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A reaction's stoichiometric vector outer product multiplied by its reaction propensity function is its contribution to the infinitesimal state-increment covariance. Summing over channels gives $B$. Its normalized form $D$ divides by the two mean copy numbers. The Fokker-Planck equation writes the associated second-derivative term with $B/2$; the covariance evolution and normalized stationary Lyapunov equation instead use $B$ and $D$ themselves. A simultaneous unit loss of two species contributes positive cross noise even when their long-time stationary covariance is negative.

For reactions with stoichiometric vectors ${\nu }_r$ and reaction propensity functions $a_r$, the probability mass function $p(x,t)$ obeysIt is the forward operator of a Markov jump process equation for the copy-number continuous-time Markov chain.

A moment hierarchy is the coupled family of equations obtained by applying a Markov jump-process generator to powers of the state. Nonlinear reaction propensity functions make a moment of one order depend on higher-order moments, so the hierarchy generally does not close after finitely many equations.

The normalized stationary fluctuation-dissipation relation for a reaction network isHere $M$ is the negative drift Jacobian matrix expressed in normalized population coordinates, and $D$ is the normalized noise matrix. More explicitly, with $S=\mathrm{diag}({\mu }_i)$, drift Jacobian $A$, stoichiometric vectors ${\nu }_r$, and stationary reaction propensity functions $a_r$, these matrices are $M=-S^{-1}AS$ and $D=S^{-1}B{S}^{-1}$, where $B={\sum }_r{a}_r{\nu }_r{\nu }_r^{\mathsf{T}}$. The linear noise approximation gives a covariance evolution $\dot{\eta }=-M\eta -\eta M^{\mathsf{T}}+D$; stationarity is the displayed Continuous Lyapunov equation.

In one dimension, the normalized restoring coefficient is $M=H/\tau =1/(2\tau )$. Squared chemical jumps determine the noise:The factor nine from a birth burst is essential; its three molecules arise in one correlated reaction event. The scalar Continuous Lyapunov equation is $2M\eta =D$, givingEquivalently, the normalized variance is $⟨s⟩/(H\mu )=2/((1/2)\mu )$. The Fano factor is approximately four. This calculation uses the prescribed mean closure and the linear noise approximation, so its small-relative-fluctuation regime is $\mu \gg 1$; it is not an exact assertion for arbitrarily small molecular copy numbers.

For the Gillespie algorithm, start from $t=0$ and an integer molecular copy number $x\ge 0$. At each step use the current state, without the mean approximation in parts (a)–(b). The reaction propensity functions are $a_{+}(x)=\lambda$ and $a_{-}(x)=\beta \sqrt{x}$, and their sum is $a_0(x)=\lambda +\beta \sqrt{x}$. Draw two independent uniform random variables $U_1,U_2$ on $(0,1)$ and setThe waiting time has an exponential distribution with rate $a_0(x)$. Keep the sample path constant on $[t,t+\Delta t)$, advance $t\leftarrow t+\Delta t$, and choose the reaction byRecord the post-jump state, recompute both reaction propensity functions, and repeat with fresh independent uniforms. At $x=0$ the death propensity is zero, so only a birth is selected and the trajectory remains nonnegative. For a prescribed terminal time $T$, if the next event would occur after $T$, keep the current state to $T$ and stop. This produces exact piecewise constant sample paths of the specified continuous-time Markov chain.

Let ${\mu }_i=⟨X_i⟩$. Applying the Markov jump-process generator to either coordinate gives the exact moment equationsIn each equation the joint reaction removes one molecule, at reaction propensity function $C{X}_1{X}_2$. Because the two reactants are different chemical species, the propensity uses their product; no same-species factorial correction is needed. The exact identity $⟨X_1{X}_2⟩={\mu }_1{\mu }_2+\mathrm{Cov}(X_1,X_2)$ shows explicitly where the equations fail to close. Neglecting this covariance relative to the product of means is the small-fluctuation moment closure. It gives the mean-field joint removal of two chemical species:The difference equation $d({\mu }_1-{\mu }_2)/dt=-\beta ({\mu }_1-{\mu }_2)$ actually holds exactly as well as after closure, because both the constant production and the joint removal terms cancel. The nonlinear moment closure will be used for the subsequent phase plane and stationary linear noise approximation calculations, as requested.

Stochastic chemical kinetics models molecular copy numbers as a continuous-time Markov chain. Each reaction has a state-dependent reaction propensity function and changes the state by its stoichiometric vector when it fires.