Nonequilibrium thermodynamics perspectives for the monotonicity of the renormalization group flow

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.