Variational Principles in Conformation Dynamics - FU Berlin, FB MI

III.Diffusion processes andapplication in one dimensionWe are now prepared to apply the variational methods we have just derived to a number ofexample systems. Our guiding example will be a diffusion process. This is a system where anumber of classical particles move under the influence of a deterministic force field, generatedby some potential energy function. If there was no further influence, this system could bemodelled by classical equations of motion. However, the particles are also subject to randomfluctuations, which interfere with the deterministic motion. The fundamental concepts ofsuch a system will be introduced below. As a result, the system trajectory can no longerbe uniquely predicted. Instead, we have a stochastic process to which our theoretical resultscan be applied.III.1. Brownian motion and stochastic differentialequationsThe fundamental model for the description of random fluctuations is Brownian motion orthe Wiener process. In1826-27,R.Brownstudiedtheirregularbehaviourofpollengrainsinwater. He and many others observed that the pathways taken by such a particle were highlyirregular. Moreover, whereas the average distance from the starting point seemed to vanish,the mean fluctuation from the average seemed to be growing linearly. These observationsmotivated the following mathematical model, [Evans, ch.3].Definition III.1: Aone-dimensionalstochasticprocessW t on a probability space Ω, definedfor t ≥ 0, iscalledaBrownianmotionoraWienerprocessifitsatisfies:(a) W 0 =0a.e.