Concept of the Energy-Dependent Temperature for Direct Langevin Dynamics Simulations of Rare Events in Systems with Arbitrary High Energy Barriers

We suggest an algorithm which allows single-stage direct Langevin dynamics simulations of transitions over arbitrarily high energy barriers. For this purpose, we propose a concept of the energy-dependent temperature (EDT): near the energy minima this temperature is high, but it tends toward room temperature for energies approaching the barrier value. In the resulting algorithm simulation time required for the computation of the escape rate over the barrier does not increase with barrier height. Switching times computed via our EDT algorithm agree very well with those obtained with the forward flux sampling (FFS). As the simulation time required by our method does not increase with the energy barrier, we achieve a very large speed-up compared even to our strongly optimized version of FFS (and all other multistage algorithms). In addition, our approach is free from the instability occurring in all multistage “climbing” methods where a product of a large number of transition probabilities between the interfaces must be computed.

Figure 1

Energy-dependent temperature (1) for $\mathrm{\Delta }E=18k_{B}T$; $a_{\mathrm{lrg}}=4$, $b_{\mathrm{cool}}=7$.

Figure 2

Markov chain and energy landscape.

Figure 3

Total transition probabilities $w_{i\to i+1}$ for FFS (open triangles) and MCH (closed circles) methods for the constant $T=300\mathrm{K}$ (a) and EDT given by Eq. (1) (b); here $\mathrm{\Delta }E/k_{B}T=38$ and $\delta E/k_{B}T=1$.

Figure 4

Ratio of probabilities $w_{i\to i+1}^{\mathrm{EDT}}/w_{i\to i+1}^{\mathrm{CT}}$ for FFS (open triangles) and MCH (full circles) methods as function of the interface energy.

Figure 5

Switching times computed by FFS (blue circles) and EDT (red crosses) (a) and the same times divided by the analytical result from [27] (b); excellent agreement between FFS and EDT is clearly demonstrated.

Figure 6

Computer times (a) and speed-up of the EDT method versus FFS. In both methods, LD with the constant time step $\mathrm{\Delta }t=0.001$ was used to achieve the accuracy of 5% for ${\tau }_{\mathrm{sw}}$.