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arXiv 2603.24151cond-mat.stat-mech
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布朗重排中顺序统计的普遍性

Universality of order statistics for Brownian reshuffling

AGH University（亚捷隆大学）；University of Melbourne（墨尔本大学）

Zdzislaw Burda, Mario Kieburg, Tomasz Maciocha

备注：
15 pages
发表信息：
Phys. Rev. E 114, 014130 (2026)
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AI总结：

研究一维布朗运动粒子在趋向无穷大的势场中，证明领导者的重排过程与γ无关，但重排时间尺度依赖于γ，且生成函数在缩放时间下独立于γ。

摘要：

我们讨论了N个相同独立粒子在一维中进行布朗运动时，粒子位置的顺序统计。我们证明在稳态下，描述领导者重排的顺序统计是普遍的，与γ无关。依赖于γ的是领导者重排的时间尺度，其与种群大小的对数呈幂律关系：t ~ (ln N)^(2(1-γ)/γ τ)，其中τ为常数级。我们推导了粒子在时间t1时第k大的x值在时间t2=t1+t时变为第j大的概率的生成函数，该生成函数在缩放时间τ下独立于γ。特别地，我们证明长列表的平均重叠系数取普遍形式erfc(√τ)。

英文摘要：

We discuss the order statistics of the particle positions of a gas of $N$ identical independent particles performing Brownian motion in one dimension in a potential that asymptotically behaves like $V(x) \sim x^γ$ for $x\rightarrow+\infty$, with a positive power $γ>0$. We show that in the stationary state, the order statistics that describe how the leaders are reshuffled are universal and independent of $γ$. What depends on $γ$ is the timescale of the leaders' reshuffling, which scales as a power of the logarithm of the population size: $t \sim (\ln N)^\frac{2(1-γ)}γ τ$, where $τ$ is of order one. We derive the probability that the particle which has the $k$th largest value of $x$ at some time $t_1$ will have the $j$th largest value at time $t_2=t_1+t$ in the form of an explicit expression for the generating function for the reshuffling probabilities for all $k\ge 1$ and $j\ge 1$. The generating function, expressed in scaled time $τ$, is independent of $γ$. In particular, we show that the average percentage overlap coefficient of leader lists takes the universal, $γ$-independent form ${\rm erfc}(\sqrtτ)$ for long lists.

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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