The Ising Population Data

Let $\chi$, $\psi$, $q$, $q_{0}$, $\mu$ and $T$ be positive real numbers, and let $\underline{a}$ and $\bar{a}$ be real numbers with $0<\underline{a}<1<\bar{a}$. These eight numbers are the parameters¶, with the roles: $\chi$ is the entropic weight, and a large $\chi$ makes deviation of the transition rates from the rest rate $1$ cheap; $\psi$ is the splitting strength, and the cost it weights is smallest when the population is evenly split; $q$ is the state-dependent observation rate and $q_{0}$ the background observation rate; $\mu$ is a convexity weight, used only in the extension of the running cost; $T$ is the horizon; and $[\underline{a},\bar{a}]$ is the range of admissible transition rates.

The Ising population data with these parameters are the objects displayed in the five clauses below, taken together. They describe a population in which each agent occupies one of two states, the two coordinates of the control are the rates of the two possible transitions, the running cost prices deviation of those rates from $1$ entropically and rewards an even split, and each agent emits observations in a channel determined by its state; the components are not separately meaningful, since the transition rates, the cost and the observation rates are the rate, cost and observation data of one controlled population, and each is accompanied by the extension of itself that the fluctuation theory differentiates.

1. (Dimensions and vectors.)¶ The state count, control dimension and channel count are the natural numbers $l=2$, $m=2$ and $\tilde{l}=2$. State $1$ is called down and state $2$ up. Write $\Delta^{2}\subset\mathbb{R}^{2}$ for the probability simplex, whose points are the population profiles $x=(x^{1},x^{2})$; $|\cdot|$ for the Euclidean norm on Euclidean space; $x\cdot y$ for the dot product; $M^{\top}$ for the transpose and $MM'$ for the product of real matrices; and $\mathbf{1}_{\{\cdot\}}$ for the indicator equal to $1$ when the subscripted condition holds and to $0$ otherwise. Put

Every difference of two population profiles is a multiple of $v$, since the two coordinates of a profile sum to $1$ and so those of a difference sum to $0$, while $\mathsf{n}$ points across the simplex; the two are orthogonal, $\mathsf{n}\cdot v=0$, and $v\cdot v=\mathsf{n}\cdot\mathsf{n}=2$.

2. (Control set and ambient open sets.)¶ The control set is the box

and the ambient open sets are

together with $U_{c}=U$ and $\tilde{U}=U$. The set $U$ is the state-side open set of both the transition-rate extension and the cost extension, $V$ the control-side open set of the transition-rate extension, and $\tilde{U}$ the open set of the observation-rate extension.

3. (Transition rates and their extension.)¶ An agent in the down state moves up at the rate given by the first control coordinate, and an agent in the up state moves down at the rate given by the second: for $\Sigma\in\Delta^{2}$ and $\alpha\in\mathcal{A}$,

neither depending on the profile $\Sigma$. These rates are affine in the control with the coefficients

and their extension is $\bar{\beta}(1,2,x,a)=a^{1}$ and $\bar{\beta}(2,1,x,a)=a^{2}$ for $(x,a)\in U\times V$.

4. (Observation rates and their extension.)¶ An agent in state $\sigma$ emits observations in the matching channel $\sigma$ at rate $q+q_{0}$ and in the other channel at the background rate $q_{0}$: for $\sigma,\upsilon\in\{1,2\}$,

again independent of the profile, with the extension $\bar{\tilde{\beta}}(\sigma,\upsilon,x)=q\,\mathbf{1}_{\{\sigma=\upsilon\}}+q_{0}$ for $x\in\tilde{U}$.

5. (Running and terminal cost and their extension.)¶ Let $\phi:\mathbb{R}\to\mathbb{R}$ be the regularised entropic rate cost with cut-offs $\underline{a}$ and $\bar{a}$, a function which is nonnegative, vanishes exactly at the rest rate $1$, and has second derivative $u^{-1}$ at every $u\in[\underline{a},\bar{a}]$, by claims The Regularised Entropic Rate Cost §profile, The Regularised Entropic Rate Cost §cost-function and The Regularised Entropic Rate Cost §sign-bounds of that lemma. The running cost and terminal cost are

where the first term charges each agent for the rate at which it is being moved, weighted by the share of the population subject to that rate, and the second is smallest when the population is evenly split. Their extension is

The term weighted by $\mu$ is written into $\bar{L}$ only, not into $L$; it vanishes at every point of $\Delta^{2}\times\mathbb{R}^{2}$, since $x^{1}+x^{2}=1$ there.

6. (Notational cautions.)¶ The parameter $\chi$ is written $\boldsymbol{\beta}$ in the source of this model, a letter reserved here for transition-rate families; $\psi$ is the negative of the interaction parameter written $\mathbf{J}$ there, and is unrelated to the auxiliary integrand written $\psi_{t}$ inside The Fluctuation Linear-Quadratic Cost Functional; the external field written $\mathbf{H}$ there is taken to be $0$ here and is not named. The open set $V$ introduced above is unrelated to the coefficient matrices written $V_{t}$ in Completion of Squares and A Priori Control Bound for the Fluctuation Cost. The cut-offs $\underline{a}$ and $\bar{a}$ are real numbers, not maps, and the overbar in $\bar{a}$ is unrelated to the overbars marking extensions in $\bar{\beta}$, $\bar{L}$, $\bar{G}$ and $\bar{\tilde{\beta}}$. The vector $\mathsf{n}$ is not a natural number, and $q$ carries no time subscript anywhere.