The master equation

Generate the full master equation matrix for the network.

Throughout this document we will utilize the following terminology:

  • An isomer is a unimolecular configuration on the potential energy surface.
  • A reactant channel is a bimolecular configuration that associates to form an isomer. Dissociation from the isomer back to reactants is allowed.
  • A product channel is a bimolecular configuration that is formed by dissociation of an isomer. Reassociation of products to the isomer is not allowed.

The isomers are the configurations for which we must model the energy states. We designate \(p_i(E, J, t)\) as the population of isomer \(i\) having total energy \(E\) and total angular momentum quantum number \(J\) at time \(t\). At long times, statistical mechanics requires that the population of each isomer approach a Boltzmann distribution \(b_i(E, J)\):

\[\lim_{t \rightarrow \infty} p_i(E, J, t) \propto b_i(E, J)\]

We can simplify by eliminating the angular momentum quantum number to get

\[p_i(E, t) = \sum_J p_i(E, J, t)\]

Let us also denote the (time-dependent) total population of isomer \(i\) by \(x_i(t)\):

\[x_i(t) \equiv \sum_J \int_0^\infty p_i(E, J, t) \ dE\]

The two molecules of a reactant or product channel are free to move apart from one another and interact independently with other molecules in the system. Accordingly, we treat these channels as fully thermalized, leaving as the only variable the total concentrations \(y_{n\mathrm{A}}(t)\) and \(y_{n\mathrm{B}}(t)\) of the molecules \(\mathrm{A}_n\) and \(\mathrm{B}_n\) of reactant channel \(n\). (Since the product channels act as infinite sinks, their populations do not need to be considered explicitly.)

Finally, we will use \(N_\mathrm{isom}\), \(N_\mathrm{reac}\), and \(N_\mathrm{prod}\) as the numbers of isomers, reactant channels, and product channels, respectively, in the system.

The governing equation for the population distributions \(p_i(E, J, t)\) of each isomer \(i\) and the reactant concentrations \(y_{n\mathrm{A}}(t)\) and \(y_{n\mathrm{B}}(t)\) combines the collision and reaction models to give a linear integro-differential equation:

\[ \begin{align}\begin{aligned}\begin{split}\frac{d}{dt} p_i(E, J, t) &= \omega_i(T, P) \sum_{J'} \int_0^\infty P_i(E, J, E', J') p_i(E', J', t) \ dE' - \omega_i(T, P) p_i(E, J, t) \\ & \mbox{} + \sum_{j \ne i}^{N_\mathrm{isom}} k_{ij}(E, J) p_j(E, J, t) - \sum_{j \ne i}^{N_\mathrm{isom}} k_{ji}(E, J) p_i(E, J, t) \\ & \mbox{} + \sum_{n=1}^{N_\mathrm{reac}} y_{n\mathrm{A}}(t) y_{n\mathrm{B}}(t) f_{in}(E, J) b_n(E, J, t) - \sum_{n=1}^{N_\mathrm{reac} + N_\mathrm{prod}} g_{ni}(E, J) p_i(E, J, t) \\\end{split}\\\begin{split}\frac{d}{dt} y_{n\mathrm{A}}(t) = \frac{d}{dt} y_{n\mathrm{B}}(t) &= \sum_{i=1}^{N_\mathrm{isom}} \int_0^\infty g_{ni}(E, J) p_i(E, J, t) \ dE \\ & \mbox{} - \sum_{i=1}^{N_\mathrm{isom}} y_{n\mathrm{A}}(t) y_{n\mathrm{B}}(t) \int_0^\infty f_{in}(E, J) b_n(E, J, t) \ dE\end{split}\end{aligned}\end{align} \]

A summary of the variables is given below:

