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Published online by Cambridge University Press: 09 June 2025
This chapter deals with advanced topics for a multivariate Langevin and Fokker–Planck dynamics. For systems with multiplicative noise it is shown that neither the drift term in the Langevin equation nor the discretization parameter can be determined uniquely. If one of the two is fixed, the other one is determined. In contrast, the Fokker–Planck equation, which contains the physically observable distribution is unique. Experimental data for a particle near a wall illustrate the relevance of space-dependent friction. Martingales are introduced for a Langevin dynamics with a nonlinear expression of entropy production as a prominent example that with Doob’s optimal stopping theorem leads to universal results of its statistics. Finally, underdamped Langevin dynamics is described by the Klein–Kramers equation, for which entropy production is determined by the irreversible currents. A multi-time-scale analysis recovers the Smoluchowski equation in the overdamped limit even in the presence of an inhomogeneous temperature for which an anomalous contribution to entropy production is found.
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