  Cold atoms, loaded into an optical lattice with double-well sites, are
considered. Pseudospin representation for an effective Hamiltonian is derived.
The system in equilibrium displays two phases, ordered and disordered. The
second-order phase transition between the phases can be driven either by
temperature or by changing the system parameters. Collective pseudospin
excitations have a gap disappearing at the phase-transition point. Dynamics of
atoms is studied, when they are loaded into the lattice in an initially
nonequilibrium state. It is shown that the temporal evolution of atoms,
contrary to their equilibrium thermodynamics, cannot be described in the
mean-field approximation, since it results in a structurally unstable dynamical
system, but a more accurate description is necessary taking account of
attenuation effects.
