一维圈闭Riesz气体线性统计的累积量和大偏差

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Pierre Le Doussal, Grégory Schehr
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引用次数: 0

摘要

我们考虑经典的困住的Riesz气体,即N个粒子在一维中\(x_i\)的位置具有排斥幂律相互作用势\(\propto 1/|x_i-x_j|^{k}\),与\(k>-2\),在形式为\(V(x) \sim |x|^n\)的外部限制势。我们关注气体的平衡吉布斯状态,对于这个状态,密度有一个有限的支持\([-\ell _0/2,\ell _0/2]\)。研究了光滑函数f(x)在大N极限下线性统计量\({{\mathcal {L}}}_N = \sum _{i=1}^N f(x_i)\)的涨落。对于一般的\(k>-2\),我们得到了\({{\mathcal {L}}}_N\)累积量的解析公式。对于长距离相互作用,例如\(k<1\),其中包括对数气体(\(k \rightarrow 0\))和库仑气体(\(k =-1\)),这些是单项\(f(x)= |x|^m\)得到的。对于短程相互作用,即\(k>1\),其中包括Calogero-Moser模型,即\(k=2\),我们计算一般f(x)的第三累积量\({{\mathcal {L}}}_N\)和单项的任意累积量\(f(x)= |x|^m\)。我们还得到了\({{\mathcal {L}}}_N\)概率分布的大偏差形式,它表现为“蒸发过渡”,其中\({{\mathcal {L}}}_N\)的波动由最大的一个\(x_i\)主导。此外,在短期情况下,我们将结果推广到(非光滑)指示函数f(x),从而获得了区间\([-L/2,L/2]\)中粒子数的全计数统计量的高阶累积量。我们特别指出,当L/2接近气体的边缘\(L/\ell _0 \rightarrow 1\)时,它们表现出一种有趣的缩放形式,我们将其与实线上互补区间的空性概率的大偏差联系起来。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Cumulants and Large Deviations for the Linear Statistics of the One-Dimensional Trapped Riesz Gas

We consider the classical trapped Riesz gas, i.e., N particles at positions \(x_i\) in one dimension with a repulsive power law interacting potential \(\propto 1/|x_i-x_j|^{k}\), with \(k>-2\), in an external confining potential of the form \(V(x) \sim |x|^n\). We focus on the equilibrium Gibbs state of the gas, for which the density has a finite support \([-\ell _0/2,\ell _0/2]\). We study the fluctuations of the linear statistics \({{\mathcal {L}}}_N = \sum _{i=1}^N f(x_i)\) in the large N limit for smooth functions f(x). We obtain analytic formulae for the cumulants of \({{\mathcal {L}}}_N\) for general \(k>-2\). For long range interactions, i.e. \(k<1\), which include the log-gas (\(k \rightarrow 0\)) and the Coulomb gas (\(k =-1\)) these are obtained for monomials \(f(x)= |x|^m\). For short range interactions, i.e. \(k>1\), which include the Calogero–Moser model, i.e. \(k=2\), we compute the third cumulant of \({{\mathcal {L}}}_N\) for general f(x) and arbitrary cumulants for monomials \(f(x)= |x|^m\). We also obtain the large deviation form of the probability distribution of \({{\mathcal {L}}}_N\), which exhibits an “evaporation transition” where the fluctuation of \({{\mathcal {L}}}_N\) is dominated by the one of the largest \(x_i\). In addition, in the short range case, we extend our results to a (non-smooth) indicator function f(x), obtaining thereby the higher order cumulants for the full counting statistics of the number of particles in an interval \([-L/2,L/2]\). We show in particular that they exhibit an interesting scaling form as L/2 approaches the edge of the gas \(L/\ell _0 \rightarrow 1\), which we relate to the large deviations of the emptiness probability of the complementary interval on the real line.

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来源期刊
Journal of Statistical Physics
Journal of Statistical Physics 物理-物理:数学物理
CiteScore
3.10
自引率
12.50%
发文量
152
审稿时长
3-6 weeks
期刊介绍: The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.
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