Brownian dynamics and stochastic differential equations SDEs

Brownian dynamics and stochastic differential equations (SDEs) 

We next consider an important application of probability theory to physical science, the theory of Brownian motion, and introduce the subject of stochastic calculus. Let us consider the jc-direction motion of a small spherical particle immersed in a Newtonian fluid. As observed by the botanist Robert Brown in the early 1800s, the motion of the particle is very irregular, and apparently random. Let Vx t) be the x-direction velocity as a function of time. For a particle of mass m and radius R in a fluid of viscosity /x, the equation of motion is 

This function should have the property Cvf—t) = Cv (t), and at r = 0 should agree with the average (V ) predicted by the Maxwell velocity distribution, 

since the velocities measured at very different times should be uncorrelated, we expect hm oo C(0 = 0. In general, we expect this correlation function to take the approximate form of an exponential decay, 

How is tv, related to the properties of the particle and fluid Let us say that the particle is moving through the fluid at some velocity, and then at time r = 0, we turn off the random 

---