Rates (pele.rates)

This module contains tools to quickly (and exactly) compute transition rates, first passage times and and commitor probabilities in a transition network. The resulting rates are exact, in the sense of Kinetic Monte Carlo, but the analysis can be orders of magnitude faster than doing a Kinetic Monte Carlo

This module can also be found as an independent package at https://github.com/js850/kmc_rates

Description of the method

The rates are computed using the New Graph Transformation (NGT) method of described in the paper

Calculating rate constants and committor probabilities for transition networks by graph transformation David Wales (2009) J. Chem. Phys., 130, 204111 http://dx.doi.org/10.1063/1.3133782

The method uses a graph renormalization method (renormalization in the sense of renormalization group theory) to compute exact Kinetic Monte Carlo rates and first passage probabilities from a reactant group A to a product group B. Each node u has an attribute tau_u which is the waiting time at that node. Each edge u -> v has an associated transition probability and P_uv. An important feature of this algorithm is that each node has a loop edge pointing back to itself and associated probability P_uu which gives the self-transitio probability. In the typical case the self-transition probabilities will all be zero initially, but will take non zero values after renormalization. The transition probabilities always satisfy sum_v P_uv = 1.

graph transformation

The algorithm is most easily described if we first assume that A, and B each contain only one node (called a, and b). In the algorithm, nodes are iteratively removed from the graph until the only two remaining nodes are a and b. Upon removing node x, in order to preserve the transition times and probabilities, the properties of the neighbors of x are all updated. For each neighbor, u, of x, the transition times are updated according to

tau_u -> tau_u + P_ux * tau_x / (1 - P_xx)

Similarly, for each pair, u and v, of neighbors of x the transition probabilities are updated according to

P_uv -> P_uv + P_ux * P_xv / (1 - P_xx)

Note that the self-transition probabilities P_uu are also updated according to the above equation.

Once the graph is reduced to only the two nodes, a, and b, the probability P_ab is interpreted as the commitor probability from a to b. That is, the probability that a trajectory starting at a will end up at b before returning to a. Similarly, the mean first passage time from a to b is simply tau_a / P_ab. Note that because the probabilies sum to 1 the mean first passage time can also be written tau_a / (1-P_aa). The transtion rate from a to b is simply the inverse of the mean first passage time. The rates and probabilites from b -> a are read from the resuling graph in the same way. The above interpretations are exact in the sense that a Kinetic Monte Carlo simulation will give the same result.

if B has more than one element

If there is more than one element in B the calculation of rates from a -> B is nearly as simple. Following the same procedure described above, all nodes except those in A or B are iteratively removed. The commitor probability from a to B is then the sum over the transition probabilities from a to b for each element b in B. This can also be written as

1 - P_aa

The mean first passage time from a to B is given by

T_aB = tau_a / (1 - P_aa)

if A has more than one element

In this, the most general case, when both A and B have more than one element, the transition rate from A to B must be computed as an average over the inverse mean first passage time for each element a in A. That is

k_AB = average( 1 / T_aB )

The computation is done in two phases. In the first phase the intermediate nodes (those not in A or in B) are all removed from the graph. In the second phase we first make a backup copy of the graph. Then for each node a in A we remove from the graph all nodes in A (except a). This allows us to compute commitor probabilities and mean first passage times (T_aB) from a to B as described in the preceding section.

If the nodes a are not all equally likely to be occupied, then the above average can be a weighted average where each node is weighted according to its equilibrium occupation probabilities.

The rates B -> A can be computed in a similar manner