Generalized Einstein relation: A stochastic modeling approach

For anomalous random walkers, whose mean square displacement behaves like $〈x^2\mathrm{}\left(t\right)\mathrm{}〉\sim t^{\delta }$ $\left(\delta \ne 1\right),$ the generalized Einstein relation between anomalous diffusion and the linear response of the walkers to an external field $F$ is studied, using a stochastic modeling approach. A departure from the Einstein relation is expected for weak external fields and long times. We investigate such a departure using the Scher-Lax-Montroll model, defined within the context of the continuous time random walk, and which describes electronic transport in a disordered system with an effective exponent $\delta <1.\mathrm{}$ We then consider a collision model which for the force free case may be mapped on a Lévy walk $(\delta >1).\mathrm{}$ We investigate the response in such a model to an external driving force and derive the Einstein relation for it both for equilibrium and ordinary renewal processes. We discuss the time scales at which a departure from the Einstein relation is expected.