processes increase the ionization rate they can increase the rate at which the charge balance adjusts to sudden changes in the electron temperature.

Equilibration Timescales {#subsec:timescales}
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One way to quantify the effect of EIMI processes on a dynamic plasma is to calculate the timescale for the charge balance to reach equilibrium following a sudden change in $T$. These timescales have been used, for example, in the analysis of spectra from supernova remnants [@Masai:ASS:1984; @Hughes:ApJ:1985; @Smith:ApJ:2010].

The method for calculating the ionization timescales has been described by @Masai:ASS:1984. Equation (\[eq:dydt\]) can be written in the form $$\frac{d\vec{y}}{dt}=n_{\mathrm{e}}\mathbf{A}(T)\vec{y}.
\label{eq:eqtime1}$$ For constant $T$ and $n_{\mathrm{e}}$, the solution to this equation is $\vec{y}(t) = \vec{y}_{0}\exp{\left[n_{\mathrm{e}}t \mathbf{A}(T)\right]}$. The exponential of a matrix is defined in terms of a Taylor expansion, which involves powers of the matrix $\mathbf{A}$. This is simplified by diagonalizing the matrix by finding its eigenvalues and eigenvectors. Doing so, one finds that the solution to Equation (\[eq:eqtime1\]) can be written as $$\vec{y}(t)=\mathbf{S} \left[ \begin{array}{ccc}
\exp{\left(\lambda_{1}n_{\mathrm{e}}t\right)} & 0 & \cdots \\
0 & \exp{\left(\lambda_{j}n_{\mathrm{e}}t\right)} & 0 \cdots \\
\vdots & 0 & \ddots 
\end{array}\right]\mathbf{S}^{-1}\vec{y}_{0},
\label{eq:eqtime2}$$ where $\mathbf{S}$ is the matrix in which each $j$th column is the eigenvector corresponding to the eigenvalue 