Nonequilibrium statistics of barrier crossings with competing pathways

Abstract

Many biological, chemical, and physical systems are underpinned by stochastic transitions between equilibrium states in a potential energy. Here, we consider such transitions in a minimal model with two possible competing pathways, both starting from a local potential energy minimum and eventually finding the global minimum. There is competition between the distance to travel in state space and the height of the potential energy barriers to be surmounted, for the transition to occur. One pathway has a higher energy barrier to go over, but requires traversing a shorter distance, whereas the other pathway has a lower potential barrier but it is substantially further away in configuration space. The most likely pathway taken depends on the available time for the transition process; when only a relatively short time is available, the most likely path is the one over the higher barrier. We find that upon varying temperature the overall most likely pathway can switch from one to the other. We calculate the statistics of where the barrier crossing occurs and the distribution of times taken to reach the potential minimum. Interestingly, while the configuration space statistics is complex, the time of arrival statistics is rather simple, having an exponential probability density over most of the time range. Taken together, our results show that empirically observed rates in nonequilibrium systems should not be used to infer barrier heights.

Figures (11)

• Figure 1: (a) Plot of the surface described by the potential $\phi(x,y)$ in Eq. \ref{['eq:potential']}, together with its projection on the $xy$-plane. (b) Contour plot of the potential $\phi(x,y)$, with the stationary points indicated.
• Figure 2: Plots of $\phi(x,y=0)$, i.e. cuts through the potential in Eq. \ref{['eq:potential']} for various temperatures (as indicated), displayed in each case as the solid line. We also present the results from Brownian dynamics (BD) simulations, sampling the density distribution $\rho(x,y=0)$, which are displayed (using $\times$ symbols) by plotting the quantity $-k_BT \ln \rho(x,y=0)+$constant [c.f. Eq. \ref{['eq:ln_rho']}]. Each is the result from a single simulation of total time 2000$\tau_B$.
• Figure 3: Probability density $\rho_{FC}(x_c)$ over the position $x_c$ where the system finally crosses the $x$-axis, having been initiated at point $A$ in Fig. \ref{['fig:contour_plot']}(b), calculated using BD simulations. Each run is terminated when it reaches the vicinity of the global minimum, point $E$. Panels (a)--(d) present results for the four different temperatures indicated and compare with the equilibrium density result in Eq. \ref{['eq:epd']}, as the solid green line in each case.
• Figure 4: The probability $\hat{P}_L$, defined in Eq. \ref{['eq:hatP']}, plotted as a function of temperature, calculated using BD simulations. $\hat{P}_L$ is the probability for the system initiated at point $A$ [see Fig \ref{['fig:contour_plot']}(b)] to move to the global minimum at $E$, while crossing the $x$-axis to the left of the origin. We also display $P_L$, the estimate based on the equilibrium density distribution, defined in Eq. \ref{['eq:P_eq']}. Of course, the corresponding right-crossing probabilities are $\hat{P}_R=1-\hat{P}_L$ and $P_R=1-P_L$.
• Figure 5: Arrival time distributions (histograms) for the four different temperatures indicated. In each case, the red symbols are the distribution irrespective of the value of $x_c$, the point on the $x$-axis where the system crossed, while the blue and orange distributions are for $x_c<0$ and $x_c>0$, respectively. In the insets, we plot $\ln p(t)$ versus $t$ using the same data as displayed in the main plots, in order to display as a straight line the exponential tails. We also display straight line fits to the tails, i.e. fits to Eq. \ref{['eq:exp_distrib']}.