Question

32. A Brownian particle is placed in a very long narrow channel. The particle is trapped in a harmonic potential V (x) = {kx2 (where k is a constant and x is the distance from the origin) and is subject to random thermal noise. In other words, we have a one-dimensional harmonic oscillator coupled to a thermal bath. At time t = 0, the particle is as the position x (0) = 20. Assuming the inertia can be neglected (the so-called overdamped limit), the equation of motion of the particle is dr(t) -=-kx (t) + f(t), 3 dt where is the friction coefficient, f(t) is the random force with (f (t)) = 0 and (f (t)f (t')) = 2kbTC8 (t – t'). (...) is an average with respect to the distribution of the realisations of the stochastic variable f (t). For convenience, we define y = 4 and & (t) = f(C). (a) Solve the equation of motion, (b) Using the result of (a) and stochastic properties of the random force, find (x (t)). (c) Using the result of (a) and stochastic properties of the random force, find (x2 (t)). (d) Using the result obtained in (c) show that in the limit k + 0 one recovers the standard diffusive behaviour with D = 2kpl being the diffusion constant.

Answer 1

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