On the possible critical behaviour of a marginally stable stellar disc
  Using hydrodynamic approach, it is shown that the properties of a marginally
stable collisionless stellar disc resemble those of a thermodynamic system
undergoing a gas--liquid phase transition. The maximum in Toomre's stability
diagram, which separates gravitationally stable and unstable states with
respect to axisymmetric perturbations, can be treated as a critical point for
this transition. Static perturbations of stellar density are explored and the
mean perturbation amplitude is considered as the order parameter of the theory.
The disc's state is assumed to change as the disc passes through the critical
point. Since the disc tends to retain hydrostatic equilibrium, structures can
be formed spontaneously, identifiable with a seed spiral structure. A power-law
scaling of the order parameter in the vicinity of the critical point has been
found. The susceptibility and other Landau--Weiss exponents similar to those in
the Van der Waals theory are calculated. The critical behaviour of marginally
stable discs at the initial stage of their evolution occurs in numerical
simulations where snapshots of stellar positions reveal stellar splinters and
crescents diverging from the disc centre. These structures can be a result of
the phase transition. In numerical simulations, these structures eventually
reduce to decaying worm-type features because of the `heating' most likely
resulting from instability of stellar orbits due to resonances. Under
favourable conditions the critical behaviour leading to the establishment of
order in a stellar disc can result in the generation of a spiral structure.
