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Kinetic monte Carlo can be used to simulate Master equation for Markov processes.For simple 1D random walk the master equation is - $$\frac{\partial P(n,t)}{\partial t}=pP(n-1,t)+qP(n+1,t)-(p+q)P(n,t)$$ where $p$ and $q$ are forward hopping rate and backward hopping rates respectively.
$P(n,t)$ is the probability of random walker to be at position $n$ at time $t$

This equation can be solved analytically but we can also solve it by Kinetic Monte Carlo method.I want some materials to follow or some guidence for the simulation . It would be really helpful if someone can provide some materials,notes or book chapters to look .

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You can calculate the position of a single walker at time $t$ using kMC. To calculate the probability of a walker being found in position n at time $t$, you will need to run many kMC simulations in order to generate $P(n,t)$.

To calculate the position of a single walker using kMC at time $t$, you will need to calculate the total rate ($=p+q$), from that calculate a time step ($\tau$) between hopping events, and then select whether to execute a forward or a backwards hop by selecting one at random, weighted according to their average rates.

Specifically for your case, $\tau = (1/(p+q)) \ln(1/r_1)$, where $r_1$ is a random number between 0 and 1. To select between executing forward or reverse rate, generate a random number $r_2$ between 0 and $(p+q)$ (inclusive of 0, excluding $p+q$). If $r_2$ is less than $p$, execute a forward hop. If $r_2$ is between $p$ and $p+q$, execute a backwards hop.

As far as general resources go, you can find information about interarrival times in Poisson distributions in a statistics book, for example, chapter 5 (specifically, section 5.3.3 in the Ninth Edition) of Sheldon Ross's classic text "Introduction to Probability Models," which should let you see how the timestep tau is derived. I would expect choosing $p$ or $q$ according to their magnitudes is fairly obvious, but you could look, for example, in the 1976 and 1977 Daniel Gillespie papers on chemical kinetics for information on this as well.

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