Le Doussal P, Monthus C, Fisher D S
CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris, France.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1999 May;59(5 Pt A):4795-840. doi: 10.1103/physreve.59.4795.
Sinai's model of diffusion in one dimension with random local bias is studied by a real space renormalization group, which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position x(t) of a particle, the probability of it not returning to the origin, and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite but large sample. We compute the distribution of the meeting time of two particles in the same environment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward. These quantities obey multifractal scaling, characterized by generalized persistence exponents theta(g) which we compute. In the presence of a small bias, the number of returns to the origin becomes finite, characterized by a universal scaling function which we obtain. The full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained, as well as detailed analytical information about correlations between directions and times of successive jumps. The two-time distribution of the positions of a particle, x(t) and x(t') with t>t', is also computed exactly. It is found to exhibit "aging" with several time regimes characterized by different behaviors. In the unbiased case, for t-t' approximately t'alpha with alpha>1, it exhibits a ln t/ln t' scaling, with a singularity at coinciding rescaled positions x(t)=x(t'). This singularity is a novel feature, and corresponds to particles that remain in a renormalized valley. For closer times alpha<1, the two-time diffusion front exhibits a quasiequilibrium regime with a ln(t-t')/ln t' behavior which we compute. The crossover to a t/t' aging form in the presence of a small bias is also obtained analytically. Rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail. Connections with the Green function of a one-dimensional Schrödinger problem and quantum spin chains are discussed.
通过实空间重整化群研究了具有随机局部偏差的一维 Sinai 扩散模型，该模型在长时间时能给出精确结果。还研究了额外小均匀偏差力的影响。我们通过解析得到了粒子位置(x(t))的分布的标度形式、粒子不返回原点的概率以及首次通过时间的分布，这些结果适用于无限样本、存在边界的情况以及有限但大的样本。我们计算了在相同环境中两个粒子相遇时间的分布。我们还通过计算诸如返回次数和向前跳跃次数的联合分布等量，得到了热平均轨迹的详细解析描述。这些量服从多重分形标度，其特征由我们计算的广义持续指数(\theta(g))表征。在存在小偏差的情况下，返回原点的次数变为有限，其特征由我们得到的通用标度函数表征。得到了热平均轨迹连续返回时间分布的完整统计信息，以及关于连续跳跃方向和时间之间相关性的详细解析信息。还精确计算了粒子在(t>t')时位置(x(t))和(x(t'))的二次分布。发现它表现出“老化”现象，具有几个由不同行为表征的时间区域。在无偏差情况下，对于(t - t'\approx t'\alpha)且(\alpha>1)，它表现出(\ln t / \ln t')标度，在重合的重标位置(x(t)=x(t'))处有奇点。这个奇点是一个新特征，对应于留在重整化谷中的粒子。对于更接近的时间(\alpha<1)，二次扩散前沿表现出具有(\ln(t - t') / \ln t')行为的准平衡区域，我们计算了该行为。还通过解析得到了在存在小偏差时向(t / t')老化形式的转变。详细表征了对应于热包在分离阱之间间歇性分裂的罕见事件，这些事件主导了一些平均可观测量。讨论了与一维薛定谔问题的格林函数和量子自旋链的联系。
