Variational Principles in Conformation Dynamics - FU Berlin, FB MI

48 A. AppendixProof of Lemma III.4. For convenience, we restrict the proof to the case that α =1and leave out all pre-factors. The case i =1was already treated when we discussed thestationary distribution, so let us check the statement for i =2. By partial integration wefind that:dx p(x, y; τ)x exp − x2= −2== e−θτν− e−2θτνdx p(x, y; τ) d dx exp − x22dx e −θτ (y − e−θτ x)νyp(x, y; τ)expdx p(x, y; τ)exp− x22dx p(x, y; τ)x exp− x22 − x22(A.24)(A.25)(A.26). (A.27)Upon multiplication by ν, weseethatthelasttermcancels,andweareleftwith: dx p(x, y; τ)x exp− x2= e −θτ y dx p(x, y; τ)exp− x2= e −θτ y exp22− y2.2(A.28)Having established the first two cases, we can prove the assertion for general i by induction.We will need the recursion relation for Hermite polynomials, which readsH i+1 (x) =xH i (x) − iH i−1 (x),(A.29)and also holds for the eigenfunctions φ i .Furthermore,similarlytothefirstcase,wecanusethat φ i (x) =(−1) i di exp(− x2 ),whichisonewaytodefinetheHermitefunctions. Thedx i 2course of the proof is then quite the same as for i =2: dx p(x, y; τ)φ i+1 (x) =(−1) i+1 dx p(x, y; τ) di+1dx exp − x2(A.30)i+1 2=(−1) i dx e −θτ (y − e−θτ x)p(x, y; τ) diνdx exp − x2(A.31)i 2= e−θτνy dx p(x, y, τ)φ i (x) − e−2θτdx p(x, y; τ)xφ i (x)ν(A.32)= e−θτνλ iyφ i (y) − e−2θτνdx p(x, y; τ)[φ i+1 (x)+iφ i−1 (x)] .(A.33)Again, multiplying the equation by ν eliminates the terms containing φ i+1 and the factore −2θτ .Weareleftwith:dx p(x, y; τ)φ i+1 (x) =e −θτ λ i yφ i (y) − ie −2θτ dxp(x, y; τ)φ i−1 (x) (A.34)= e −θτ λ i yφ i (y) − ie −2θτ λ i−1 φ i−1 (y). (A.35)