Transition path sampling (TPS) is a rare-event sampling method used in computer simulations of rare events: physical or chemical transitions of a system from one stable state to another that occur too rarely to be observed on a computer timescale.
For example, an initially unfolded protein will vibrate for a long time in an open-string configuration before undergoing a transition and fold on itself.
All the relevant information can then be extracted from the ensemble, such as the reaction mechanism, the transition states, and the rate constants.
One of the path times is chosen at random, the momenta p are modified slightly into p + δp, where δp is a random perturbation consistent with system constraints, e.g. conservation of energy and linear and angular momentum.
A new trajectory is then simulated from this point, both backward and forward in time until one of the states is reached.
The TPS rate constant calculation can be improved in a variation of the method called Transition interface sampling (TIS).
Being a rare event, the flux is very small and practically impossible to compute with a direct simulation.
The quantities PA(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 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.
TPS/TIS as normally implemented can be acceptable for non-equilibrium calculations provided that the interfacial fluxes are time-independent (stationary).