problems at the beginning of the simulations, when the statistics for $N_{\alpha,\beta}$ is not accurate enough. To overcome this problem we set $\pi_{\alpha}=cst$ $\forall \alpha=1,\ldots,N$ at the beginning when statistics is insufficient to solve Eq. . Using more educated guesses is possible too. Also, it is worth noting that $\nu_{\alpha,\beta}$ and $\pi_{\alpha}$ can be monitored on-the-fly to assess their convergence as a function of the length of the simulation, and the actual simulations of the replicas in each cell can be performed in parallel; only the re-entry events require communication. Finally, we should stress that the procedure above relies on the ability to count the successive points at which a trajectory crosses the edges of the cells. This may lead to difficulties if the dynamics is governed by a stochastic differential equation, in which case these crossing points may form a fractal set. It leads to no difficulty, however, if some components of the trajectory are smooth and the cells are defined accordingly. We will come back to this issue later in the illustrative example section.

#### Rate calculation. {#rate-calculation. .unnumbered}

Let us now come to the question of how to compute the reaction rate between a reactant and product 