Nonequilibrium dynamics of symmetry breaking in $\lambda {\Phi }^4$ theory

The time evolution of O$\mathrm{}\left(N\right)\mathrm{}$ symmetric $\lambda ({\Phi }^2{)}^2$ scalar field theory is studied in the large $N$ limit. In this limit the $〈{\mathbf{\Phi }}_i〉$ mean field and two-point correlation function $〈{\mathbf{\Phi }}_i{\mathbf{\Phi }}_j〉$ evolve together as a self-consistent closed Hamiltonian system, characterized by a Gaussian density matrix. The static part of the effective Hamiltonian defines the true effective potential $U_{\mathrm{eff}}$ for configurations far from thermal equilibrium. Numerically solving the time evolution equations for energy densities corresponding to a quench in the unstable spinodal region, we find results quite different from what might be inferred from the equilibrium free energy potential $F$. Typical time evolutions show effectively irreversible energy flow from the coherent mean fields to the quantum fluctuating modes, due to the creation of massless Goldstone bosons near threshold. The plasma frequency and collisionless damping rate of the mean fields are calculated in terms of the particle number density by a linear response analysis and compared with the numerical results. Dephasing of the fluctuations leads also to the growth of an effective entropy and the transition from quantum to classical behavior of the ensemble. In addition to casting some light on fundamental issues of nonequilibrium quantum statistical mechanics, the general framework presented in this work may be applied to a study of the dynamics of second order phase transitions in a wide variety of Landau-Ginsburg systems described by a scalar order parameter.