Problem set 1:

(a):

A particle is subject to a harmonic force with force constant f and is in contact with a heat bath at temperature T. The mobility of the particle is $\mu$ and it has position x₀, velocity v₀ at time t=0.

• Write down Kramers equation for the problem
• Find $\langle v(t)\rangle$ and $\langle x(t)\rangle$.
• find the covariance matrix with components
  
  $$\langle x(t)^2\rangle -\langle x(t)\rangle^2;\;\langle x(t)v(... ...e v(t)\rangle; \; \langle v(t)^2\rangle -\langle v(t)\rangle^2$$
  
  

(b):

An overdamped pendulum is subject to a weak constant torque $\tau$ and satisfies the Fokker-Planck equation

$$\frac{\partial P(\theta,t)}{\partial t}=\frac{1}{\gamma} \le... ...in\theta-\tau)P+kT\frac{\partial^2P}{ \partial\theta^2}\right]$$

Here $\theta$ is the angle of the pendulum with respect to the vertical,M is the mass of the pendulum, l its length and $\gamma$ a suitably choosen damping term. The torque is weak enough that the pendulum will not rotate without thermal noise. In the steady state the probability curent will not be zero. Instead the steady state is characterized by periodic boundary conditions $P_s(\theta+2\pi)=P_s(\theta)$. Find an expression for the average rate at which the pendulum rotates.

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