  We study the deuteron production via the deuteron pion and nucleon catalysis
reactions, $\pi p n \leftrightarrow \pi d$ and $N p n \leftrightarrow N d$, by
employing stochastic multi-particle reactions in the hadronic transport
approach SMASH for the first time. This is an improvement compared to previous
studies, which introduced an artificial fake resonance $d'$ to simulate these
$3 \leftrightarrow 2$ reactions as a chain of $2\leftrightarrow 2$ reactions.
The derivation of the stochastic criterion for multi-particle reactions is
presented in a comprehensive fashion and its implementation is tested against
an analytic expression for the scattering rate and the equilibrating particle
yields in box calculations. We then study Au + Au collisions at
$\sqrt{s_{\mathrm{NN}}} = 7.7$ GeV, where we find that multi-particle
collisions substantially reduce the time required for deuterons to reach
partial chemical equilibrium with nucleons. Subsequently, the final yield of
$d$ is practically independent from the number of $d$ at particlization,
confirming the results of previous studies. The mean transverse momentum and
the integrated elliptic flow as a function of centrality are rather insensitive
to the exact realization of the $2\leftrightarrow 3$ reactions.
