论文标题
与随机重置的自相似高斯过程的大偏差的凝结过渡
Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting
论文作者
论文摘要
我们研究了区域$ a（t）= \ int_0^t x（τ）\，dτ$在自相似的高斯过程（SGP）$ x（τ）$的情况下，带有hurst指数$ h> 0 $（例如，标准的或标准的棕色运动或随机的动作，或随机的流程，或随机的过程），或者是$ $ $ $ $ $ $ r $ r $ r $ r $ r $ r $。 $ a（t）$ scale的典型波动为$ \ sim \ sqrt {t} $对于大$ t $，在此规模上，分布是高斯，正如人们从中央限制定理中期望的那样。在这里，我们的主要重点是$ a（t）$的非典型波动。在长期限制$ t \ to \ infty $中，我们发现该区域的完整分布采用$ p_ {r} \ left（a | t \ right）\ sim \ sim \ exp \ left [-t^αφ\ left（a/t^ββ\ weft（a/t^β\ right）\ right] （2H+3）/（4H+4）$在中等大波动方面，以及不同的异常缩放表格$ p_ {r} \ left（a | t \ right）\ sim \ sim \ sim \ sim \ exp \ left [-t病\ left（a/t \ left（a/t^{a/t+weft（a/t）关联的费率功能$φ（y）$和$ψ（w）$取决于$ h $，并且完全找到。值得注意的是，$φ（y）$具有奇异性，我们将其解释为一阶动力冷凝转换，而$ψ（w）$表现出二阶动力学相变，上面的重置事件的数量将停止。 $ y = 0 $周围的$φ（y）$的抛物线行为正确地描述了$ a（t）$的典型高斯波动。尽管有这些异常的尺度，我们发现分布的所有累积物$ p_ {r} \ left（a | t \ right）$及时线性地生长，$ \ langle a^n \ rangle_c \ rangle_c \ oft c_n \ c_n \ c_n \，t $，在长期限制中。对于重置Brownian运动的情况（对应于$ H = 1/2 $），我们开发了一个递归方案来计算系数$ C_N $，并使用它来计算前6个非散布累积物。
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.
