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Probabilistic Interpretation of the Einstein Relation

R. Hilfer{}^{{1}} and A. Blumen{}^{2}
{}^{1}Department of Physics, University of California, Los Angeles, California 90024
{}^{2}Physikalisches Institut, Universität Bayreuth, D-8580 Bayreuth, Federal Republic of Germany

We present a probabilistic picture for the Einstein relation which holds for arbitrarily connected structures. The diffusivity is related to mean first-passage times, while the conductance is given a direct-passage probability. The fractal Einstein relation is an immediate consequence of our result. In addition, we derive a star-triangle transformation for Markov chains und calculate the exact values of the fracton (spektral) dimension for treelike structures. We point to the relevance of the probabilistic interpretation for simulation and experiment.