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

Fix an integer $d\ge 1$, a side length $L>0$ in metres, a mass $m>0$ in kilograms, and a smooth even periodic pair potential $W\in C^{\mathrm{\infty }}({\mathbb{T}}_L^d;\mathbb{R})$ in joules, where ${\mathbb{T}}_L^d:=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^d$ is the flat position torus and even means $W(-q)=W(q)$. For each finite integer $N\ge 1$, the phase space is $(T^{\ast }{\mathbb{T}}_L^d{)}^N$ with canonical coordinates $z_i=(q_i,p_i)$, $q_i\in {\mathbb{T}}_L^d$ in metres and $p_i\in {\mathbb{R}}^d$ in kilogram-metres per second. In position charts obtained by lifting torus coordinates to ${\mathbb{R}}^d$, define

${\theta }_N:={\sum }_{i=1}^N{\sum }_{a=1}^d{p}_i^{a}d{q}_i^a,{\omega }_N:=-d{\theta }_N={\sum }_{i=1}^N{\sum }_{a=1}^{d}d{q}_i^a\wedge d{p}_i^a\mathrm{.}$

Overlaps of these position charts are translations by vectors in $L{\mathbb{Z}}^d$, with the same momentum coordinates, so $d{q}_i^a$ and $p_i^a$ agree on overlaps. Thus the displayed forms glue globally on $(T^{\ast }{\mathbb{T}}_L^d{)}^N$; ${\omega }_N$ is closed and nondegenerate by its displayed coordinate matrix. Both forms have action units of joule-seconds. Use the Hamiltonian convention ${\iota }_{X_{H_N}}{\omega }_N=d{H}_N$ of Hamiltonian phase, complete-domain flow and reference measure. The finite-particle Hamiltonian, with energy in joules, is

$H_N(q_1,\dots ,q_N,p_1,\dots ,p_N)={\sum }_{i=1}^N\frac{\mathrm{\mid }{p}_i{\mathrm{\mid }}^2}{2m}+\frac{1}{N}{\sum }_{1\le i<j\le N}W(q_i-q_j)\mathrm{.}$

The interaction force is $K(q):=-{\mathrm{\nabla }}_{q}W(q)$, in newtons. It is bounded and globally Lipschitz on the compact torus; evenness of $W$ makes $K$ odd and $K(0)=0$. Hamilton's equations in the canonical convention are

${\dot{q}}_i=\frac{p_i}{m},{\dot{p}}_i=\frac{1}{N}{\sum }_{j\mathrm{\ne }i}K(q_i-q_j)\mathrm{.}$

For any chosen momentum scale $P_0>0$ in kilogram-metres per second, define the time scale ${\tau }_0:=mL\mathrm{/}{P}_0$ in seconds, dimensionless variables $x_i:=q_i\mathrm{/}L\in {\mathbb{T}}^d:=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^d$, $v_i:=p_i\mathrm{/}{P}_0$, and $s:=t\mathrm{/}{\tau }_0$. Define the dimensionless potential $\mathrm{\Phi }(x):=mW(Lx)\mathrm{/}{P}_0^2$ and force $k(x):=-{\mathrm{\nabla }}_x\mathrm{\Phi }(x)$. Then the equations become

$x_i^{\mathrm{\prime }}=v_i,v_i^{\mathrm{\prime }}=\frac{1}{N}{\sum }_{j\mathrm{\ne }i}k(x_i-x_j),$

where prime denotes derivative with respect to dimensionless time $s$. Here $\mathrm{\parallel }Dk{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{x\in {\mathbb{T}}^d}\mathrm{\parallel }Dk(x){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$, with operator norm induced by the Euclidean norm, and $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_x\mathrm{\mid }k(x)\mathrm{\mid }$; the constants $A:=\mathrm{\parallel }Dk{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $B:=\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}$ are finite and dimensionless. This is the selected smooth periodic mean-field scaling; it is not the hard-sphere Boltzmann-Grad scaling.

The canonical cotangent forms are defined locally above and glued by the torus translation charts. The current draft Hamiltonian phase, complete-domain flow and reference measure supplies the convention ${\iota }_{X_H}\omega =dH$, the maximal Hamiltonian-flow terminology, and the volume/reference conventions $\mathrm{\Omega }={\omega }^n\mathrm{/}n!$ and $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}{a}_0^n$ for a fixed action scale $a_0>0$ (with its stated countable-choice assumption where its density measure is used). That supplier is a definition, so it has no proof claim; its displayed hypotheses and draft status have been inspected. Here $n=dN$, and the locally specified symplectic form and Hamiltonian above give the displayed finite-particle equations. No completeness or probability preparation is asserted by this definition.