Selected mean-field particle model and iid preparation

Adopt Smooth periodic finite-N and scaled mean-field phase data as the selected finite-particle model: for each $N\ge 1$, the phase points $(q_i,p_i)$ evolve by its displayed Hamilton equations with smooth even periodic potential $W$ and interaction coefficient $1\mathrm{/}N$. This is a model assumption with $q_i$ in metres, $p_i$ in kilogram-metres per second, $m$ in kilograms, $W$ in joules, and force $K=-\mathrm{\nabla }W$ in newtons.

For the propagation-of-chaos preparation branch, give $Z={\mathbb{T}}^d\times {\mathbb{R}}^d$ its product topology and the Borel sigma algebra generated by its open sets. Choose a Borel probability ${\mu }_0$ and a compact set $K\subset Z$ with ${\mu }_0(K)=1$. Put $\mathrm{\Omega }=Z^{\mathbb{N}}$ with product sigma algebra $\mathcal{B}(Z{)}^{\otimes \mathbb{N}}$. Under the full Axiom of Choice, Ionescu-Tulcea construction of a Markov chain applied to the constant kernels $K_n(z_0,\dots ,z_n,A)={\mu }_0(A)$ supplies a probability $P$ on $\mathrm{\Omega }$. Let $Z_i(0):\mathrm{\Omega }\to Z$ be the coordinate projections, and use the first $N$ coordinates for the finite-$N$ preparation. For every finite list of indices $i_1<\cdots <i_r$ and Borel sets $A_1,\dots ,A_r\subseteq Z$, this construction gives

$P(Z_{i_1}(0)\in A_1,\dots ,Z_{i_r}(0)\in A_r)={\prod }_{j=1}^r{\mu }_0(A_j)\mathrm{.}$

Thus each coordinate has law ${\mu }_0$ and the coordinates are mutually independent. Here “iid” means exactly this common-marginal and finite-rectangle factorization property. The cited construction uses Choice to choose a compatible infinite coordinate sequence and again to choose successive coordinates in its continuity-at-the-empty-set argument; this postulate does not weaken that assumption.

These assumptions define the selected physical model and preparation. They do not assert that a real gas, hard-sphere system, or experiment realizes them, and they do not deduce the assumptions from mechanics or observation.