Smooth periodic finite-N and scaled mean-field phase data — Examples

By [F3], the position density $L^{-d}$ is normalized on ${\mathbb{T}}_L^d$. Scaling $p=\sqrt{2}{\sigma }_{p}u$ and applying [F4] gives ${\int }_{\mathbb{R}}{e}^{-p^2\mathrm{/}(2{\sigma }_p^2)}dp=\sqrt{2\pi }{\sigma }_p$; Tonelli [F5] factors the $d$-dimensional integral, so ${\int }_{{\mathbb{R}}^d}{g}_{{\sigma }_p}(p)dp=1$. Therefore $F_2^0$ integrates to one relative to $d{q}_{1}d{q}_{2}d{p}_{1}d{p}_2$. It is symmetric and smooth; every momentum derivative is a polynomial times a Gaussian, so it has all rapid-tail bounds required by [F2]. Its one-particle marginal is $f_1(0,q,p)=L^{-d}{g}_{{\sigma }_p}(p)$.

Use the $k=1,N=2$ case of [F2]: ${\mathrm{\partial }}_t{f}_1+\frac{p_1}{m}\cdot {\mathrm{\nabla }}_{q_1}{f}_1+\frac{1}{2}{\mathrm{\nabla }}_{p_1}\cdot \int K(q_1-q_2)F_2(t,z_1,z_2)d{z}_2=0.$ At $t=0$, the streaming derivative vanishes because $f_1(0)$ is independent of $q_1$. Also ${\int }_{{\mathbb{T}}_L^d}K(q_1-q_2)L^{-d}d{q}_2=0$: by [F3], this is the normalized fundamental-cube integral of a periodic derivative of $W$, which is zero in each coordinate by the fundamental theorem of calculus and periodicity. Thus the force average in the marginal equation is identically zero before its momentum divergence, and ${\mathrm{\partial }}_t{f}_1(0)=0$.

The $N=2$ joint Liouville equation in [F2] has interaction coefficient $1\mathrm{/}2$. The initial position derivatives vanish, and ${\mathrm{\nabla }}_{p_i}{g}_{{\sigma }_p}(p_i)=-(p_i\mathrm{/}{\sigma }_p^2)g_{{\sigma }_p}(p_i)$. Consequently ${\mathrm{\partial }}_t{F}_2(0)=-\frac{1}{2}K(q_1-q_2)\cdot ({\mathrm{\nabla }}_{p_1}-{\mathrm{\nabla }}_{p_2})F_2^0=\frac{1}{2{\sigma }_p^2}K(q_1-q_2)\cdot (p_1-p_2)F_2^0\mathrm{.}$ Because $W$ is nonconstant, its gradient is nonzero at some point; continuity gives an open set of differences where $K\mathrm{\ne }0$. For each such difference, the condition $K\cdot (p_1-p_2)\mathrm{\ne }0$ holds on a nonempty open momentum set, and $F_2^0$ is strictly positive there. Hence the joint derivative is nonzero on a nonempty open set.

If the evolved joint density remained the product of its two one-particle marginals on a time interval around zero, differentiating that product at $t=0$ would give ${\mathrm{\partial }}_t{f}_1(0)\otimes f_1(0)+f_1(0)\otimes {\mathrm{\partial }}_t{f}_1(0)=0$ by step 2.1. Step 2.2 gives a nonzero joint derivative on an open set, a contradiction. At a point in that set, differentiability makes the difference between the joint density and the product density nonzero for all sufficiently small nonzero times. Thus correlations appear immediately from this initially factorized law.