Complete finite-N flow and permutation-invariant Liouville law

Lift the torus positions periodically to ${\mathbb{R}}^{dN}$. The force bound $K_{\ast }:=\mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{\infty }}$ gives $\mathrm{\mid }{p}_i(t)-p_i(0)\mathrm{\mid }\le K_{\ast }\mathrm{\mid }t\mathrm{\mid }$ and $\mathrm{\mid }{q}_i(t)-q_i(0)-(t\mathrm{/}m)p_i(0)\mathrm{\mid }\le K_{\ast }{t}^2\mathrm{/}(2m)$ for each particle. Thus on every finite time interval the lifted trajectory stays in a bounded state cylinder. The lifted vector field from [F1] is smooth with bounded first derivatives there, so [F3] gives a common local continuation interval; repeating this fixed extension across the finite interval gives a global lifted solution, and periodicity makes it a well-defined torus solution. Reversing time gives existence on negative intervals. Local smooth dependence from [F3], composed over finitely many local intervals, makes every global finite-time map smooth; uniqueness makes ${\mathrm{\Phi }}_{-t}^N$ its inverse.

With the symplectic form in [F1] and Hamiltonian sign convention in [F2], the Hamiltonian in [F1] defines the canonical Hamiltonian vector field on this cotangent phase space. Apply [F4] on each local time interval of step 1.1 and compose the volume-preserving local maps; this gives $({\mathrm{\Phi }}_t^N{)}^{\ast }{\mathrm{\Omega }}_N={\mathrm{\Omega }}_N$ for every real $t$. The Hamiltonian vector field is equivariant under every permutation $\sigma$ of particle labels because the kinetic sum and even pair-potential sum are invariant. Hence $\sigma {\mathrm{\Phi }}_t^N(z)$ and ${\mathrm{\Phi }}_t^N(\sigma z)$ solve the same initial-value problem; uniqueness from [F3] gives equality.

Along characteristics $z(t)={\mathrm{\Phi }}_t^N(z_0)$, the derivative of $F_N(t,z(t))=F_N^0(z_0)$ is zero. The chain rule therefore gives ${\mathrm{\partial }}_t{F}_N+X_{H_N}\cdot \mathrm{\nabla }{F}_N=0$. Substituting the Hamilton equations from [F1] and grouping the two terms for each pair $i<j$, using $K(q_j-q_i)=-K(q_i-q_j)$, yields ${\mathrm{\partial }}_t{F}_N+{\sum }_{i=1}^N\frac{p_i}{m}\cdot {\mathrm{\nabla }}_{q_i}{F}_N+\frac{1}{N}{\sum }_{1\le i<j\le N}K(q_i-q_j)\cdot ({\mathrm{\nabla }}_{p_i}-{\mathrm{\nabla }}_{p_j})F_N=0.$ The rapid tails justify the displayed derivatives as classical ones on every finite time slab.

On each finite time slab the rapid bounds on all phase derivatives make ${\mathrm{\partial }}_t{F}_N$ integrable and supply one integrable majorant for its time difference quotients; [F6] therefore permits differentiation under the $d{\mathrm{\Omega }}_N$ integral. Integrate the equation in step 3.1 over all phase variables. The $q_i$-derivative terms have zero integral by periodicity. For each force term, $K(q_i-q_j)$ is independent of momenta, so integration by parts in $p_i$ and $p_j$ has zero boundary terms by the rapid tails. Fubini [F5] justifies these coordinate integrations and their order. Hence $\frac{d}{dt}\int F_N(t)d{\mathrm{\Omega }}_N=0$; its value is $1$ at $t=0$ by [A2], so $F_N(t)$ remains normalized. The same permutation identity from step 2.1 and [A2] give $F_N(t,\sigma z)=F_N(t,z)$. Finally, the vector field derivatives of every order are bounded globally: its first derivative has entries $1\mathrm{/}m$ and $DK$, and higher derivatives are bounded derivatives of the periodic force. The variational equations and [F3] bound every finite state derivative of ${\mathrm{\Phi }}_{-t}^N$ on the time slab. Since $\mathrm{\mid }{p}_i(t)-p_i(0)\mathrm{\mid }\le K_{\ast }\mathrm{\mid }t\mathrm{\mid }$, the momentum norms differ by at most $\sqrt{N}{K}_{\ast }\mathrm{\mid }t\mathrm{\mid }$; the chain rule transfers every rapid-decay bound of $F_N^0$ to $F_N(t)$ uniformly on the slab.