Variational Principles in Conformation Dynamics - FU Berlin, FB MI

6 II. The variational principleregardless of τ, [Sarich, Noé, Schütte, 2010, ch.2.2]Thefunctionµ is called the invariantmeasure, theinvariant density or the stationary distribution of the process.Also, we assume the transition kernel to satisfy the detailed balance condition, i.e.µ(x)p(x, y; τ) =µ(y)p(y, x; τ) ∀x, y ∈ Ω. (II.5)Detailed balance is an important assumption that is justified in many applications fromphysics. In a system not satisfying Equation II.5, therewouldbeapathwaywhereonedirectionis preferred compared to the other. This would allow the process to produce work,that is, to convert thermal energy into work. This is impossible in physical systems due tothe second law of thermodynamics, [Prinz et al, 2011, sec. 2.A].We see that Equation II.3 defines the action of a lag time dependent integral operator P(τ)on suitable probability densities ρ. Let us call it the propagator of the system, which hasthe following important property:Lemma II.1 (Chapman-Kolmogorov equation): For any τ 1 ,τ 2 > 0, wehaveP(τ 1 + τ 2 )=P(τ 1 )P(τ 2 ).(II.6)Proof. First, we observe that the Markov property and time-homogeneity imply for x, z ∈Ω,1p(x, z; τ 1 + τ 2 )=P(X τ1 +τ 2= z|X 0 = x) =P(X 0 = x) P(X τ 1 +τ 2= z,X 0 = x) (II.7)1=dy P(X τ1 +τP(X 0 = x)2= z|X τ2 = y)P(X τ2 = y, X 0 = x) (II.8)Ω= dy p(y, z; τ 1 ) P(X τ 2= y, X 0 = x)= dy p(y, z; τ 1 )p(x, y; τ 2 ).ΩP(X 0 = x)Ω(II.9)Using this, we show that for a function ρ,P(τ 1 + τ 2 )ρ(z) = dx p(x, z; τ 1 + τ 2 )ρ(x) ==ΩΩΩdxdy p(y, z; τ 1 )P(τ 2 )ρ(y) =P(τ 1 )P(τ 2 )ρ(z).dy p(y, z; τ 1 )p(x, y; τ 2 )ρ(x)(II.10)(II.11)