二维自仿射随机势中的随机游动：小外力下反常扩散相的性质
Random walk in a two-dimensional self-affine random potential : properties of the anomalous diffusion phase at small external force
摘要：
本文研究了粒子在二维Hurst指数为$H=1/2$的自仿射随机势中的随机行走问题，其中存在一个外力$H=1/2$.我们提出了数值结果的第一次通过时间的统计，满足封闭落后的主方程。我们发现在外力作用下$0<F<F_c$的有限区域内存在一个零速度相，动力学遵循反常扩散定律$ x(t) \sim ξ(F) \ t^{μ(F)} $ .反常指数$0<μ(F)<1$和关联长度$ξ(F)$随$F$连续变化。在消失力$F \to 0$的极限下，我们测量了下列幂律：反常指数以$μ(F) \propto F^a$与$a \simeq 0.6$（而不是$d=1$维中的$a=1$）的形式消失，关联长度以$ξ(F) \propto F^{-ν}$与$ν\simeq 1.29$（而不是$d=1$维中的$ν=2$）的形式发散。因此，我们的主要结论是，动力学重整化到一个有效的有向陷阱模型，其中的陷阱的特点是由一个典型的长度$ξ(F)$沿力的方向，并通过一个典型的障碍$1/μ(F)$。这些陷阱的线性尺寸和深度比尺寸$d=1$“更小”的事实意味着粒子使用横向方向来找到较低的势垒。
英文摘要：
We consider the random walk of a particle in a two-dimensional self-affine random potential of Hurst exponent $H=1/2$ in the presence of an external force $F$. We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force $0<F<F_c$, where the dynamics follows the anomalous diffusion law $ x(t) \sim ξ(F) \ t^{μ(F)} $. The anomalous exponent $0<μ(F)<1$ and the correlation length $ξ(F)$ vary continuously with $F$. In the limit of vanishing force $F \to 0$, we measure the following power-laws : the anomalous exponent vanishes as $μ(F) \propto F^a$ with $a \simeq 0.6$ (instead of $a=1$ in dimension $d=1$), and the correlation length diverges as $ξ(F) \propto F^{-ν}$ with $ν\simeq 1.29$ (instead of $ν=2$ in dimension $d=1$). Our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model, where the traps are characterized by a typical length $ξ(F)$ along the direction of the force, and by a typical barrier $1/μ(F)$. The fact that these traps are 'smaller' in linear size and in depth than in dimension $d=1$, means that the particle uses the transverse direction to find lower barriers.
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