The rule for a subdiffusive particle in an extremely diverse environment

We consider a subdiffusive continuous time random walker in an inhomogeneous environment. Each microscopic random time is drawn from a waiting time probability density function (WT-PDF) of the form: φ(t;k)∼k(1+kt)1+β, 0<β⩽10<β⩽1. The parameter k   is a random quantity also, and is drawn from a PDF, p(k)=1−γk˜(k˜k)γ, 0⩽γ<10⩽γ<1, for a cutoff parameter k˜. We show that the effective WT-PDF, ψ(t)ψ(t), obtained by averaging φ(t;k)φ(t;k) with p(k)p(k), exhibits a transition in the rule that governs the power of ψ(t)ψ(t). ψ(t)ψ(t) obeys, ψ(t)∼1t1+μ, and μ   is given by two different formula. When, 1−γ>β1−γ>β, μ=βμ=β, but otherwise, μ=1−γμ=1−γ. The rule for the scaling of ψ(t)ψ(t) reflects the competition between two different mechanisms for subdiffusion: subdiffusion due to the heavily tailed φ(t;k)φ(t;k) for individual jumps, and subdiffusion due to the collective effect of an environment made of many slow local regions. These two different mechanisms for subdiffusion are not additive, and compete each other. The reported transition is dimension independent, and disappears when the power β   is also distributed, in the range, 0<β⩽10<β⩽1. Simulations exemplified the transition, and implications are discussed.