Question

part b and c

In class we derived a Fokker-Planck equation for the velocity distribution P(et) starting from the assumption of small random changes in velocity at each time step f.(t) where f(t) is chosen from a distribution WU: ). Einstein's original approach to Brownian motion had a different starting point, focusing on position differences at each time step x(t + Δt)-x(t) + E(t) where £(t) is a random displacement chosen from some distribution W(E). Underlying this ap- proach is the assumption that each successive random displacement is independent, Le. 2(t + Δ)-x(t) and ξ(t + Δ)-. z(t + 2Δ)-2(t + Δ) are unco this can only be true if the corresponding velocities at r(t) and r(t + Δ) are uncorrelated Einstein's theory did not explicitly treat velocities, but we know from the results in class that (m(t + Δt)ur(t)) (kBT/M) exp(,"7Δt/M). Hence Einstein's approach is a good approxi mation assuming Δ), M/7. It will describe dynamics at coarser time scales than the Fokker- Planck equation from lecture. rrelated. We know that Derive a Fokker-Planck equation for the position distribution P(a,t) using Einstein's ap- proach. Follow the same reasoning as shown in class, but with W(E) instead of W(i). For the first and second moments (E) and (2) of the distribution w(6), use the class results for (0(t +an-2 (t))) and ((x(t + Δ)-(t)) ), taking the limit Δ> M/r. In the end you should derive the following dynamical equation ar? where D -kyT/y. This is the Fokker-Planck equation for the position of a Brownian particle in the absence of a potential (otherwise known as the diffusion equation). b) Solve Eq. (3) for the distribution p(z, t; zo) with initial condition p(r,0;ro) = δ(z-zo) Hint: Assume P(r, t; zo) is a Gaussian with arbitrary time-dependent mean and variance, and determine what those two quantities have to be in order for Eq. (3) to be satisfied. c) Using your solution from part b, calculate the mean squared displacement (MSD) Ar2() (x(t) - x(0)?). You can define a rough measure of "mean velocity" over the time interval between 0 and t as: olt)Armt)/t. In the limit t 0, compare the result of Einstein's theory for o(t) to the result from the theory presented in class. Show that Einstein's result gives nonsense as t 0, while the class theory (which is applicable to smaller time scales) gives the expected Maxwell-Boltzmann value. This underscores the fact that Einstein's approach should only be used when t » M/γ

Answer 1

Given that: - b) P (x, 0: x_0) = 6 (x - x_0) To solve equation, we can do the Fourier transform So, p (x, t) = integral^infinity_-infinity P^bar (R^bar, t) e^iR^bar x^bar d e with P^bar (R^bar, t) = 1/(2 pi)^3 mintegral^infinity__infinity P (x^bar, t) e^-iR^bar x^bar dx So, for 1 dimensional = 1/(2 pi)^3 integral^infinity_-infinity d t^bar e^-AR^-2 t e^-iR^bar x^bar = P (x, t) P (y, t), P (2, t) = 1/(4 pi D t)^3/2 e^-x^2/4 delta t \r\n From Lange in equation x (t + delta t) - x (t) = sigma (t) = delta x = F (x)/r delta t integral^t + delta t_t dt' sigma (t) = F (x)/v delta t + N^bar (t) = Here N (t) is a Gaussian distributed no with zero mean & variance 2 delta t So, nu^bar (t) = delta x/t = F (x)/nu So, Now for limit delta_1 = limn_delta t rightarrow 0