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

The even pair potential $u$ represents repulsion: because it decreases for positive separation, the force $-u^{\prime }(x_i-x_j)$ pushes particles apart. Marginalizing the $N$-particle Fokker-Planck equation givesUnder 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 equationThe convolution term is the collective repulsive drift generated by the population density.