Interacting Ensemble Geometry — Examples

On the ordered region $x_1<x_2$, allowed pairs satisfy $x_2\ge x_1+a$, so their volume is ${\int }_0^{L-a}(L-x_1-a)d{x}_1=(L-a{)}^2\mathrm{/}2$. The coordinate-swap map preserves product Lebesgue measure, so the region $x_2<x_1$ has the same volume; the diagonal has zero product measure. Thus the full labeled allowed volume is $(L-a{)}^2$, and division by $2!$ gives $Z(L,2;0)=(L-a{)}^2\mathrm{/}2$.

Since $0<a<L$, $(L-a{)}^2\mathrm{/}2<L^2\mathrm{/}2$, the unrestricted ideal-gas two-particle canonical term. The strict deficit is caused by the forbidden separation band and remains at beta zero by [F2].

The $N$-particle term without the field is $z^N{V}^N\mathrm{/}N!$, by the product volume [F1] and the $1\mathrm{/}N!$ convention [given]. Summing over $N\ge 0$ gives ${\mathrm{\Xi }}_0={\sum }_N(zV{)}^N\mathrm{/}N!=e^{zV}$ by [F2].

With the field, each term is multiplied by $e^{-\beta cN}$, so ${\mathrm{\Xi }}_c={\sum }_N(z{e}^{-\beta c}V{)}^N\mathrm{/}N!=e^{z{e}^{-\beta c}V}$ by [F2]. Therefore $V^{-1}\log {\mathrm{\Xi }}_0=z$ and $V^{-1}\log {\mathrm{\Xi }}_c=z{e}^{-\beta c}$. If $c\mathrm{\ne }0$, these differ because $z,\beta >0$.

Order the coordinates $0\le x_1<\cdots <x_N\le L$. The exclusion constraints are $x_{i+1}-x_i\ge a$. Apply the coordinatewise translation $x_i\mapsto y_i=x_i-(i-1)a$; each one-coordinate translation preserves Lebesgue measure [F5], so product measure is preserved. The ordered region becomes $0\le y_1\le \cdots \le y_N\le S$, where $S=L-(N-1)a$. The cube $[0,S{]}^N$ splits, up to equality sets of zero product measure, into $N!$ coordinate-order regions. Product-measure uniqueness makes these permutation chambers equal in measure, so the ordered region has volume $S^N\mathrm{/}N!$. The full labeled allowed volume is $N!$ times this, and canonical division by $N!$ gives $Z(L,N;\beta )=S^N\mathrm{/}N!$.

If $L=(N-1)a$, then $S=0$ and the transformed region is a null set; its integral is zero. If $L>(N-1)a$, then $S>0$ and the ordered region has positive product volume. Thus the maximal-density endpoint does not carry a positive canonical partition at exact close packing; the interior-only theorem [F2] makes no contrary assertion.

Let $P,R$ be stationary laws of finite specific entropy and let $P_{\theta }=\theta P+(1-\theta )R$. For a bounded cube $G$, the binary-label chain rule [F3] gives $\theta D(P_G\mathrm{\parallel }{Q}_G)+(1-\theta )D(R_G\mathrm{\parallel }{Q}_G)=D((P_{\theta }{)}_G\mathrm{\parallel }{Q}_G)+I(S;X_G),$ where $S$ is the mixture label and $X_G$ the observed configuration. Since $0\le I(S;X_G)\le H(S)<\mathrm{\infty }$, divide by $\mathrm{\mid }G\mathrm{\mid }$ and take the cube limit [F2]; $H(S)\mathrm{/}\mathrm{\mid }G\mathrm{\mid }\to 0$, so $I(P_{\theta })=\theta I(P)+(1-\theta )I(R)$. The expectation defining $\rho$ is affine. For finite energy densities, $E_{P_{\theta }}{H}_G=\theta E_P{H}_G+(1-\theta )E_R{H}_G$ for every $G$; dividing by volume and taking limits gives affinity of $\mathrm{\Phi }$. Thus $J$ is affine on finite mixtures of finite-$J$ laws.

Applying step 1.1 to $P_0,P_1$ gives $J(P_{\theta })=\theta J(P_0)+(1-\theta )J(P_1)=m$, so every mixture is a minimizer. Put $P_{2n}=P_0$ and $P_{2n+1}=P_1$. The even and odd subsequences converge locally to $P_0$ and $P_1$. Since distinct laws differ on a bounded-region measurable event, its indicator is a bounded local test by [F4] with different expectations; the full sequence therefore has no ${\tau }_L$ limit. Both accumulation points are minimizers.