Dentz Marco, Le Borgne Tanguy, Lester Daniel R, de Barros Felipe P J
Spanish National Research Council, IDAEA, CSIC, 08034 Barcelona, Spain.
Geosciences Rennes, UMR No. 6118, Université de Rennes 1, CNRS, Rennes, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Sep;92(3):032128. doi: 10.1103/PhysRevE.92.032128. Epub 2015 Sep 21.
We study the scaling behavior of particle densities for Lévy walks whose transition length r is coupled with the transition time t as |r|∝t^{α} with an exponent α>0. The transition-time distribution behaves as ψ(t)∝t^{-1-β} with β>0. For 1<β<2α and α≥1, particle displacements are characterized by a finite transition time and confinement to |r|<t^{α} while the marginal distribution of transition lengths is heavy tailed. These characteristics give rise to the existence of two scaling forms for the particle density, one that is valid at particle displacements |r|≪t^{α} and one at |r|≲t^{α}. As a consequence, the Lévy walk displays strong anomalous diffusion in the sense that the average absolute moments 〈|r|^{q}〉 scale as t^{qν(q)} with ν(q) piecewise linear above and below a critical value q_{c}. We derive explicit expressions for the scaling forms of the particle densities and determine the scaling of the average absolute moments. We find that 〈|r|^{q}〉∝t^{qα/β} for q<q_{c}=β/α and 〈|r|^{q}〉∝t^{1+αq-β} for q>q_{c}. These results give insight into the possible origins of strong anomalous diffusion and anomalous behaviors in disordered systems in general.
我们研究了 Lévy 行走中粒子密度的标度行为，其跃迁长度 r 与跃迁时间 t 耦合，满足|r|∝t^α，其中指数α>0。跃迁时间分布表现为ψ(t)∝t^(-1-β)，β>0。对于1<β<2α且α≥1，粒子位移的特征是具有有限的跃迁时间且局限于|r|<t^α，而跃迁长度的边际分布是重尾的。这些特征导致粒子密度存在两种标度形式，一种在粒子位移|r|≪t^α时有效，另一种在|r|≲t^α时有效。因此，Lévy 行走表现出强烈的反常扩散，即平均绝对矩〈|r|^q〉的标度为t^(qν(q))，其中ν(q)在临界值q_c之上和之下是分段线性的。我们推导了粒子密度标度形式的显式表达式，并确定了平均绝对矩的标度。我们发现，对于q<q_c =β/α，〈|r|^q〉∝t^(qα/β)，对于q>q_c，〈|r|^q〉∝t^(1 +αq -β)。这些结果深入了解了一般无序系统中强反常扩散和反常行为的可能起源。
