# First passage time for Kinetic Monte Carlo model

## Main Question or Discussion Point

Hi all!
I have a problem on finding the first passage time for kinetic monte carlo model.
Suppose I have a linear kinetic model for n states:
S1<->S2<->S3<->...<->Sn
where all the rate constants k_ij for transition between any two states are known. Is there any general way to find out the analytical solution for the (probability density) of the first passage time from state 1 to any state i?
Many thanks!

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One approach is to consider a modified version of the process where state i is an absorbing state (i.e. no transitions out of state i) which will have the same first passage time distribution for state i. Since i is absorbing, the cdf of the first passage time will equal the probability that the process is in state i at time t, which can be solve in terms of a matrix exponential.

One approach is to consider a modified version of the process where state i is an absorbing state (i.e. no transitions out of state i) which will have the same first passage time distribution for state i. Since i is absorbing, the cdf of the first passage time will equal the probability that the process is in state i at time t, which can be solve in terms of a matrix exponential.
Thanks for your comment. So I try $P(t)=P(0)e^{Tt}$, where T is the transition matrix with state i being the absorption state, t is the time and P(t) is the probability vector of different states with norm equal 1. I solved the equation in matlab numerically and I observed some strange things:
1. the norm of P(t) keep increasing in time without bound, do I have to renormalized it at each time? $\overline{P(t)}=P(t)/norm(P(t))$
2. if I do the renormalization, P(t) become stationary after some time t, but the probability of the absorption state is not 1 in stationary solution. Since there is only one state keep absorbing, I would expect at the end I should certainly find the system in the absorbing state. What is wrong with my solution?

Renormalisation shouldn't be necessary and if the transition matrix is written correctly, it shouldn't have any positive eigenvalues. Perhaps show your solution for n=i=2?

My system is actually more complicated than a linear markov chain, in the sense that the transition matrix is not tridiagonal. But I think theory still applies. What happened is the transition matrix for original system has all its eigenvalues <0 but by doing the tricks that modifying one of the state to be absorbing state, one of the eigenvalues now become positive. Do I miss something important or does the trick just not apply?

Ok, I have been stupid about it. Since I have changed one state to absorbing state, the total change of probability for all the states is not zero now, $\sum\frac{dP_i}{dt}≠0$. Hence the probability vector keeps increasing due to one and only one positive eigenvalue. But it is still strange that the time it takes for the probability of being in absorbing state i increases to 1 (so the cdf of first passage time reaches 1 as well) so quickly. The result does not agree with the simulation result from stochastic model of the same system....

I can't say much without seeing more details but the transition matrix of the modified process should be the same except with a row of zeros for the absorbing state - that shouldn't break the law of conservation of probability.