arXiv 2202.035462022-02cond-mat.stat-mechmath-phmath.MP

具有随机重置的自相似高斯过程大偏差中的凝聚跃迁

Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

Naftali R. Smith, Satya N. Majumdar

备注：: 21 pages, 6 figures
发表信息：: J. Stat. Mech. (2022) 053212
DOI：: 10.1088/1742-5468/ac6f04

摘要：

我们研究了具有Hurst指数$H>0$的自相似高斯过程（SGP）$x(τ)$下面积$A(t)= \int_0^t x(τ)\, dτ$的波动（例如，标准或分数布朗运动，或随机加速过程），随机以速率$r$重置到原点。$A(t)$的典型波动尺度为大型$t$的$\sim \sqrt{t}$，在此尺度上，分布为高斯分布，正如中心极限定理所预期的那样。在这里，我们主要关注的是$A(t)$的巨大波动。在长时间的限制$t\to\infty$，我们发现，该地区的全部分布采取的形式$P_{r}\left(A|t\right)\sim\exp\left[-t^αΦ\left(A/t^β\right)\right]$与异常指数$α=1/(2H+2)$和$β= (2H+3)/(4H+4)$的制度中的中等大的波动，和一个不同的异常标度形式$P_{r}\left(A|t\right)\sim\exp\left[-tΨ\left(A/t^{\left(2H+3\right)/2}\right)\right]$的制度非常大的波动。相关的速率函数$Φ(y)$和$Ψ(w)$依赖于$H$，并被精确地找到。值得注意的是，$Φ(y)$有一个奇点，我们解释为一阶动力学凝聚过渡，而$Ψ(w)$表现出二阶动力学相变以上的重置事件的数量不再是广泛的。$Φ(y)$围绕原点$y=0$的抛物线行为正确地描述了$A(t)$的典型高斯波动。尽管有这些异常标度，我们发现，所有的累积量分布$P_{r}\left(A|t\right)$线性增长的时间，$\langle A^n\rangle_c\approx c_n \, t$，在长时间限制。对于重置布朗运动（对应于$H=1/2$）的情形，我们发展了一种递归格式来精确计算系数$c_n$，并利用它来计算前6个非零累积量。

英文摘要：

We study the fluctuations of the area $A(t)= \int_0^t x(τ)\, dτ$ under a self-similar Gaussian process (SGP) $x(τ)$ with Hurst exponent $H>0$ (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate $r$. Typical fluctuations of $A(t)$ scale as $\sim \sqrt{t}$ for large $t$ and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of $A(t)$. In the long-time limit $t\to\infty$, we find that the full distribution of the area takes the form $P_{r}\left(A|t\right)\sim\exp\left[-t^αΦ\left(A/t^β\right)\right]$ with anomalous exponents $α=1/(2H+2)$ and $β= (2H+3)/(4H+4)$ in the regime of moderately large fluctuations, and a different anomalous scaling form $P_{r}\left(A|t\right)\sim\exp\left[-tΨ\left(A/t^{\left(2H+3\right)/2}\right)\right]$ in the regime of very large fluctuations. The associated rate functions $Φ(y)$ and $Ψ(w)$ depend on $H$ and are found exactly. Remarkably, $Φ(y)$ has a singularity that we interpret as a first-order dynamical condensation transition, while $Ψ(w)$ exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of $Φ(y)$ around the origin $y=0$ correctly describes the typical, Gaussian fluctuations of $A(t)$. Despite these anomalous scalings, we find that all of the cumulants of the distribution $P_{r}\left(A|t\right)$ grow linearly in time, $\langle A^n\rangle_c\approx c_n \, t$, in the long-time limit. For the case of reset Brownian motion (corresponding to $H=1/2$), we develop a recursive scheme to calculate the coefficients $c_n$ exactly and use it to calculate the first 6 nonvanishing cumulants.

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