Finite reversible stochastic rate model

Assume the Axiom of Choice The Axiom of Choice. A finite reversible stochastic rate model consists of an integer $r\ge 2$, the finite state set $E=\{1,\dots ,r\}$, real energy parameters $e_a\in \mathbb{R}$, an inverse-energy parameter $\beta >0$, and finite transition rates $w_{ab}\ge 0$ for distinct $a,b\in E$. When physical units are assigned, $e_a$ is measured in joules, $\beta$ in joules${}^{-1}$, and $w_{ab}$ in seconds${}^{-1}$; this mathematical definition does not assert a physical realization. Define

$Z:={\sum }_{a\in E}{e}^{-\beta e_a},{\pi }_a:=Z^{-1}{e}^{-\beta e_a}\mathrm{.}$

Then $Z$ is finite and positive, and $\pi$ is a strictly positive probability vector. Require $w_{ab}>0$ exactly when $w_{ba}>0$, require the undirected graph with edge $\{a,b\}$ whenever $w_{ab}>0$ to be connected, and impose detailed balance ${\pi }_a{w}_{ab}={\pi }_b{w}_{ba}$. Define the row generator $Q$ by $Q_{ab}=w_{ab}$ for $a\mathrm{\ne }b$ and $Q_{aa}=-q_a$, where $q_a:={\sum }_{b\mathrm{\ne }a}{w}_{ab}$. Since $r\ge 2$ and the graph is connected, every $q_a>0$. Thus $(Qf{)}_a={\sum }_{b\mathrm{\ne }a}{w}_{ab}(f_b-f_a)$ for a column observable $f$, and a row probability vector evolves by $p^{\mathrm{\prime }}=pQ$.

For any observable $f:E\to \mathbb{R}$, write $\pi (f):={\sum }_{a\in E}{\pi }_a{f}_a$. For $f,g:E\to \mathbb{R}$, define $⟨f,g{⟩}_{\pi }:={\sum }_{a\in E}{\pi }_a{f}_a{g}_a$. Since $E$ is finite and every ${\pi }_a>0$, all real functions on $E$ belong to this finite-dimensional real inner-product space, denoted $L^2(E,\pi )$; equality in it is pointwise equality.

The associated jump-chain construction waits an exponential time of rate $q_a$ in state $a$ and then selects state $b\mathrm{\ne }a$ with probability $w_{ab}\mathrm{/}{q}_a$, using fresh independent random choices at successive jumps. Here an exponential time of rate $q>0$ has survival probability $\mathrm{Pr}(\tau >t)=e^{-qt}$ for $t\ge 0$. The Ionescu–Tulcea construction Ionescu-Tulcea construction of a Markov chain supplies the countable random sequence under the stated Choice assumption. The theorem on this page proves nonexplosion, identifies the transition semigroup, and establishes the positive reversible spectral gap. This is a finite mathematical rate model, not a derivation of rates from Hamiltonian mechanics.