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For jumps $r$ from state $x$ at nonnegative rates $W(x,r)$, the Fokker-Planck diffusion matrix in the infinitesimal-covariance convention isAssume the second jump moments are finite. Over a short interval $dt$, the conditional covariance of the state increment is $B(x)dt+o(dt)$. Subtracting the product of the mean increments changes only order-$d{t}^2$ terms. The matrix is symmetric and positive semidefinite because $v^{\mathsf{T}}Bv={\sum }_{r}W(x,r)(v\cdot r{)}^2\ge 0$ for every real vector $v$.

With this definition, its term in the Fokker-Planck equation is $\frac{1}{2}{\sum }_{i,j}{\partial }_i{\partial }_j(B_{ij}p)$. It is therefore $B/2$, not $B$, that corresponds to a scalar diffusion coefficient multiplying ${\partial }_x^{2}p$ in one dimension. Unit births and deaths at rates $b,d$ give drift $b-d$ and increment variance $B=b+d$: opposite mean displacements cancel in the drift but both contribute positively to the variance. The chemical diffusion matrix applies the same outer-product construction to reaction channels.

For $\beta >0$, symmetry of the unique stationary mean-field state gives $\mu =⟨X_1⟩=⟨X_2⟩$ and the balance $\lambda =\beta \mu +C{\mu }^2$. The fraction of the molecular removal flux using the joint reaction and the mean molecular lifetime are consequentlyIn particular $0\le E<1$, $C\mu =E/\tau$, and $\beta =(1-E)/\tau$.

The total loss flux of species one is $v_1^{-}={\mu }_1(\beta +C{\mu }_2)$, whereas its production flux $v_1^{+}=\lambda$ is constant. The reaction-rate elasticities are therefore $H_{11}=1$ and $H_{12}=C{\mu }_2/(\beta +C{\mu }_2)=E$ at the symmetric state. Species two gives $H_{22}=1$, $H_{21}=E$. All chemical jumps in either component have unit absolute size, so their molecular-flux-weighted reaction event sizes satisfy $⟨r_1⟩=⟨r_2⟩=1$. With $M_{ij}=H_{ij}/{\tau }_i$, this provesThe same matrix follows from $M=-S^{-1}AS$ with $S=\mu I$ and the drift Jacobian matrix in part (b); its positive eigenvalues are the restoring rates.

For the chemical diffusion matrix, each component receives noise from one unit birth, its own unit death, and the joint unit loss. The unnormalized diagonal noise intensity isOnly the joint reaction changes both components in one event. Its stoichiometric vector is $(-1,-1)$, so $B_{12}=B_{21}=(-1)(-1)C{\mu }^2=C{\mu }^2$. The positive sign reflects simultaneous decreases, rather than independent death events. Divide by the product of the means to normalize:Thus the chemical diffusion matrix in the normalized stationary fluctuation-dissipation relation for a reaction network isThe unequal roles of $M$ and $D$ matter: $M$ describes restoration of perturbations, while $D$ records the covariance of the immediate stochastic reaction increments.