Fluctuation-dissipation relations for steady-state systems

The fluctuation-dissipation (FD) theorem is a fundamental result for systems near thermodynamic equilibrium. It states that the nonequilibrium relaxation dynamics is related to the spontaneous fluctuation at equilibrium. Recently we have shown that the dynamics of a dissipative system described by stochastic differential equations can be mapped to that of a thermostated Hamiltonian system, with a nonequilibrium steady state of the former corresponding to the equilibrium state of the latter. In present manuscript, the corresponding FD theorem is derived in the way parallel to the procedure for deriving the near equilibrium FD theorem, based on this mapping. The analytical results in the present approach are in good agreement with numerical results. We find some previous results are special cases of the current relations. We also suggest further studies exploiting the analogy between a general dissipative system appearing in other science branches and a Hamiltonian system.