Conditional finite-state equilibration and fluctuation response

Assume the adopted finite thermal rate model Adopted finite reversible bath rates and field coupling, including its finite energy states, detailed-balance rates, temperature parameter and stationary Gibbs preparation for the response branch. For any initial microscopic probability law, the model's dimensionless relative information $\mathcal{D}(p_t\mathrm{\parallel }\pi )$ is nonincreasing and tends to zero, and every bounded state observable has an almost-sure time average equal to its Gibbs mean. For real state observables $A,B:E\to \mathbb{R}$, any finite horizon $t_f>0$, and a continuous field $h:[0,t_f]\to \mathbb{R}$, the equilibrium-response branch with the postulated symmetric field-dependent rates has, for $0\le t\le t_f$, first-order response $\delta ⟨A(t)⟩={\int }_0^{t}h(s)[-\beta \frac{d}{du}{C}_{BA}(u){\mid }_{u=t-s}]ds,$ where $\delta ⟨A(t)⟩:={{\mathrm{\partial }}_{ϵ}{\sum }_a{p}_a^{ϵ}(t)A_a\mid }_{ϵ=0}$, $p^{ϵ}(0)=\pi$, and $p^{ϵ}$ solves the finite forward equation for the postulated controlled rates. The correlation is $C_{BA}(u)=⟨(B-\pi (B)),P_u(A-\pi (A)){⟩}_{\pi }$. For a constant field amplitude $g$ (in joules per unit of $B$), define the normalized perturbed equilibrium vector by ${\pi }_a^g:=\frac{{\pi }_a{e}^{\beta g{B}_a}}{{\sum }_{c\in E}{\pi }_c{e}^{\beta g{B}_c}}$ and define its susceptibility per unit field by ${\chi }_{AB}:={{\mathrm{\partial }}_g{\pi }^g(A)\mid }_{g=0}$. Then ${\chi }_{AB}=\beta {\mathrm{Cov}}_{\pi }(A,B)$ and, when $g=ϵh$ for constant $h$, the derivative with respect to dimensionless $ϵ$ is $h{\chi }_{AB}$. Here ${\pi }^g(A):={\sum }_a{\pi }_a^g{A}_a$ and ${\mathrm{Cov}}_{\pi }(A,B):=\pi (AB)-\pi (A)\pi (B)$. If physical units are assigned to $A$ and $B$, write them as $[A]$ and $[B]$: $C_{BA}$ has units $[A][B]$, while $\beta hd{C}_{BA}\mathrm{/}du$ has units $[A]$ per second, so the time integral and the derivative with respect to dimensionless $ϵ$ have units $[A]$. These are conditional predictions of this finite stochastic model. They concern its dimensionless relative-information functional and state observables; no experimental observation, calibrated rate, material-specific claim or universal thermodynamic law is asserted.