# Generalized Kinetic Monte Carlo Framework for Organic Electronics
## 2 Kinetic Monte Carlo Method  
Monte Carlo (MC) simulations refer to a broad class of numerical algorithms utilizing random numbers to solve real-world problems. The first Monte Carlo methods were developed during the 1940s within the Manhattan project to study neutron diffusion in fissionable materials. During that time, the Metropolis algorithm [29] originated, which is until today a convenient technique to study equilibrium properties of physical systems. However, the Metropolis algorithm is not capable of modeling the time-dependent evolution of a system. The kinetic Monte Carlo method is a specific MC method that allows propagating a physical system from state to state including the corresponding time information of the propagation steps. Therefore, kMC simulations are often termed simulated experiments because they essentially mimic the explicit evolution of a real-world system. The step by step evolution, where each time-step depends only on the previous one, fulfills the properties of a Markov chain and makes the kMC algorithm a sequential algorithm. kMC methods have gained large popularity in a variety of fields due to their ease of implementation and the increasing computational power of modern computers. The modern version of the kMC method is based on the work of [PERSON] [30] and [PERSON] et al. [31]. For further information for the reader, an illustrative description of the method was provided by [PERSON] [32]. To model a system, one has to identify the spatial dimensions and the time scales on which the processes determining the properties of the system occur. The kMC method commonly is referred to as a simulation technique at the mesoscale, in between molecular dynamics (MD) on the microscopic and hydrodynamic models on the macroscopic scale. In comparison to MD, which typically resolves systems on the timescale on which molecular vibrations and rotations occur (of the order of femtoseconds), the kMC method is based on the concept of coarse-graining the system dynamics. A system is characterized by a set of long-term states with discrete transitions between them (so-called infrequent events); all transitions on smaller timescales are neglected as long as they do not change a long-term state (illustrated in Figure 1a). Coarse-graining allows one to neglect a sufficient amount of (fast) processes and, thus, reach much longer overall simulation times up to seconds. Only transitions between long-term states determine the dynamic evolution of the system. It needs to be carefully considered which processes are implemented in the model and which ones are combined in long-term states. This way, the kMC can accomplish a tradeoff between accuracy and simulation time. For the application of the kMC method, the system to be modeled needs to be characterized by a set of long-time system states \(\{i\}\). These can be, for example, the entirety of positions and energies of all particles currently in the system. For a system being in a certain state \(i\), a set of possible states \(j\) exists to which a system may evolve (illustrated in Figure 1b). Every transition is accompanied by a transition rate \(a_{ij}\) (given in units of \(s^{-1}\)), describing the physical process leading to the transition \(i\to j\). The magnitude of the rates is proportional to their probability of execution, i.e., larger rates (faster processes) are more likely to occur than smaller rates (slower processes). The temporal evolution of discretized transitions between states is phenomenologically described by the master equation [33]:  
\[\frac{\mathrm{d}p_{i}(t)}{\mathrm{d}t}=-\sum_{j}a_{ij}p_{i}(t)+\sum_{j}a_{ji}p_ {j}(t)\;, \tag{1}\]  
where \(p_{i/j}(t)\) gives the probability to find the system in state \(i/j\) at time \(t\). An analytic or numerical solution of the master equation is often not feasible. The kMC method provides a numerical approach to essentially execute the master equation to stochastically propagate the system between its long-term states.