Past exam of the kinetic theory course of the University of Oxford / 2023 / 1 / c / Solution

Put $z_i=({\mathbf{x}}_i,{\mathbf{v}}_i)$ and $d{z}_i=d^3{x}_i{d}^3{v}_i$. Indistinguishability gives an exchangeable ensemble. Define unit-normalized marginal distributionsIntegrating the dissipative Liouville equation over $z_2,\dots ,z_N$ removes their boundary fluxes. The $N-1$ identical pair terms remain:This is the first equation of a BBGKY hierarchy, with no independence assumed yet.

The additional physical assumption is a mean-field closure for velocity alignment: neglect pair correlations, $p_2(z,z_{\star },t)\simeq p_1(z,t)p_1(z_{\star },t)$. Initial independence alone does not make this exact at later times, because the particles interact. In the usual number convention, $f=N{p}_1$ and the two-particle distribution function is $f_2=N(N-1)p_2$. Consequently the closed equation at finite $N$ isNeglecting the relative $1/N$ correction gives the intended mean-field alignment equation:Equivalently, the closure $f_2\simeq f{f}_{\star }$ incorporates this large-$N$ approximation. It cannot be an exact finite-$N$ identity with $\int f=N$, since $\int f_{2}d{z}_{\star }=(N-1)f$ whereas $\int f{f}_{\star }d{z}_{\star }=Nf$.

There is also a consistent finite-$N$ convention yielding the boxed equation directly under the independence closure: define $\tilde{f}=(N-1)p_1$ instead. Its total normalization of a probability distribution is $N-1$, rather than the physical particle number $N$. Thus the displayed source equation needs a stated normalization of a probability distribution convention, or the large-$N$ approximation, in addition to neglect of correlations. In the remaining parts we use the displayed mean-field alignment equation itself; its moment identities do not depend on the total normalization of a probability distribution.