, we then have an estimate for the probability flux from $B_\beta$ into $B_\alpha$: $\pi_{\beta} \nu_{\beta,\alpha}$. Consistently, the probability that the trajectory enters cell $B_\alpha$ by coming from $B_\beta$ is simply: $$\label{eq:probflux}
  \PP_{\partial B_{\beta} \cap \partial B_{\alpha}}=
  \frac{\pi_{\beta} \nu_{\beta,\alpha}}
  {\sum_{\beta'\neq\alpha} \pi_{\beta'} \nu_{\beta',\alpha} }, 
  \quad (\beta\neq \alpha),$$ if $B_\alpha$ and $B_\beta$ have a common edge and $\PP_{\partial
  B_{\beta} \cap \partial B_{\alpha}}=0$ otherwise. This expression gives us the desired probability to pick the edge $\partial B_{\beta}
\cap \partial B_{\alpha}$ for re-entry into $B_\alpha$.

Summarizing, the algorithm to perform simulations restricted inside the cells for systems at nonequilibrium steady state is as follows:

-   Denoting by $\zz_{\alpha}(t)\in B_\alpha$ the current state of the replica in cell $B_\alpha$, let $\zz_{\alpha}^\star$ be the state of the system produced from $\zz_{\alpha}(t)$ after one timestep $\mathit{\Delta} t$ by a standard (i.e. unrestricted) integrator for the system (e.g. velocity-Verlet if the system is a molecular dynamics (MD) system). If $\zz^\star_{\alpha}\in
      B_{\alpha}$, set $$\label{eq:reject}
        \zz_{\alpha}(t+\mathit{\Delta} t) = \zz^\star_{\alpha}.$$ Otherwise, if $\zz^\star_{\alpha}\in B_{\beta}$ with $\beta \neq \alpha$:

    -   Store the point $\zz_{\alpha}^\star$ in a databank of entry points from $B_{\alpha}$ into $B_{\beta}$. In other words, assuming that the databank contains already $m$ points $\zz_{\alpha,\beta}^{k}$ with $k=1,2,\ldots,m$, set $\zz_{\alpha,\beta}^{m+1}=\zz_{\alpha}^\star$.

    -   Update 