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Mapping between dissipative and Hamiltonian systems

Theoretical studies of nonequilibrium systems are complicated by the lack of a general framework. In this work we first show that a transformation introduced by Ao recently (J. Phys. A {\bf 37}, L25 (2004)) is related to previous works of Graham (Z. Physik B {\bf 26}, 397 (1977)) and Eyink {\it et al.} (J. Stat. Phys. {\bf 83}, 385 (1996)), which can also be viewed as the generalized application of the Helmholtz theorem in vector calculus. We then show that systems described by ordinary stochastic differential equations with white noise can be mapped to thermostated Hamiltonian systems. A steady-state of a dissipative system corresponds to the equilibrium state of the corresponding Hamiltonian system. These results provides a solid theoretical ground for corresponding studies on nonequilibrium dynamics, especially on nonequilibrium steady state. The mapping permits the application of established techniques and results for Hamiltonian systems to dissipative non-Hamiltonian systems, those for thermodynamic equilibrium states to nonequilibrium steady states. We discuss several implications of the present work.