Evolution of inhomogeneous condensates after phase transitions

Using the O(4) linear $\sigma$ model, we address the topic of nonequilibrium relaxation of an inhomogeneous initial configuration due to quantum and thermal fluctuations. The space-time evolution of an inhomogeneous fluctuation of the condensate in the isoscalar channel decaying via the emission of pions in the medium is studied within the context of disoriented chiral condensates. We use out of equilibrium closed time path methods in field theory combined with the amplitude expansion. We give explicit expressions for the asymptotic space-time evolution of an initial inhomogeneous configuration including the contribution of thresholds at zero and nonzero temperature. At nonzero temperature we find new relaxational processes due to thermal cuts that have no counterpart in the homogeneous case. Within the one-loop approximation, we find that the space-time evolution of such inhomogeneous configuration out of equilibrium is effectively described in terms of a rapidity-dependent temperature $T\left(\vartheta \right)=\frac{T}{cosh\left[\vartheta \right]}$ as well as a rapidity-dependent decay rate $\Gamma (\vartheta , T(\vartheta \left)\right)$. This rate is to be interpreted as the production minus absorption rate of pions in the medium and approaches the zero-temperature value at large rapidities. An initial configuration localized on a bounded region spreads and decays in spherical waves with slower relaxational dynamics at large rapidity.