KMC algorithm¶

• Form and maintain a list of all possible rates $r_j$ in system

  $$r_j\;\mbox{for all}\;\; j=1,2,\ldots,M$$

  $M$ – number of possible events

• Calculate the cummulative function

  $$R_m=\sum_{j=1}^{m}r_j, \;\;\mbox{for all}\;\; m=1,2,\ldots,M$$

• Generate random number $\rho_1\in[0,1)$

• Select event $i$

  $$\begin{split}R_{i-1}\leq \rho_1 R_M < R_i\end{split}$$
  

• Carry out event $i$

• Generate random number $\rho_2\in(0,1)$

• Compute time increment $\Delta t$

  $$\Delta t=-\frac{ln(\rho_2)}{R_M}$$

• Update time: $t=t+\Delta t$

• Update list of events and goto next step