Path Summation Formulation of the Master Equation

Markovian dynamics, modeled by the kinetic master equation, has wide ranging applications in chemistry, physics, and biology. We derive an exact expression for the probability of a Markovian path in discrete state space for an arbitrary number of states and path length. The total probability of paths repeatedly visiting a set of states can be explicitly summed. The transition probability between states can be expressed as a sum over all possible paths connecting the states. The derived path probabilities satisfy the fluctuation theorem. The paths can be the starting point for a path space Monte Carlo procedure which can serve as an alternative algorithm to analyze pathways in a complex reaction network.

Figure 1

Repeated visits to the same set of states are common in a path. The total probability of these paths can be computed. Here states $i$ through $i+R$ are repeatedly visited. The sum of these paths is given in Eq. (18).

Figure 2

Path sampling can be implemented by shooting from an existing path and comparing the probabilities of paths. Here, a new path (dotted arrows) is generated from an old path (solid arrows) by growing 2 segments from a shared state. These paths connect regions $A$ and $B$. Metropolis Monte Carlo acceptance criterion will allow the paths to diffuse through path space according to their probabilities.