Variational Principles in Conformation Dynamics - FU Berlin, FB MI

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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)
A.1. Diffusion in a quadratic potential 49By assumption, we have that e −θτ λ i−1 = λ i . With one more application of the recursionrelation, we arrive at the final result:dx p(x, y; τ)φ i+1 (x) =e −θτ λ i [yφ i (y) − iφ i−1 (y)](A.36)= e −θτ λ i φ i+1 (y), (A.37)which proves both the assertions about the eigenfunctions and the corresponding eigenvalues.The pre-factors are then chosen to assure that the eigenfunctions are normalized.
Page 1: Freie Universität BerlinInstitut f
Page 5: Table of contentsI. Introduction 3I
Page 8 and 9: 4 I. IntroductionFigure I.1.: An in
Page 10 and 11: 6 II. The variational principlerega
Page 12 and 13: 8 II. The variational principleIn E
Page 14 and 15: 10 II. The variational principleThe
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Page 18 and 19: 14 II. The variational principleFor
Page 20 and 21: 16 II. The variational principleSin
Page 22 and 23: 18 II. The variational principleThe
Page 25 and 26: III.Diffusion processes andapplicat
Page 27 and 28: III.2. Diffusion process 23time ∆
Page 29 and 30: III.3. Numerical considerations 25W
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Page 33 and 34: III.5. Two-well potential 29molecul
Page 35 and 36: IV.Application to moleculesIn this
Page 37 and 38: IV.1. The example system 33221.81.8
Page 39 and 40: IV.2. Simulation 35Quantity Symbol
Page 41 and 42: IV.3. Application of the Roothan-Ha
Page 47 and 48: V. Summary and outlookWe have prese
Page 49 and 50: A. AppendixA.1. Diffusion in a quad
Page 51: A.1. Diffusion in a quadratic poten
Page 57 and 58: Bibliography[Behrends, 2011] Ehrhar

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