2006/12/07 by Cesar Augusto Sangaletti Tercariol, César Augusto Sangaletti Terçariol, Rodrigo Silva Gonzalez +6
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0612188
12 pages and 3 figures
arxiv created 2006/12/07 · openalex publication_date 2006/12/07 · arxiv updated 2009/12/01 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Consider N points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory μ, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding μ steps. We have obtained analytically the probability PN(μ) = (1 - 2-μ)N - μ- 1 that all N points are visited in this open system, with N ≫ μ≫ 1. The expression for PN(μ) evaluated in the mentioned limit is valid even for small N and leads to a transition region centered at μ1 = ln N/ln 2 and with width ε= e/ln2. For μ< μ1 - ε/2, the walker gets trapped in cycles and does not fully explore the system. For μ> μ1 + ε/2 the walker explores the whole system. In both cases the walker presents diffusive behavior. Nevertheless, in the intermediate regime μ∼ μ1 ± ε/2, the walker presents anoumalous diffusion behavior. Since the intermediate region increases as ln N and its width is constant, a sharp transition is obtained for one-dimensional large systems. The walker does not need to have full memory of its trajectory to explore the whole system, it suffices to have memory of order μ1.