Phase transition for 1-D Ising model with competing interactions

A Gibbs measure is a mathematical idealization of an equilibrium state of a physical system which consists of a very large number of interacting components. In the language of Probability Theory, a Gibbs measure is simply the distribution of a stochastic process which, instead of being indexed by the time, is parametrized by the sites of a spatial lattice, and has the special feature of admitting prescribed versions of the conditional distributions with respect to the configurations outside finite regions. Then the physical phenomenon of phase transition should be reflected by the non-uniqueness of the Gibbs measures for fixed configurations outside finite regions. As shown by Ising for 1-D homogeneous Ising model there does not exist effect of phase transition.
In this presentation we consider 1-D Ising model with competing nearest-neighbour and next-nearest-neighbour interactions and show that its phase diagram contain paramagnetic, ferromagnetic, antiferromagnetic and modulated phases.
Also we prove that on the set of ferromagnetic phases one can reach a phase transition.