We are interested in the study of the motion of tracer particles in random environments, realised either by random fields or by interacting particle systems with stochastic dynamics. When a tracer particle, initially performing a diffusive motion, is submitted to a uniform external field it starts drifting and reaches an asymptotic velocity. The Einstein Relation suggests that in the limit as the external field vanishes the velocity becomes proportional to the field, the constant of proportionality being the self-diffusivity of the tracer. Such a relation is believed to be generally valid for time reversible systems.
Until recently, rigorous proofs existed only for very few models and were in general ad-hoc. During the last two years however, tools are being developed that aim towards a unifying approach. We propose to study some open problems related to specific models (simple exclusion processes, divergence-free random fields) and work towards a general proof of the Einstein Relation with only mild assumptions on the system other than time reversibility.
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