The even pair potential $u$ represents repulsion: because it decreases for positive separation, the force $-u'(x_i-x_j)$ pushes particles apart. Marginalizing the $N$-particle <Fokker-Planck equation> gives
$$
\boxed{\partial_tp(x_1)=D\partial_{x_1}^2p
-\partial_{x_1}[\chi c'(x_1)p]
+\frac{N-1}{N}\partial_{x_1}
\int u'(x_1-x_2)P_2(x_1,x_2)dx_2.}
$$
Under the <mean-field approximation>, propagation of chaos gives $P_2(x_1,x_2)\simeq p(x_1)p(x_2)$. Taking $N\to\infty$ yields the closed nonlocal equation
$$
\boxed{\partial_tp=\partial_x\left[
D\partial_xp-\chi c'p+p\,\partial_x(u*p)
\right].}
$$
The convolution term is the collective repulsive drift generated by the population density.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-356/3/d/solution