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Information theory and renormalization group flows
Statistical Mechanics · Submitted 12 Oct 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present a possible approach to the study of the renormalization group (RG) flow based entirely on the information theory. | contribution 1.00 | 0.88 | 0.08 | hedged possibility 0.94 |
| s1 | 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. | method 0.74 | 0.89 | 0.07 | asserted 1.00 |
| s2 | Its positivity results in the monotonic decrease of the informational entropy under renormalization. | result 0.66 | 0.81 | 0.14 | asserted 1.00 |
| s3 | 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. | interpretation 0.69 | 0.64 | 0.09 | asserted 0.69 |
| s4 | Only some size-independent additive part of the entropy could possibly provide the required Lyapunov function. | interpretation 0.67 | 0.53 | 0.11 | hedged possibility 1.00 |
| s5 | 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. | contribution 0.97 | 0.93 | 0.09 | hedged possibility 0.98 |
| s6 | 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. | result 0.98 | 0.88 | 0.51 | asserted 1.00 |
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The paper
Original 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.
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References
How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it
Cites it as background 4 papers
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In this context, the importance of considering the geometry of the theory space is perhaps epitomized by Zomolodchikov's $C$-theorem [ref] , and the study of information loss along RG flows in the space of couplings has attracted much attention [ref] .
Renormalization group flows from the Hessian geometry of quantum effective actions · Introduction · 0.82
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The task of developing an information theoretic framework for the RG has been attempted by several authors (see, e.g., [ref] for a selection), however, there are still several major remaining obstructions.
Renormalisation as an inference problem · Introduction · 0.92
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possible connections to information theory have been explored in several different approaches [ref] .
Minimizing couplings in renormalization by preserving short-range mutual information · Introduction · 0.99
What it cites · 3 sentences without a confident label
Cites as background 3 papers
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Renormalization group (RG), which is a powerful instrument for analyzing different strongly coupled systems [ref] (see also [ref] for more recent reviews), is usually based upon an appropriate division of the whole set of variables into two subsets: so called “fast” and “slow” variables with subsequent elimination of the fast ones.
An Introduction to the Nonperturbative Renormalization Group · Introduction · 0.99
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Much work have been done in this direction and the $c$-theorem was studied in detail with its possible generalizations to higher dimensions (see e.g. [ref] and references therein and also [ref] for quite recent development).
Towards the F-Theorem: N=2 Field Theories on the Three-Sphere · Introduction · 0.98
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Much work have been done in this direction and the $c$-theorem was studied in detail with its possible generalizations to higher dimensions (see e.g. [ref] and references therein and also [ref] for quite recent development).
On the c-theorem in more than two dimensions · Introduction · 0.98
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