Explicit, spontaneous and dynamical breakdown of the time reversal symmetr

The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial, external manner. When they are incorporated in the dynamics by means of a generalized $\epsilon$-prescription then two different, less obvious symmetry breaking mechanisms are identified, a spontaneous symmetry breaking in closed dynamics and a dynamical symmetry breaking in effective dynamics, both resulting from the non-uniform convergence as $\epsilon\to0$. The energy balance equation is derived for harmonic systems and the soft irreversibility is proven for stationary, stable initial conditions. Two time arrows are constructed, one points away in time from the initial conditions and the other distinguishes the cause from its effect. It is argued that the effective theories do not support runaway solutions and can be acausal.