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arXiv 2311.18366cond-mat.stat-mech
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被困在$|x|$势中的布朗粒子所扫过面积涨落的异常标度

Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential

Naftali R. Smith

备注：
18 pages, 4 figures
发表信息：
Physica A 650, 129987 (2024)
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AI总结：

本文研究$|x|$势中布朗粒子扫过面积的涨落，发现典型涨落为高斯型，而大偏差呈现两种异常标度，并精确求出速率函数，揭示其奇点对应一阶和三级动力学相变。

摘要：

我们研究在一维布朗运动$x(t)$处于捕获势$\sim |x|$中时，长时间$T\to\infty$下面积$A=\int_0^T x(t) dt$的涨落。我们发现$A$的典型涨落服从高斯分布，其方差随时间线性增长（在大$T$时），分布的所有高阶累积量也是如此。然而，$A$的大偏差并不由“通常”的标度（即大偏差原理）描述，而是由两种不同的异常标度行为描述：$A$的中等大偏差服从异常标度$P\left(A;T\right)\sim e^{-T^{1/3}f\left(A/T^{2/3}\right)}$，而非常大的偏差则表现为$P\left(A;T\right)\sim e^{-TΨ\left(A/T^{2}\right)}$。我们精确求出了相应的速率函数$f$和$Ψ$。这两个函数各含有一个奇点，我们将其分别解释为一阶和三级动力学相变。我们通过刻画系统达到给定非典型$A$值的最可能情景，揭示了这些显著行为的起源。我们通过研究绝对面积$B=\int_0^T|x(t)| dt$扩展了分析，并将其推广到更高空间维度，重点关注三维这一特殊情况。

英文摘要：

We study the fluctuations of the area $A=\int_0^T x(t) dt$ under a one-dimensional Brownian motion $x(t)$ in a trapping potential $\sim |x|$, at long times $T\to\infty$. We find that typical fluctuations of $A$ follow a Gaussian distribution with a variance that grows linearly in time (at large $T$), as do all higher cumulants of the distribution. However, large deviations of $A$ are not described by the ``usual'' scaling (i.e., the large deviations principle), and are instead described by two different anomalous scaling behaviors: Moderately-large deviations of $A$, obey the anomalous scaling $P\left(A;T\right)\sim e^{-T^{1/3}f\left(A/T^{2/3}\right)}$ while very large deviations behave as $P\left(A;T\right)\sim e^{-TΨ\left(A/T^{2}\right)}$. We find the associated rate functions $f$ and $Ψ$ exactly. Each of the two functions contains a singularity, which we interpret as dynamical phase transitions of the first and third order, respectively. We uncover the origin of these striking behaviors by characterizing the most likely scenario(s) for the system to reach a given atypical value of $A$. We extend our analysis by studying the absolute area $B=\int_0^T|x(t)| dt$ and also by generalizing to higher spatial dimension, focusing on the particular case of three dimensions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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