Phase transitions out of equilibrium: Domain formation and growth.

We study the dynamics of phase transitions out of equilibrium in weakly coupled scalar field theories. We consider the case in which there is a rapid supercooling from an initial symmetric phase in thermal equilibrium at temperature Ti > Tc to a final state at low temperature Tf ≈ 0. In particular we study the formation and growth of correlated domains out of equilibrium. It is shown that the dynamics of the process of domain formation and growth (spinodal decomposition) cannot be studied in perturbation theory, and a non-perturbative self-consistent Hartree approximation is used to study the long time evolution. We find in weakly coupled theories that the size of domains grow at long times as ξD(t) ≈ √