Variable Meaning
\(p_i(E, J, t)\) Population distribution of isomer \(i\)
\(y_{n\mathrm{A}}(t)\) Total population of species \(\mathrm{A}_n\) in reactant channel \(n\)
\(\omega_i(T, P)\) Collision frequency of isomer \(i\)
\(P_i(E, J, E', J')\) Collisional transfer probability from \((E', J')\) to \((E, J)\) for isomer \(i\)
\(k_{ij}(E, J)\) Microcanonical rate coefficient for isomerization from isomer \(j\) to isomer \(i\)
\(f_{im}(E, J)\) Microcanonical rate coefficient for association from reactant channel \(m\) to isomer \(i\)
\(g_{nj}(E, J)\) Microcanonical rate coefficient for dissociation from isomer \(j\) to reactant or product channel \(n\)
\(b_n(E, J, t)\) Boltzmann distribution for reactant channel \(n\)
\(N_\mathrm{isom}\) Total number of isomers
\(N_\mathrm{reac}\) Total number of reactant channels
\(N_\mathrm{prod}\) Total number of product channels

The above is called the two-dimensional master equation because it contains two dimensions: total energy \(E\) and total angular momentum quantum number \(J\). In the first equation (for isomers), the first pair of terms correspond to collision, the second pair to isomerization, and the final pair to association/dissociation. Similarly, in the second equation above (for reactant channels), the pair of terms refer to dissociation/association.

We can also simplify the above to the one-dimensional form, which only has \(E\) as a dimension:

\[ \begin{align}\begin{aligned}\begin{split}\frac{d}{dt} p_i(E, t) &= \omega_i(T, P) \int_0^\infty P_i(E, E') p_i(E', t) \ dE' - \omega_i(T, P) p_i(E, t) \\ & \mbox{} + \sum_{j \ne i}^{N_\mathrm{isom}} k_{ij}(E) p_j(E, t) - \sum_{j \ne i}^{N_\mathrm{isom}} k_{ji}(E) p_i(E, t) \\ & \mbox{} + \sum_{n=1}^{N_\mathrm{reac}} y_{n\mathrm{A}}(t) y_{n\mathrm{B}}(t) f_{in}(E) b_n(E, t) - \sum_{n=1}^{N_\mathrm{reac} + N_\mathrm{prod}} g_{ni}(E) p_i(E, t) \\\end{split}\\\begin{split}\frac{d}{dt} y_{n\mathrm{A}}(t) = \frac{d}{dt} y_{n\mathrm{B}}(t) &= \sum_{i=1}^{N_\mathrm{isom}} \int_0^\infty g_{ni}(E) p_i(E, t) \ dE \\ & \mbox{} - \sum_{i=1}^{N_\mathrm{isom}} y_{n\mathrm{A}}(t) y_{n\mathrm{B}}(t) \int_0^\infty f_{in}(E) b_n(E, t) \ dE\end{split}\end{aligned}\end{align} \]

The equations as given are nonlinear, both due to the presence of the bimolecular reactants and because both \(\omega_i\) and \(P_i(E, E')\) depend on the composition, which is changing with time. The rate coefficients can be derived from considering the pseudo-first-order situation where \(y_{n\mathrm{A}}(t) \ll y_{n\mathrm{B}}(t)\), and all \(y(t)\) are negligible compared to the bath gas \(\mathrm{M}\). From these assumptions the changes in \(\omega_i\), \(P_i(E, E')\), and all \(y_{n\mathrm{B}}\) can be neglected, which yields a linear equation system.

Except for the simplest of unimolecular reaction networks, both the one-dimensional and two-dimensional master equation must be solved numerically. To do this we must discretize and truncate the energy domain into a finite number of discrete bins called grains. This converts the linear integro-differential equation into a system of first-order ordinary differential equations:

