Changes of graph structure of transition probability matrices indicate the slowest kinetic relaxations

Graphs of the most probable transitions for a transition probability matrix, $e^{\tau K}$, i.e., the time evolution matrix of the transition rate matrix $K$ over a finite time interval $\tau$, are considered. We study how the graph structures of the most probable transitions change as functions of $\tau$, thereby elucidating that a kinetic threshold ${\tau }_g$ for the graph structures exists. Namely, for $\tau <{\tau }_g$, the number of connected graph components is constant. In contrast, for $\tau \ge {\tau }_g$, recombinations of most probable transitions over the connected graph components occur multiple times, which introduce drastic changes into the graph structures. Using an illustrative multifunnel model, we show that the recombination patterns indicate the existence of the eigenvalues and eigenvectors of the slowest relaxation modes quite precisely. We also devise an evaluation formula that enables us to correct the values of eigenvalues with high accuracy from the data of merging processes. We show that the graph-based method is valid for a wide range of kinetic systems with degenerate, as well as nondegenerate, relaxation rates.

Figure 1

Saddle connectivity graph for a four-funnel model [37]. Vertical upward lines at $i=1,2,\cdots ,48$ starting at $E_i$ represent states of $i$. Horizontal lines at $E=E_{ij}$ connecting $i$ and $j$ represent saddles $ij$. Most probable transitions $j\to i$ are also shown by (red) arrows from $j$ to $i$. (See the text.)

Figure 2

Eigenvectors ${\mathit{v}}_2$, ${\mathit{v}}_3$, and ${\mathit{v}}_4$ of transition rate matrix $K$ for the four-funnel model of Fig. 1 at $\beta =5$ are plotted in (a), (b), and (c), respectively. In each plot, the equilibrium, ${\mathit{v}}_1$, is also plotted using circles connected with a dashed line for comparison. Here, ${\mathit{v}}_1$ is normalized as ${\sum }_{i=1}^n{\left({\mathit{v}}_1\right)}_i=1$, and ${\mathit{v}}_k$ ($k\ge 2$) are scaled such that the components satisfying ${\left({\mathit{v}}_k\right)}_i>0$ approximately agree with ${\left({\mathit{v}}_1\right)}_i$.

Figure 3

Transition graph for the four-funnel model of Fig. 1, where all possible transitions between the saddles of $ij$ are represented by the edges connecting vertices $i$ and $j$.

Figure 4

Most probable path graphs for (a) $K$ and $T\left(\tau \right)$ with $\tau =0.0001$ (both are the same graph) and for $T\left(\tau \right)$ with (b) $\tau =0.1$, (c) $\tau =1$, (d) $\tau =4.197$. Each graph of (a)–(d) has four connected graph components, which we call the metabasin. (See the text.)

Figure 5

Most probable path graphs of $T\left(\tau \right)$ for (a) $\tau =4.198$, (b) $\tau =10.34$, (c) $\tau =10.35$, (d) $\tau =27.75$, and (e) $\tau =27.76$, which show the important recombinations of most probable transitions that lead the merging processes of metabasins.

Figure 6

Schematic illustration of the changes of most probable transitions as a function of $\tau$. Suppose the initial probability distribution is the local equilibrium in the rightmost basin, which is represented by the blue filled bell-shaped curve in each figure. The most probable transitions from the rightmost state at various times of (a) $\tau ={\tau }_a$, (b) ${\tau }_b$, and (c) ${\tau }_c$ (${\tau }_a<{\tau }_b<{\tau }_c$) are indicated by curved arrows.

Figure 7

For the evaluation of ${\lambda }_2$, ${\left[T\left(\tau \right)\right]}_{j_a,j_0}$ and ${\left[T\left(\tau \right)\right]}_{j_b,j_0}$ with $j_0=23$, $j_a=39$, and $j_b=14$ are plotted by blue and orange lines, respectively, as functions of $\tau$. The intersection at $\tau ={\tau }_2=27.76$ corresponds to the merging process: ${\mathrm{MB}}_{\{1,2\}}$ and ${\mathrm{MB}}_{\{3,4\}}\to {\mathrm{MB}}_{\left\{\right\{1,2\},\{3,4\left\}\right\}}$. The local equilibration time ${\tau }_a$, ${\left({\mathbf{p}}_a\right)}_{j_a}$, ${\left({\mathbf{p}}_a\right)}_{j_b}$, ${\left({\mathbf{p}}_b\right)}_{j_a}$, and ${\left({\mathbf{p}}_b\right)}_{j_b}$, which are necessary for the evaluation of Eq. (15), are shown in this figure.

Figure 8

Eigenvectors of transition rate matrix $K^{″}$ for the modified four-funnel model of Eq. (21): (a) ${\mathit{v}}_4^{\prime }$, (b) ${\mathit{v}}_3^{\prime }$, and (c) ${\mathit{v}}_2^{\prime }$. In each plot, the equilibrium, ${\mathit{v}}_1$, is also plotted using circles connected with a dashed line for comparison. Here, ${\mathit{v}}_k^{\prime }$ ($k\ge 2$) are scaled such that the components satisfying ${\left({\mathit{v}}_k^{\prime }\right)}_i<0$ approximately agree with $-{\left({\mathit{v}}_1\right)}_i$.