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

The pair $(X_t,V_t)$ is a hybrid-state process: position is continuous and velocity is discrete. It is Markov because its future law is fixed by the present position and velocity. Position $X_t$ alone is generally non-Markovian, since its future displacement depends on the hidden persistent velocity.

The two transport equations areFor $p=p_{+}+p_{-}$ and $j=s(p_{+}-p_{-})$ they becomeEliminating $j$ gives the closed chemotactic telegraph equation

Take ${\lambda }_0\to \infty$ withfixed, so $s$ and $b$ scale as $\sqrt{{\lambda }_0}$. The rapidly relaxing flux satisfies $j\simeq -D{p}_x+\chi c^{\prime }p$, and thereforeZero stationary flux gives $p_s(x)\propto e^{\chi c(x)/D}$. A stationary probability density on $\mathbb{R}$ exists exactly when this exponential is integrable, in addition to the positivity condition $|c^{\prime }|<{\lambda }_0/b$. The limiting process is

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.