\[\begin{split}\frac{d}{dt} \begin{bmatrix} \mathbf{p}_1 \\ \mathbf{p}_2 \\ \vdots \\ y_{1\mathrm{A}} \\ y_{2\mathrm{A}} \\ \vdots \end{bmatrix} = \begin{bmatrix} \mathbf{M}_1 & \mathbf{K}_{12} & \ldots & \mathbf{F}_{11} \mathbf{b}_1 y_{1\mathrm{B}} & \mathbf{F}_{12} \mathbf{b}_2 y_{2\mathrm{B}} & \ldots \\ \mathbf{K}_{21} & \mathbf{M}_2 & \ldots & \mathbf{F}_{21} \mathbf{b}_1 y_{1\mathrm{B}} & \mathbf{F}_{22} \mathbf{b}_2 y_{2\mathrm{B}} & \ldots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\ (\mathbf{g}_{11})^T & (\mathbf{g}_{12})^T & \ldots & h_1 & 0 & \ldots \\ (\mathbf{g}_{21})^T & (\mathbf{g}_{22})^T & \ldots & 0 & h_2 & \ldots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \end{bmatrix} \begin{bmatrix} \mathbf{p}_1 \\ \mathbf{p}_2 \\ \vdots \\ y_{1\mathrm{A}} \\ y_{2\mathrm{A}} \\ \vdots \end{bmatrix}\end{split}\]

The diagonal matrices \(\mathrm{K}_{ij}\) and \(\mathrm{F}_{in}\) and the vector \(\mathrm{g}_{ni}\) contain the microcanonical rate coefficients for isomerization, association, and dissociation, respectively:

\[\begin{split}(\mathbf{K}_{ij})_{rs} &= \begin{cases} \frac{1}{\Delta E_r} \int_{E_r - \Delta E_r/2}^{E_r + \Delta E_r/2} k_{ij}(E) \, dE & r = s \\ 0 & r \ne s \end{cases} \\ (\mathbf{F}_{in})_{rs} &= \begin{cases} \frac{1}{\Delta E_r} \int_{E_r - \Delta E_r/2}^{E_r + \Delta E_r/2} f_{in}(E) \, dE & r = s \\ 0 & r \ne s \end{cases} \\ (\mathbf{g}_{ni})_r &= \frac{1}{\Delta E_r} \int_{E_r - \Delta E_r/2}^{E_r + \Delta E_r/2} g_{ni}(E) \, dE\end{split}\]

The matrices \(\mathbf{M}_i\) represent the collisional transfer probabilities minus the rates of reactive loss to other isomers and to reactants and products:

\[\begin{split}(\mathbf{M}_i)_{rs} = \begin{cases} \omega_i \left[ P_i(E_r, E_r) - 1 \right] - \sum_{j \ne i}^{N_\mathrm{isom}} k_{ij}(E_r) - \sum_{n=1}^{N_\mathrm{reac} + N_\mathrm{prod}} g_{ni}(E_r) & r = s \\ \omega_i P_i(E_r, E_s) & r \ne s \end{cases}\end{split}\]

The scalars \(h_n\) are simply the total rate coefficient for loss of reactant channel \(n\) due to chemical reactions:

\[h_n = - \sum_{i=1}^{N_\mathrm{isom}} \sum_{r=1}^{N_\mathrm{grains}} y_{n\mathrm{B}} f_{in}(E_r) b_n(E_r)\]

The interested reader is referred to any of a variety of other sources for alternative presentations, of which an illustrative sampling is given here [Gilbert1990] [Baer1996] [Holbrook1996] [Forst2003] [Pilling2003].

[Gilbert1990]R. G. Gilbert and S. C. Smith. Theory of Unimolecular and Recombination Reactions. Blackwell Sci. (1990).
[Baer1996]T. Baer and W. L. Hase. Unimolecular Reaction Dynamics. Oxford University Press (1996).
[Holbrook1996]K. A. Holbrook, M. J. Pilling, and S. H. Robertson. Unimolecular Reactions. Second Edition. John Wiley and Sons (1996).
[Forst2003]W. Forst. Unimolecular Reactions: A Concise Introduction. Cambridge University Press (2003).
[Pilling2003]M. J. Pilling and S. H. Robertson. Annu. Rev. Phys. Chem. 54, p. 245-275 (2003). doi:10.1146/annurev.physchem.54.011002.103822