  Sometimes the dynamics of a physical system is described by non-Hamiltonian
equations of motion, and additionally, the system is characterized by
long-range interactions. A concrete example is that of particles interacting
with light as encountered in free-electron laser and cold-atom experiments. In
this work, we study the relaxation dynamics to non-Hamiltonian systems, more
precisely, to systems with interactions of both Hamiltonian and non-Hamiltonian
origin. Our model consists of $N$ globally-coupled particles moving on a circle
of unit radius; the model is one-dimensional. We show that in the infinite-size
limit, the dynamics, similarly to the Hamiltonian case, is described by the
Vlasov equation. In the Hamiltonian case, the system eventually reaches an
equilibrium state, even though one has to wait for a long time diverging with
$N$ for this to happen. By contrast, in the non-Hamiltonian case, there is no
equilibrium state that the system is expected to reach eventually. We
characterize this state with its average magnetization. We find that the
relaxation dynamics depends strongly on the relative weight of the Hamiltonian
and non-Hamiltonian contributions to the interaction. When the non-Hamiltonian
part is predominant, the magnetization attains a vanishing value, suggesting
that the system does not sustain states with constant magnetization, either
stationary or rotating. On the other hand, when the Hamiltonian part is
predominant, the magnetization presents long-lived strong oscillations, for
which we provide a heuristic explanation. Furthermore, we find that the
finite-size corrections are much more pronounced than those in the Hamiltonian
case; we justify this by showing that the Lenard-Balescu equation, which gives
leading-order corrections to the Vlasov equation, does not vanish, contrary to
what occurs in one-dimensional Hamiltonian long-range systems.
