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Regular version of the site

Article

Information theory and renormalization group flows

Physica A: Statistical Mechanics and its Applications. 2012. Vol. 391. No. 1-2. P. 62-77.
Apenko S.M.

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 entropy is an extensive
quantity and 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.