A Tale of Three Approaches: Dynamical Phase Transitions for Weakly Bound Brownian Particles

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Lucianno Defaveri, Eli Barkai, David A. Kessler
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Abstract

We investigate a system of Brownian particles weakly bound by attractive parity-symmetric potentials that grow at large distances as \(V(x) \sim |x|^\alpha \), with \(0< \alpha < 1\). The probability density function P(xt) at long times reaches the Boltzmann–Gibbs equilibrium state, with all moments finite. However, the system’s relaxation is not exponential, as is usual for a confining system with a well-defined equilibrium, but instead follows a stretched exponential \(e^{- \textrm{const} \, t^\nu }\) with exponent \(\nu =\alpha /(2+\alpha )\), as we announced recently in a short letter. In turn, the stretched exponential relaxation is related to large-deviation theory, which is studied from three perspectives. First, we propose a straightforward and general scaling rate-function solution for P(xt). This rate function displays anomalous time scaling and a dynamical phase transition. Second, through the eigenfunctions of the Fokker–Planck operator, we obtain, using the WKB method, more complete solutions that reproduce the rate function approach and provide important pre-exponential corrections. Finally, we show how the alternative path-integral formalism allows us to recover the same results, with the above rate function being the solution of the classical Hamilton–Jacobi equation describing the most probable path. Properties such as parity, the role of initial conditions, and the dynamical phase transition are thoroughly studied in all three approaches.

三种方法的故事:弱约束布朗粒子的动态相变
我们研究了一个布朗粒子系统,该系统被吸引的宇称对称势弱束缚,在很远的距离上增长为\(V(x) \sim |x|^\alpha \),其中\(0< \alpha < 1\)。概率密度函数P(x, t)在长时间内达到Boltzmann-Gibbs平衡态,且所有矩都是有限的。然而,系统的弛豫不是指数的,就像通常具有明确定义的平衡的约束系统一样,而是遵循指数\(\nu =\alpha /(2+\alpha )\)的拉伸指数\(e^{- \textrm{const} \, t^\nu }\),正如我们最近在一封简短的信中宣布的那样。拉伸后的指数弛豫与大偏差理论有关,并从三个方面进行了研究。首先,我们提出了P(x, t)的一个简单而通用的标度速率函数解。该速率函数显示了异常的时间标度和动态相变。其次,通过Fokker-Planck算子的特征函数,我们使用WKB方法获得了更完整的解,这些解再现了速率函数方法并提供了重要的指数前校正。最后,我们展示了替代路径积分形式如何允许我们恢复相同的结果,上面的速率函数是描述最可能路径的经典汉密尔顿-雅可比方程的解。在这三种方法中,对宇称、初始条件的作用和动态相变等性质进行了深入的研究。
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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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