Variational Principles in Conformation Dynamics - FU Berlin, FB MI

III.3. Numerical considerations 25We find:D∆µ(x) = k BT mγZi= k BTmγZ∂ ∂V (x)−β exp (−βV (x))∂x i ∂x i−β∆V (x)exp(−βV (x)) + i= − 1 [∆V (x)µ(x)+∇V (x) ·∇µ(x)]mγ (III.11)= − 1 ∇· (∇V (x)µ(x)) .mγ (III.12)(III.9) 2 ∂V (x)β 2 exp (−βV (x))∂x i(III.10)We note that µ has to be well-defined for this result to make sense, i.e. the integral definingZ has to exist. This mainly requires that V (x) →∞fast enough as |x| →∞. If thestate space is dynamically connected, which we will always assume, this can also be shownto be sufficient for the process to be ergodic and meet the requirements from chapter II,[Sarich, Noé, Schütte, 2010, ch. 2.2]. Therefore,wehavejustshownthatµ is the stationarydistribution of the diffusion process, and we can now apply the foregoing theory.III.3. Numerical considerationsKnowing the invariant distribution of a diffusion process helps us a lot, because our computationalresult for the first eigenfunction can be compared to the function exp(−β∇V (x)),and should be almost equal to it up to a pre-factor. However, little more can be computedin advance, except for a few simple systems. In particular, the transition kernel p(x, y; τ) isunknown to us. But in order to apply the variational methods, we need to perform a numericalsimulation of the process, which means that if the current simulation step is x ∈ Ω, weneed to draw the next step with time lag ∆t from the distribution p(x, · ;∆t). Wethereforeapproximate this distribution by a very common procedure, the so-called Euler-Maruyamamethod. It consists of combining a deterministic explicit Euler step with a normally distributedrandom step. If the current simulation position is x, thedeterministicEulerstepisx → x − ∆t∇V (x).(III.13)This approximates the position that the process would attain if it was unperturbed. Butsince it is subject to normally distributed noise which, after time ∆t, hasvariance2D∆t,we add a normally distributed random number with standard deviation √ 2D∆t. Altogether,