We discuss classical lattice-gas models without periodic ground states
. To prove the thermodynamic stability of such systems we have to cons
truct nonperiodic equilibrium states which are small perturbations of
ground states. We review partial results in this direction and discuss
possibilities of constructing a model with nonperiodic equilibrium st
ates. It would be a microscopic (toy) model of a quasicrystal, a syste
m of interacting particles with a stable nonperiodic equilibrium state
.