## Kinetic Monte Carlo The difference between the previous Monte Carlo and kinetic one is, we now have a rate content. $\Gamma_i =\Gamma_{0i} e^{-\beta\Delta E}$ >[!Note] > Here, the Boltzmann factor is something controls the rate. $\Gamma_{0i}$ is the effective attempt frequency, unit $[s^{-1}]$, for the event of kind $i$. Although it is theoretically calculatable, but extremely difficult. We typically use the standard value, $10^{13}$. And some kinetic Mount Cardo's rate related. We have to identify different process. Say A, B, … that could occur at current, i.e., $i^{th}$ configuration, and we can compute a total rate. $R=\sum_i \Gamma_i$ these $\Gamma$ get different value and computed as $\Gamma_0e^{-\beta\Delta E}$ ![[Drawing 2023-09-24 17.53.03.excalidraw.svg]] Then we'll guess which one will happen and how long it will take, we generate two random number from $(0,1]$, $\rho_1$, $\rho_2$. >[!Notice] > Note that there will definitely be an event that will happen. So we don't consider probability of happening. Use $\rho_1$ to determine which will happen, by $\sum_{i=1}^{l-1} \Gamma_i < \rho_1R < \sum_{i=1}^l \Gamma _i$ >[!Note] > This is just like throwing something in the long belt above and "hit" one event. Then execute the event identified and update the time with $\Delta t = - \frac{\ln\rho_2}{ R}$. From the time that evolves according to Poisson distribution, $P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}$, and for $k=1$ just $\lambda e^{-\lambda}$. With time updated, we get its configuration. >[!Info] >More on wiki https://en.wikipedia.org/wiki/Kinetic_Monte_Carlo