The Ising Population Model Instantiates the Data of the Fluctuation Theory

Fix parameters $\chi$, $\psi$, $q$, $q_{0}$, $\mu$, $T$, $\underline{a}$, $\bar{a}$ as in The Ising Population Data (i) and adopt the Ising population data with those parameters: the control set $\mathcal{A}$, the open sets $U$, $V$, $U_{c}$, $\tilde{U}$, the transition-rate coefficients $(\beta_{0},\beta_{1})$ with the rates $\beta$ and their extension $\bar{\beta}$, the observation rates $\tilde{\beta}$ with their extension $\bar{\tilde{\beta}}$, the costs $(L,G)$ with their extension $(\bar{L},\bar{G})$, the function $\phi$, and the vectors $v$, $\mathsf{n}$. Write $\Lambda$ for the Lipschitz constant of an affine-controlled transition-rate family, $B$ for the rate bound of a transition-rate family, $K$ for the derivative bound of a twice continuously differentiable extension of such a family, $\tilde{B}$ and $\tilde{K}$ for the corresponding constants of an observation-rate family and of its twice continuously differentiable extension, and $K_{c}$ for the second-derivative bound of a twice continuously differentiable extension of population cost data. Adopt the coordinate and partial-derivative notation $\partial_{i}$ of the extension definitions, so that $\partial_{1},\partial_{2}$ differentiate in the state coordinates $x^{1},x^{2}$ and $\partial_{3},\partial_{4}$ in the control coordinates $a^{1},a^{2}$. Then the following hold.

1. (The control set.)(i) $\mathcal{A}$ is a nonempty convex subset of $\mathbb{R}^{2}$, compact for the topology determined by the Euclidean distance; thus hypothesis (A) of Quadratic Growth of the Mean-Field Hamiltonian in the Control along a Stationary Mean-Field Triple holds. Moreover $\sup_{a\in\mathcal{A}}|a|=\sqrt{2}\,\bar{a}$, and, with

$$\varrho_{\mathcal{A}}=\min\{1-\underline{a},\ \bar{a}-1\}>0 ,$$

every $a\in\mathbb{R}^{2}$ with $|a-(1,1)|\le\varrho_{\mathcal{A}}$ lies in $\mathcal{A}$.

2. (The transition-rate family and its drift.)(ii) The pair $(\beta_{0},\beta_{1})$ is an affine-controlled transition-rate family on $2$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda=0$. The transition-rate family that The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data associates with it is the family $\beta$ of clause The Ising Population Data (iv), and it is a transition-rate family on $2$ states with control set $\mathcal{A}$ and rate bound $B=\bar{a}$. Its aggregate state drift is

$$b(\Sigma,\alpha)=\bigl(\Sigma^{2}\alpha^{2}-\Sigma^{1}\alpha^{1}\bigr)\,v\qquad(\Sigma\in\Delta^{2},\ \alpha\in\mathcal{A}).$$

3. (The aggregate fluctuation covariance.)(iii) The aggregate fluctuation covariance of $\beta$ is

$$\Theta(\Sigma,\alpha)=\bigl(\Sigma^{1}\alpha^{1}+\Sigma^{2}\alpha^{2}\bigr)\,vv^{\top}\qquad(\Sigma\in\Delta^{2},\ \alpha\in\mathcal{A}).$$

4. (The rate extension.)(iv) The triple $(U,V,\bar{\beta})$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K=1$. Its only nonvanishing partial derivatives of order one or two are $\partial_{3}\bar{\beta}(1,2,\cdot)=1$ and $\partial_{4}\bar{\beta}(2,1,\cdot)=1$ on $U\times V$. Its extended aggregate state drift is

$$\bar{b}(x,a)=\bigl(x^{2}a^{2}-x^{1}a^{1}\bigr)\,v\qquad\bigl((x,a)\in U\times V\bigr).$$

5. (The observation-rate family.)(v) The family $\tilde{\beta}$ is an observation-rate family on $2$ states with $2$ observation channels and rate bound $\tilde{B}=q+q_{0}$. Its aggregate observation drift is $\tilde{b}^{\upsilon}(\Sigma)=q\,\Sigma^{\upsilon}+q_{0}$ for $\Sigma\in\Delta^{2}$ and $\upsilon\in\{1,2\}$, and satisfies $\tilde{b}^{\upsilon}(\Sigma)\ge q_{0}>0$ for all such $\Sigma$ and $\upsilon$; thus hypothesis (OC) of Injection Certificates on the Trimmed Synthetic Copy for the Family of N-Agent Solutions: the Law-Transported Van Trees Certificate Hypothesis Holds under Deterministic Initial States, Uniform Observation Positivity and a C^2 Observation Extension holds with $\underline{b}=q_{0}$.

