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The Einstein relation, relating the mobility of a Browninan particle in response to a field to the diffusion constant in zero field, is the simplest realisation of a fluctutation-dissipation relation. Here we consider a general formulation of the Einstein relation for the steady states of stochastic models with a finite state space. We show how an Einstein relation always holds if the unperturbed steady state satisfies detailed balance. More generally, we consider nonequilibrium steady states where detailed balance does not hold and show how a generalisation of the Einstein relation may be derived in certain cases. In particular, for the asymmetric simple exclusion process and a driven diffusive dimer model, the external perturbation creates and annihilates particles thus breaking the particle conservation of the unperturbed model. |
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