对这些模型的解析和数值分析，我们发现当幂律指数为时，方差表现出反常的标度性。对于幂律指数小于的计数过程，这种异常标度不会发生：这表明如果我们只考虑期望值的标准差，任何异常行为都不会被检测到。在这种情况下，我们认为，反常标度出现在高阶累积量。最后，多体粒子相互作用通过软核相互作用的边界条件在克努森气体中使用的数值模拟进行了研究。我们观察到，方差标度变得正常，即使幂律指数的边界条件。
For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have revealed that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is. For a counting process with the power-law exponent smaller than, this anomalous scaling does not take place: this indicates that if we only consider the standard deviation from the expectation, any anomalous behavior will not be detected. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is.