  Based on the relativistic Vlasov-Uehling-Uhlenbeck transport model, which
includes relativistic scalar and vector potentials on baryons, we consider a
$N-\Delta-\pi$ system in a box with periodic boundary conditions to study the
effects of energy conservation in particle production and absorption processes
on the equilibrium properties of the system. The density and temperature of the
matter in the box are taken to be similar to the hot dense matter formed in
heavy ion collisions at intermediate energies. We find that to maintain the
equilibrium numbers of $N$, $\Delta$ and $\pi$, which depend on the mean-field
potentials of $N$ and $\Delta$, requires the inclusion of these potentials in
the energy conservation condition that determines the momenta of outgoing
particles after a scattering or decay process. We further find that the baryon
scalar potentials mainly affect the $\Delta$ and pion equilibrium numbers,
while the baryon vector potentials have considerable effects on the effective
charged pion ratio at equilibrium. Our results thus indicate that it is
essential to include in the transport model the effect of potentials in the
energy conservation of a scattering or decay process, which is ignored in most
transport models, for studying pion production in heavy ion collisions.
