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Published online by Cambridge University Press: 09 June 2025
The overdamped Langevin equation for a particle in a potential and, possibly, subject to a nonconservative force is introduced. The corresponding Fokker–Planck equation, the Smoluchowski equation, is derived. In a time-independent potential, any initial distribution finally approaches the equilibrium one. For a constant external force and periodic boundary condition like the motion along a ring, a nonequilibrium steady state is established. As an application, the Kramers escape from a meta-stable well can be discussed. The mean local velocity and the path integral representation are introduced. Thermodynamic quantities like work, heat, and entropy production are identified along individual trajectories and their ensemble averages are determined. Their distributions are shown to obey detailed fluctuation relations. A master integral fluctuation relation can be specialized to yield inter alia the Jarzynski relation, the integral fluctuation relation for entropy production, and the Hatano–Sasa relation.
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