Variational Principles in Conformation Dynamics - FU Berlin, FB MI

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28 III. Diffusion processes and application in one dimension0.450.4Estimated φ1Exact φ10.250.2Estimated φ2Exact φ20.350.150.30.1φ1(x)0.250.2φ2(x)0.050−0.050.15−0.10.1−0.150.05−0.20−4 −3 −2 −1 0 1 2 3 4x−0.25−4 −3 −2 −1 0 1 2 3 4xSecond eigenvalue λ210.980.960.940.920.90.880.860.840.82Estimated second eigenvalueExact second eigenvalueImplied time scale t210.990.980.970.960.950.940.930.920.910.80 50 100 150 200Time lag τ0.90 50 100 150 200Time lag τFigure III.3.: Results obtained from applying the Roothan-Hall method to diffusion in a harmonicpotential. A simulation of 20 million time steps of length ∆t =10 −3was used with thirteen Gaussian test functions centred at −3, −2, −1, −0.8,−0.4, −0.2, 0, 0.2, 0.4, 0.8, 1, 2, 3. The variances were set to 0.5 for allcentres between −0.8 and 0.8, andto1 for the remaining functions. (a) Firsteigenfunction φ 1 .(b)Secondeigenfunctionφ 2 .(c)Secondeigenvalueλ 2 .(d)Estimated second implied time scale t 2 .Exactvalueist 2 =1.
III.5. Two-well potential 29molecular simulations, a harmonic potential is often used to model bonds between atoms,as well as bond angles. The harmonic potential will therefore frequently show up in applications.III.5. Two-well potentialAnother important example is a two-well potential. In general, this is some potential functiondisplaying two minimum positions which are separated by a significant energy barrier, andwhich rises to infinity both to the left and to the right of the minima. For our calculations,we choose V (x) =k(x 4 − 2x 2 +1) for some k>0, whichhastwominimaatx = −1and x =1.TheenergyfunctionandtheinvariantdistributionaredisplayedinFigure III.4aand Figure III.4b. Thetwo-wellpotentialisagoodmodelforcertainmolecularinteractionswhere a certain degree of freedom favours two characteristic positions. The regions aroundthe potential minima correspond to metastable states, we therefore expect one dominantslow process apart from the stationary process. We apply the Roothan-Hall method usingthirteen Gaussian functions centred around the two minima, with uniform variance equalto 0.5. In order to evaluate the results, we also computed the eigenvalues of an MSMtransition matrix. Here, we used a fine discretization of the state space into 100 sets, mostof them situated close to the potential minima. A comparison of the results can be seenin Figure III.4. Most importantly, we see that the implied time scales are estimated verywell. Both methods converge quickly to about the same value. The stationary distributionis very well approximated, and we obtain a convincing result for the second eigenfunction,clearly displaying a characteristic sign change between the two metastable regions. Thecomputational effort required by the Roothan-Hall method is much smaller, since we onlyneed to compute an eleven by eleven matrix instead of a 100 × 100 matrix.
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
Page 16 and 17: 12 II. The variational principlesim
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
Page 31: III.4. Diffusion in a quadratic pot
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 and 52: A.1. Diffusion in a quadratic poten
Page 53: A.1. Diffusion in a quadratic poten
Page 57 and 58: Bibliography[Behrends, 2011] Ehrhar

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