Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

A.7 The Fluctuation-Dissipation-TheoremThis is given byDefinition A.27 (Langevin equation as a first Order System).with W denoting the Wiener process.˙q k = 1 m p k, (A.10)ṗ k = −∇U({q k }) − γ m P k + σẆ(t), (A.11)With p k = mv k , derivation of (A.10) one more time with respect to time t and insertion of (A.11)in it leads to¨q k (t) = 1 m ṗk(t),⇐⇒ ¨q k (t) = 1 [−∇U ({q k }) − γ ]mm p k(t) + σẆ(t) ,⇐⇒ ¨q k (t) = − 1 m ∇U ({q k}) − γ m 2 p k(t) + σ mẆ(t),⇐⇒ ˙v k (t) = − 1 m ∇U ({q k}) − γ m 2 mv k(t) + σ mẆ(t),⇐⇒ m ˙v k (t) = −∇U ({q k }) − γ m v k(t) + σẆ(t).This is finally the classical general Langevin equation we already know.A.7 The Fluctuation-Dissipation-TheoremThe Fluctuation-Dissipation-Theorem (FDT) is maybe one of the most universal relations in mathematicalphysics, relating both in classical statistical physics and quantum theory the reversiblefluctuations with the irreversible dissipation of energy into heat within a system.Historically, Einstein and Smoluchowski first derived a FDT from their considerations on thephenomenon of Brownian motion. The Einstein-Smoluchowski-Relation correlates the diffusioncoefficient D (depending on the fluctuations) of a Brownian particle, suspended in water, with itsmobility µ (depending on the dissipation)D = µk B T.(A.12)A general FDT was formulated first in the year 1928 by Harry Nyquist as an explanation for the”Johnson–Nyquist noise” [60] which is the electronic noise, generated by the thermal agitation ofthe charge carriers (usually the electrons) inside an electrical conductor at equilibrium, occurring109