Variational Principles in Conformation Dynamics - FU Berlin, FB MI

III.4. Diffusion in a quadratic potential 2780.470.3560.350.25V(x)4µ(x)0.230.1520.110.050−4 −3 −2 −1 0 1 2 3 4x0−4 −3 −2 −1 0 1 2 3 4xFigure III.2.: Quadratic potential V (x) =0.5x 2 and its invariant distribution.that this is one of the rare cases where all important quantities can be determined analytically.Using stochastic integrals and Ito’s formula, we derive an explicit expression for this process inLemma A.1. Usingthisexpression,wethenshowthatiftheprocessstartswithadistributionρ 0 ,theexpectationvalueattimet is given by E [ρ t ]=e −θt E [ρ 0 ],wherewehavedefinedθ := bmγ .Moreover,thevarianceevolvesaccordingtoω2 [ρ t ]=e −2θt ω 2 [ρ 0 ]+(1− e −2θt )α.Regardless of the initial distribution, the process quickly approaches the stationary Gaussiandistribution. Even the complete set of eigenfunctions can be found analytically:Lemma III.4: The propagator eigenfunctions are given by the appropriately scaled Hermitepolynomials, multiplied with the invariant measure: αφ i (x) =i−1 x(i − 1)! H i−1 √α µ(x). (III.15)The corresponding eigenvalues are λ i (τ) =e −θ(i−1)τ .Proof. We also show the proof of this Lemma in the appendix.Since there is only one potential minimum, there are no metastable states. Unless b is verysmall or mass and friction are very large, the global relaxation time −τ = τ = mγlog λ 2is(τ) θτ bshort, the process quickly equilibrates to its invariant distribution.Since all these analytic quantities are at hand, we can compare them to the results obtainedfrom the Roothan-Hall method. As shown in Figure III.3, arelativelysmallnumericaleffortleads to a good approximation of the eigenfunctions as well as the corresponding eigenvaluesand implied time scales. Though this example may be a simple one, it is still important. In