Variational Principles in Conformation Dynamics - FU Berlin, FB MI

II.The variational principleII.1. The transfer operatorWe consider a time-dependent stochastic process X t , t ≥ 0, onausuallyhigh-dimensionaland continuous state space Ω. LettheprocesssatisfytheMarkov property, thatis,forallx 0 ,x 1 ,...,x n and y ∈ Ω, t 0 >t 1 >...>t n ≥ 0 and τ>0, wehaveP (X t0 +τ = y|X t0 = x 0 ,X t1 = x 1 ,...,X tn = x n )=P (X t0 +τ = y|X t0 = x 0 ) .(II.1)In other words, the future behaviour of the process only depends on the current state andnot on the past. A more detailed and more precise formulation of the Markov property canbe found in [Behrends, 2011, ch. 2]. Letusconsiderprocesseswhicharetime-homogeneous,meaning that the conditional probability P(X t0 +τ = y|X t0 = x) in the above equation doesnot depend on t 0 ,butonlyonthetimedifferenceτ and on the positions x, y. Inthiscase,the process allows us to define a function p:p(x, y; τ) :=P(X t+τ = y|X t = x) =P(X τ = y|X 0 = x).(II.2)We shall call this function the transition kernel. Itenablesustounderstandtheevolutionofprobability densities in time. Suppose that, at time t =0,thesystemisdistributedaccordingto a distribution ρ 0 .Theprobabilitythat,aftertimeτ>0, thesystemtransitionedfromapoint x to another point y, isgivenbytheproductoftheprobabilityofbeingatx at timet =0and the conditional probability to go from x to y in time τ. Consequently,thedensityρ τ belonging to time τ, evaluatedaty, isobtainedfromintegratingthisproductoverallpossible states x:ρ τ (y) = dx p(x, y; τ)ρ 0 (x).(II.3)ΩFurthermore, we assume the process to be sufficiently ergodic, whichmeansinessencethatthe system cannot be decomposed into two dynamically independent components, and everystate of the system will be visited infinitely often over an infinite run of the process. In thiscase, there is a unique probability distribution µ :Ω→ R, whichisinvariantintime,i.e.dx p(x, y; τ)µ(x) =µ(y),(II.4)Ω