Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

3 Theory3.9.1 Langevin EquationAs pointed out in the last section, it was Paul Langevin the french physicist who in the year 1906wrote down a differential equation which describes the behavior of a single Brownian particle,suspended in water. With Brownian particle it is meant that the particle is big (massive) in comparisonto the molecules of water, in which it is suspended. The following motivation of theLangevin equation is based on [34, chapter 8].A physical and mathematical elegant way to represent the Langevin equation in the scope of a firstorder system of stochastic differential equations is delivered in the appendix (see section A.6 onpage 108).Let m be the mass of the accordant particle, and let us assume that the particle has a velocity v(t),depending on time t. Thus this velocity will change in time, according to the collisions with themolecules of water (see figure 3.7 (a)). Obviously these collisions are random in their occurrenceand their overall effect is of dual character, namely an average retarding force on the one hand andon the other hand a stochastic force f (t), fluctuating around this mean retarding force, as we seein figure 3.7 (b). Modeling of the retarding force via a friction term f f ric (t) = −mγv(t) deliverstogether with the stochastic force f (t) the Langevin equation˙v(t) = −γv(t) + 1 f (t) (3.36)mwhich is a special type of a more general stochastic differential equation (SDE).(a)(b)Figure 3.7: (a) Brownian Movement and (b) Stochastic Force f (t) plotted versus time t. (from[34, p. 418]).The stochastic force is assumed to be a random variable with the following properties:〈 f (t)〉 = 0, (3.37)〈 f (t) f (t ′ )〉 = φ(τ), (3.38)32