Information theory and renormalization group flows

TL;DR

This paper explores a novel information-theoretic approach to understanding renormalization group flows, focusing on entropy and mutual information to analyze flow irreversibility and system changes.

Contribution

It introduces an information-theoretic framework for RG flows, emphasizing entropy and mutual information, and investigates their roles in flow irreversibility.

Findings

Average information loss is a conditional entropy of integrated-out variables.

Positivity of entropy decrease does not guarantee RG irreversibility.

Mutual information can serve as a Lyapunov function indicating flow irreversibility.

Abstract

We present a possible approach to the study of the renormalization group (RG) flow based entirely on the information theory. The average information loss under a single step of Wilsonian RG transformation is evaluated as a conditional entropy of the fast variables, which are integrated out, when the slow ones are held fixed. Its positivity results in the monotonic decrease of the informational entropy under renormalization. This, however, does not necessarily imply the irreversibility of the RG flow, because the entropy explicitly depends on the total number of degrees of freedom, which is reduced. Only some size-independent additive part of the entropy could possibly provide the required Lyapunov function. We also introduce a mutual information of fast and slow variables as probably a more adequate quantity to represent the changes in the system under renormalization and evaluate it for some simple systems. It is shown that for certain real space decimation transformations the positivity of the mutual information directly leads to the monotonic growth of the entropy per lattice site along the RG flow and hence to its irreversibility.