Existence and construction of dynamical potential in nonequilibrium processes without detailed balance

The existence of a dynamical potential with both local and global meanings in general nonequilibrium processes has been controversial. Following an earlier heuristic argument in a letter by one of us, in the present paper we show rigorously its existence for a generic class of situations in physical and biological sciences. The local dynamical meaning of this potential function is demonstrated via a special stochastic differential equation and its global steadystate meaning via a novel and explicit form of the Fokker - Planck equation. We also give a procedure to obtain the special stochastic differential equation for any given Fokker - Planck equation. No detailed balance condition is required in our demonstration. For the first time we obtain here a formula to describe the noise- induced shift in drift force compared to the steady- state distribution, a phenomenon extensively observed in numerical studies.