We study the response of generating functionals to a variation of parameters
(couplings) in equilibrium systems i.e. in quantum field theory (QFT) and
equilibrium statistical mechanics. These parameters can be either physical ones
such as coupling constants or artificial ones which are intentionally
introduced such as the renormalization scale in field theories. We first derive
general functional flow equations for the generating functional
(grand-canonical potential) W[J] of the connected diagrams. Then, we obtain
functional flow equations for the one-particle irreducible (1PI) vertex
functional (canonical potential) Γ[ϕ] by performing the Legendre
transformation. By taking the functional derivatives of the flow equations, we
can obtain an infinite hierarchical equations for the 1PI vertices. We also
point out that a Callan-Symanzik type equation holds among the vertices when
partition function is invariant under some changes of the parameters. After
discussing general aspects of parameter response, we apply our formalism to
several examples and reproduce the well-known functional flow equations. Our
response theory provides us a systematic and general way to obtain various
functional flow equations in equilibrium systems.Comment: 24 page