Each net crossing of a periodic cell advances the unwrapped particle by $2\pi$. The stationary mean velocity is therefore
$$
\boxed{v_s=2\pi J_0
=\frac{2\pi D(1-e^{-2\pi F_0/D})}
{\int_{M_0}^{M_1}u(x)dx}.}
$$
For $F_0>0$ the numerator is positive, so motion is on average toward increasing $x$, down the tilted potential.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-356/2/c/solution