# What is kinetic Monte Carlo?¶

Kinetic Monte Carlo (kMC) is a simulation technique that can be used to investigate the kinetics of chemical reactions. Kinetics can be seen as transitions between different chemical states, which obeys the master equation:

$$\dfrac{\text{d}P_\alpha}{\text{d}t} = \sum_\beta W_{\alpha\beta}P_\beta - W_{\beta\alpha}P_\alpha \\$$

where $$\alpha, \beta$$ are the states defined by the site-occupations (e.g. CO on site 1, CO on site 2, site 3 empty ,…) , $$W_\alpha\beta$$ is the transition rate from state $$\beta$$ to state $$\alpha$$, and $$P_\alpha$$ is the probability for being in state $$P_\alpha$$. The equation defines a system of coupled differential equations, with one equation for each $$\alpha$$.

KMC solves this system of equations by randomly generating transitions between states. The transitions are generated by reactive events, which for example can be $$\mathrm{O_2}$$ dissociative adsorption proceeding on sites number 1 and 3, where site 1 and 3 are neighboring sites. The time of occurrence of a reactive event (i) is in MonteCoffee generated according to the first-reaction method:

$$t^\text{occ}_i = t^\text{gen}_i-\dfrac{\text{ln}\,u}{k_i},\quad u \in [0,1[ \\$$

where $$t^\text{occ}_i$$ is the time of occurrence, $$t^\text{gen}_i$$ is the time the event was generated (simulation time), $$k_i$$ is the rate constant of the reaction-step, and $$u$$ is a random uniform deviate.