Bray AJ
Department of Physics and Astronomy, The University, Manchester M13 9PL, United Kingdom.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jul;62(1 Pt A):103-12. doi: 10.1103/physreve.62.103.
The Langevin equation for a particle ("random walker") moving in d-dimensional space under an attractive central force and driven by a Gaussian white noise is considered for the case of a power-law force, F(r) approximately -r(-sigma). The "persistence probability," P0(t), that the particle has not visited the origin up to time t is calculated for a number of cases. For sigma>1, the force is asymptotically irrelevant (with respect to the noise), and the asymptotics of P0(t) are those of a free random walker. For sigma<1, the noise is (dangerously) irrelevant and the asymptotics of P0(t) can be extracted from a weak noise limit within a path-integral formalism employing the Onsager-Machlup functional. The case sigma=1, corresponding to a logarithmic potential, is most interesting because the noise is exactly marginal. In this case, P0(t) decays as a power law, P0(t) approximately t(-straight theta) with an exponent straight theta that depends continuously on the ratio of the strength of the potential to the strength of the noise. This case, with d=2, is relevant to the annihilation dynamics of a vortex-antivortex pair in the two-dimensional XY model. Although the noise is multiplicative in the latter case, the relevant Langevin equation can be transformed to the standard form discussed in the first part of the paper. The mean annihilation time for a pair initially separated by r is given by t(r) approximately r(2) ln(r/a) where a is a microscopic cutoff (the vortex core size). Implications for the nonequilibrium critical dynamics of the system are discussed and compared to numerical simulation results.
考虑一个在幂律力(F(r)\approx -r^{-\sigma})作用下、受高斯白噪声驱动且在(d)维空间中运动的粒子（“随机游走者”）的朗之万方程。针对多种情况计算了粒子在时间(t)之前未到达原点的“持续概率”(P_0(t))。对于(\sigma>1)，该力在渐近意义上（相对于噪声）无关紧要，(P_0(t))的渐近行为与自由随机游走者的相同。对于(\sigma<1)，噪声（危险地）无关紧要，(P_0(t))的渐近行为可通过采用昂萨格 - 马赫卢普泛函的路径积分形式体系中的弱噪声极限来提取。(\sigma = 1)的情况对应于对数势，最为有趣，因为噪声恰好处于边缘状态。在这种情况下，(P_0(t))按幂律衰减，(P_0(t)\approx t^{-\theta})，其中指数(\theta)连续依赖于势的强度与噪声强度的比值。(d = 2)时的这种情况与二维(XY)模型中涡旋 - 反涡旋对的湮灭动力学相关。尽管在后一种情况下噪声是乘性的，但相关的朗之万方程可转化为本文第一部分讨论的标准形式。一对初始间距为(r)的平均湮灭时间由(t(r)\approx r^2\ln(r/a))给出，其中(a)是微观截断（涡旋核心尺寸）。讨论了该系统非平衡临界动力学的相关影响，并与数值模拟结果进行了比较。
