Sub-diffusion by random walks in quenched random environments

Hugo A. Araújo

doi:10.1016/j.physa.2026.131771

Sub-diffusion by random walks in quenched random environments

2026-06-18

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DOI：10.1016/j.physa.2026.131771

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We study random walks in quenched random environments and, by deploying the Fock space approach, we derive a heuristic statistical argument to state the condition for the emerging of sub-diffusion. In particular, let r∈[0,1] be the probability for a walker to remain in the actual position and f(r) its density function, then the process is sub-diffusive when limr→1f(r)=+∞ . Actually, we state a criterion for the second derivative in time of the mean squared displacement to be negative and this result is indeed general. In fact, within the terminology of trap models, it holds for any distribution of the energy barriers ρ(E)∝−φb(E)dlnφ/dE , with b∈(0,1) , where φ(E) is the site-dependent transition rate. The classical trap model is recovered when φ(E)=e−E/T , where T is the temperature, but the same condition holds, for example, also when φ(E)∝E−α such that ρ(E)∝E−αb−1 for any α∈(0,1) , namely when a scale for the energy barriers does not exist.

Sub-diffusion by random walks in quenched random environments

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