  A gas-liquid type of phase transition is found based on the particle dynamics
on radius-$R$ circle in which the coordinate appears as the angle-variable of
1D XY-model. Due to the specific appearance of compact-space radius (volume) in
the present interpretation of XY-model, the ground-state develops a minimum at
some critical radius, leading to the multi-valued Gibbs energy similar to
systems with first-order phase transition.
