  We investigate the monotonicity of the renormalization group (RG) flow from
the perspectives of nonequilibrium thermodynamics. Applying the
Martin-Siggia-Rose formalism to the Wilsonian RG transformation, we incorporate
the RG flow equations manifestly in an effective action, where all coupling
functions are dynamically promoted. As a result, we obtain an emergent
holographic dual effective field theory, where an extra dimension appears from
the Wilsonian RG transformation. We observe that Becchi-Rouet-Stora-Tyutin
(BRST)-type transformations play an important role in the bulk effective
action, which give rise to novel Ward identities for correlation functions
between the renormalized coupling fields. As generalized
fluctuation-dissipation theorems in the semiclassical nonequilibrium dynamics
can be understood from the Ward identities of such BRST symmetries, we find
essentially the same principle for the RG flow in the holographic dual
effective field theory. Furthermore, we discuss how these ``nonequilibrium work
identities" can be related to the monotonicity of the RG flow, for example, the
$c-theorem$. In particular, we introduce an entropy functional for the
dynamical coupling field and show that the production rate of the total entropy
functional is always positive, indicating the irreversibility of the RG flow.
