Generalized Einstein relation: A stochastic modeling approach

For anomalous random walkers, whose mean square displacement behaves like [x(2)(t)]similar to t(delta) (delta not equal 1), 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. We then consider a collision model which for the force free case may be mapped on a Levy walk(delta>1). 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.