  We critically examine the applicability of the effective potential within
dynamical situations and find, in short, that the answer is negative. An
important caveat of the use of an effective potential in dynamical equations of
motion is an explicit violation of energy conservation.
  An \emph{adiabatic} effective potential is introduced in a consistent
quasi-static approximation, and its narrow regime of validity is discussed. Two
ubiquitous instances in which even the adiabatic effective potential is not
valid in dynamics are studied in detail: parametric amplification in the case
of oscillating mean fields, and spinodal instabilities associated with
spontaneous symmetry breaking. In both cases profuse particle production is
directly linked to the failure of the effective potential to describe the
dynamics. We introduce a consistent, renormalized, energy conserving dynamical
framework that is amenable to numerical implementation. Energy conservation
leads to the emergence of asymptotic highly excited, entangled stationary
states from the dynamical evolution. As a corollary, decoherence via dephasing
of the density matrix in the adiabatic basis is argued to lead to an emergent
entropy, formally equivalent to the entanglement entropy. The results suggest
novel characterization of asymptotic equilibrium states in terms of order
parameter vs. energy density.
