Berlin 2018 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 60: Anomalous Diffusion (joint session DY/BP)

DY 60.11: Talk

Thursday, March 15, 2018, 12:45–13:00, BH-N 334

Self-trapping self-repelling random walks — •Peter Grassberger — Juelich Research Center, Juelich, Germany

The model studied in this talk is a seemingly minor modification of the “true self-avoiding walk" (TSAW) model of Amit, Parisi, and Peliti in two dimensions. The walks in it are self-repelling up to a characteristic time T^* (which depends on various parameters), but spontaneously (i.e., without changing any control parameter) become self-trapping after that. For free walks, T^* is astronomically large, but on finite lattices the transition is easily observable. In the self-trapped regime, walks are subdiffusive and intermittent, spending longer and longer times in small areas until they escape and move rapidly to a new area. In spite of this, these walks are extremely efficient in covering finite lattices, as measured by average cover times.

Basically, the phenomenon is due to a delicate balance between two opposite effects of landscape gradients and roughnesses on random walks: While a gradient enhances diffusion, roughness slows it down. Initially, the walker creates a hill by depositing debris, and the hill gradient wins. But the hill surface is also rough. When roughness becomes too large, the walk becomes subdiffusive which increases further the roughness, leading finally to catastrophic trapping.

The phenomenon seems to be described by scaling laws, and some exponent and critical parameter values seem to be simple rationals. In addition, the deposited debris forms (on square lattices and for some parameter values) non-trivial patterns that suddenly re-arrange at sharp times.