equation for the i - th particle is 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: 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: 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, ˙ x$_{i}$ ( t ) = v − 1 b ∂V$_{ext}$ ( x$_{i}$ ) ∂x$_{i}$ − 1 b ∂ ∂x$_{i}$ [ V$_{int}$ ( x$_{i,i}$$_{−}$$_{1}$ ) + V$_{int}$ ( x$_{i,i}$$_{+1}$ )] + m b Γ$_{i}$( t ) . (5) 〈 Γ$_{i}$ 〉 = 0 , (3) 〈 Γ$_{i}$( t )Γ$_{j}$( t $^{′}$) 〉 = 2 bk$_{B}$T m 2 δ$_{i,j}$δ ( t − t $^{′}$) , (4) b = 6 πµR, (2) m ¨ x$_{i}$ ( t ) = − b [ ˙ x$_{i}$ ( t ) − v ] − ∂V$_{ext}$ ( x$_{i}$ ) ∂x$_{i}$ − ∂ ∂x$_{i}$ [ V$_{int}$ ( x$_{i,i}$$_{−}$$_{1}$ ) + V$_{int}$ ( x$_{i,i}$$_{+1}$ )] + m Γ$_{i}$( t ) , (1) 5