  The existence of fluctuations together with interactions leads to
scale-dependence in the couplings of quantum field theories for the case of
quantum fluctuations, and in the couplings of stochastic systems when the
fluctuations are of thermal or statistical nature. In both cases the effects of
these fluctuations can be accounted for by solutions of the corresponding
renormalization group equations. We show how the renormalization group
equations are intimately connected with the effective action: given the
effective action we can extract the renormalization group equations; given the
renormalization group equations the effects of these fluctuations can be
included in the classical action by using what is known as improved
perturbation theory (wherein the bare parameters appearing in tree-level
expressions are replaced by their scale-dependent running forms). The improved
action can then be used to reconstruct the effective action, up to finite
renormalizations, and gradient terms.