6. (The observation-rate extension.)(vi) The pair $(\tilde{U},\bar{\tilde{\beta}})$ is a twice continuously differentiable extension of $\tilde{\beta}$ with derivative bound $\tilde{K}=0$; all its partial derivatives of order one and two vanish. Its extended aggregate observation drift is

$$\bar{\tilde{b}}^{\upsilon}(x)=q\,x^{\upsilon}+q_{0}\,(x^{1}+x^{2})\qquad(x\in\tilde{U},\ \upsilon\in\{1,2\}).$$

7. (The population cost data.)(vii) The pair $(L,G)$ is population cost data on $2$ states with control dimension $2$, with lower bounds $C_{L}=C_{G}=0$. Moreover $L(x,a)\ge0$ for every $(x,a)\in\Delta^{2}\times\mathbb{R}^{2}$, and $L(x,a)=0$ holds if and only if $x=\bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr)$ and $a=(1,1)$. For every $x\in\Delta^{2}$ the map $a\mapsto L(x,a)$ is convex on $\mathbb{R}^{2}$; in particular $L$ is convex in the control on $\mathcal{A}$. Finally, hypothesis (LipC) of Pointwise-in-Time Tracking of the Mean-Field Flow and Cost along the Realized Control of the Controlled N-Agent Dynamics holds with

$$K_{L}=\sqrt{2}\,\Bigl(\frac{2\bar{a}^{2}}{\chi\,\underline{a}}+\psi\Bigr),\qquad K_{G}=0 .$$

8. (The cost extension.)(viii) The triple $(U_{c},\bar{L},\bar{G})$ is a twice continuously differentiable extension of $(L,G)$ with second-derivative bound

$$K_{c}=\psi+2\mu+\frac{2(\bar{a}+1)}{\chi\,\underline{a}} .$$

All partial derivatives of $\bar{G}$ vanish, and the partial derivatives of $\bar{L}$ on $U_{c}\times\mathbb{R}^{2}$ are

$$\partial_{1}\bar{L}=\chi^{-1}\phi(a^{1})-\psi\,(x^{2}-x^{1})+2\mu\,(x^{1}+x^{2}-1),\qquad \partial_{2}\bar{L}=\chi^{-1}\phi(a^{2})+\psi\,(x^{2}-x^{1})+2\mu\,(x^{1}+x^{2}-1),$$ $$\partial_{3}\bar{L}=\chi^{-1}x^{1}\phi'(a^{1}),\qquad \partial_{4}\bar{L}=\chi^{-1}x^{2}\phi'(a^{2}),$$

and, for the second order,

$$\partial_{1}\partial_{1}\bar{L}=\partial_{2}\partial_{2}\bar{L}=\psi+2\mu,\qquad \partial_{2}\partial_{1}\bar{L}=\partial_{1}\partial_{2}\bar{L}=2\mu-\psi,$$ $$\partial_{3}\partial_{1}\bar{L}=\partial_{1}\partial_{3}\bar{L}=\chi^{-1}\phi'(a^{1}),\qquad \partial_{4}\partial_{2}\bar{L}=\partial_{2}\partial_{4}\bar{L}=\chi^{-1}\phi'(a^{2}),\qquad \partial_{4}\partial_{1}\bar{L}=\partial_{1}\partial_{4}\bar{L}=\partial_{3}\partial_{2}\bar{L}=\partial_{2}\partial_{3}\bar{L}=0,$$ $$\partial_{3}\partial_{3}\bar{L}=\chi^{-1}x^{1}\phi''(a^{1}),\qquad \partial_{4}\partial_{4}\bar{L}=\chi^{-1}x^{2}\phi''(a^{2}),\qquad \partial_{4}\partial_{3}\bar{L}=\partial_{3}\partial_{4}\bar{L}=0 ,$$

all evaluated at the point $(x,a)$.