content_image/361240.jpg content_image/361239.jpg content_image/361238.jpg content_image/361237.jpg content_image/361236.jpg
equation for the i - th particle is 

<img src='content_image/361240.jpg'>

where x$_{i}$ , ˙ x$_{i}$ and ¨ x$_{i}$ are, respectively, the position, velocity and acceleration of par- ticle i , x$_{i,j}$ ≡ x$_{j}$ − x$_{i}$ , and v is the terminal velocity of a particle in the moving fluid, in the absence of the periodic external potential V$_{ext}$ . V$_{int}$ is the (strongly repulsive) potential between neighboring particles. (We assume that the range of V$_{int}$ to be short enough that only neighboring particles interact.) The first term on the right side of Eq. (1) represents damping of the particle velocity relative to the fluid. For a sphere of radius R , Stokes’ Law gives: 

<img src='content_image/361239.jpg'>

where µ is the fluid viscosity. Here it is important to stress that the fluid is three dimensional although we treat the particle motion as one dimensional. The final term is a random noise with the following properties: 

<img src='content_image/361238.jpg'>

<img src='content_image/361237.jpg'>

where k$_{B}$ = 1 . 3806504 × 10 − $^{23}$J/K is Boltzmann’s constant, T is temperature, and m is the particle mass. It is convenient to ignore the inertial term m ¨ x$_{i}$ ( t ) in Eq. (1), since the observational times of interest (microseconds or greater) are much larger than the relaxation time of the velocity, m/b ∼ 10 − $^{8}$s, for our choice of parameters. Then the velocity of particle i follows, 

<img src='content_image/361236.jpg'>

