Transition Path Sampling - Transition Interface Sampling

Transition Interface Sampling

The TPS rate constant calculation can be improved in a variation of the method called Transition interface sampling (TIS). In this method the transition region is divided in subregions using interfaces. The first interface defines state A and the last state B. The interfaces are not physical interfaces but hypersurfaces in the phase space.

The rate constant can be viewed as a flux through these interfaces. The rate k_AB is the flux of trajectories starting before the first interface and going through the last interface. Being a rare event, the flux is very small and practically impossible to compute with a direct simulation. However, using the other interfaces between the states, one can rewrite the flux in terms of transition probabilities between interfaces

where P_A(i + 1|i) is the probability for trajectories, coming from state A and crossing interface i, to reach interface i + 1. Here interface 0 defines state A and interface n defines state B. The factor Φ_1,0 is the flux through the interface closest to A. By making this interface close enough, the quantity can be computed with a standard simulation, as the crossing event through this interface is not a rare event any more.

Remarkably, in the formula above there is no Markov assumption of independent transition probabilities. The quantities P_A(i + 1|i) carry a subscript A to indicate that the probabilities are all dependent on the history of the path, all the way from when it left A. These probabilities can be computed with a path sampling simulation using the TPS shooting move. A path crossing interface i is perturbed and a new path is shot. If the path still starts from A and crosses interface i, is accepted. The probability P_A(i + 1|i) follows from the ratio of the number of paths that reach interface i + 1 to the total number of paths in the ensemble.

Theoretical considerations show that TIS computations are at least twice as fast as TPS, and computer experiments have shown that the TIS rate constant can converge up to 10 times faster. A reason for this is due to TIS using paths of adjustable length and on average shorter than TPS. Also, TPS relies on the correlation function C(t), computed by summation of positive and negative terms due to recrossings. TIS instead computes the rate as an effective positive flux, the quantity k_AB is directly computed as an average of only positive terms contributing to the interface transition probabilities.