Variational Principles in Conformation Dynamics - FU Berlin, FB MI

I. IntroductionStochastic dynamical systems play an important role in many branches of mathematicsand applications. Loosely speaking, such a system is a superposition of a deterministicdynamical system with random fluctuations, often called “noise“. A typical example is thesimulation of molecules (molecular dynamics, MD). In principle, one could model a moleculeas a classical physical system by assigning to each atom its three spatial and three momentumcoordinates and propagate these coordinates using some dynamical model. Since the moleculecan typically exchange heat with its surroundings, energy is not conserved, which means thatclassical Hamiltonian dynamics is not a suitable dynamical model. Instead, the exchange ofheat is usually modelled by some random influence. As an example, one can use Smoluchowskidynamics: If we denote the number of atoms by N and the 3N-dimensional position vectorby x, thetimeevolutionisgivenbythestochasticdifferentialequationẋ(t) = 1mγ ∇V (x(t)) + √ 2DdW t .(I.1)Here, V is the potential energy function of the system, D is the diffusion constant, γ denotesfriction and W t is 3N-dimensional Brownian motion. We will come back to this model inmore detail at a later point, but note that this equation defines a stochastic process insteadof a deterministic dynamical system. Instead of asking for a deterministic position in the3N-dimensional state space, we ask for the probability to find the system in a certain regionof the state space.Molecular systems very often display so called conformations, ormetastablestates. Thismeans that while the system still oscillates and fluctuates, the overall geometry remains thesame for long times, [Schütte, Huisinga, Deuflhard, Fischer, 1999]. Only occasionally, transitionsfrom one conformation to another can be observed. An example of this phenomenonis shown in Figure I.1. A quantity of interest would be the average waiting time until suchatransitionoccurs,becauseitwouldhelptounderstandtheoverallbehaviourofsuchamolecule.It has also been shown by [Schütte, Huisinga, Deuflhard, Fischer, 1999] thatthetimeevolutionof probability densities described above can be computed by the action of a linearintegral operator, called the propagator. Moreover, the spectral properties of this operator