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Product-form stationary distributions for deficiency zero chemical reaction networks, arXiv:0803.3042.
Feinberg, M. (1987). “Chemical reaction network structure and the stability of complex isothermal reactors: I. The deficiency zero and deficiency one theorems”. Chemical Engineering Science 42 (10): 2229–2268. doi:10.1016/0009-2509(87)80099-4.
Feinberg, M. (1989). “Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity”. Chemical Engineering Science 44 (9): 1819–1827. doi:10.1016/0009-2509(89)85124-3.
Schlosser, P. M. and M. Feinberg (June 1994). “A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions”. Chemical Engineering Science 49 (11): 1749–1767. doi:10.1016/0009-2509(94)80061-8.
Feinberg, M. (December 1995). “The existence and uniqueness of steady states for a class of chemical reaction networks”. Archive for Rational Mechanics and Analysis 132 (4): 311–370. doi:10.1007/BF00375614.
(Chemical Species). $S = \{S_1, ..., S_m\}$
(kth Reaction). $v_k, v_k'\in Z_{\geq 0}^m$. $v_k$, consumed. $v_k'$, created.
(Complex). Each $v_k$, $v_k'$ is termed a complex.
Notation for reactions. $v_k\rightarrow v_k'$. Source or reactant complex to the product complex.
The set of all complexes. $\{v_k\}:= \bigcup_k(\{v_k\}\cup\{v_k'\})$.
Notation for chemical reaction networks. Species: $S=\{S_i\}$. Complexes: $C=\{v_k\}$. Reactions: $R=\{v_k\rightarrow v_k'\}$.
(Chemical Reaction Network). $\{S, C, R\}$.
(Weakly reversible). $v_k\rightarrow v_k'\in R$, $\Rightarrow$ $\exists$ a sequence of reactions starting at $v_k'$ and leading to $v_k$.
(Reversible). $v_k\rightarrow v_k'\in R$, $\Rightarrow$ $\exists$ $v_k'\rightarrow v_k\in R$.
(Induced Directed Graph). Each $\{S, C, R\}$ gives rise uniquely to a directed graph, with nodes $C$ and edges in $R$.
(Linkage Class). Each connected component of the resulting graph is termed a linkage class of the graph.
(Connected Component). A subgraph in which any two components are connected.
(Connected Graph). A graph that is itself connected has exactly one connected component, consisting of the whole graph.
(Stoichiometric Subspace). $S= \text{span}\{v_k'-v_k|v_k\rightarrow v_k'\in R\}$. Dim($S$)$:=s$.
(Compatibility Classes). Pick $c\in R^m$. $c+S$ is the stoichiometric compatibility class. $(c+S)\cap R^m_{\geq 0}$ is the positive stoichiometric compatibility class.
(Related Fact). For stochastic and deterministic models, the state of the system remains within a single stoichiometric compatibility class for all time, assuming one starts in that class.
(Deficiency). $\delta = |C|-l-s$. $l$ is the number of linkage classes of the graph. $s$ is Dim($S$)$:=s$. Note that we have thus far in our examples to date, consider the case with $l=1$.
(States). $S=\{1,...,k\}$
(Transitions). A set of transitions $T$. We consider $\tau \in T$ such that.
$\tau$ has inputs $m(\tau) = (m_1(\tau), ..., m_k(\tau))\in N^k$
$\tau$ has outputs $n(\tau) = (n_1(\tau), ..., n_k(\tau))\in N^k$
Let $x_i(t)$ be the amount of concentration of the ith state $(i=1,...,k)$
We can also arrive at a master equation. Now say we have a probability,
$\psi_{i_1...i_k}$ of having $i_1$ things in state 1, $i_k$ things in state $k$ etc. We write
$\psi = \sum \psi_{i_1...i_k}z_1^{i_1}\cdots z_k^{i_k}$
and $a_i^\dagger \psi = z_i \psi$, $a_i\psi = \frac{\partial}{\partial z_i} \psi$ then the master equation says
$\frac{d}{dt}\psi = H\psi$
Each transition corresponds to an operator. There is a term $a_1^{\dagger n_1}\cdots a_k^{\dagger n_1} a_1^{m_1}\cdots a_k^{m_k}$
In general, the master equation is written as