# Perturbatively inaccessible intermediate or large coupling phases of classical non-quantum theories

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Are there classical (ie. non-quantum) theories with inaccessible coupling regions, which can be shown to exist from extrapolating their RG flows derived from weak coupling ranges? Since these intermediate coupling ranges can neither be studied through weakly perturbing exact theories nor through effective theories of bound states, how else are they studied?

Background: For quantum systems like quantum phase transitions, QCD etc, much has been written about inaccessible coupling ranges (especially the intermediate coupling regions in BCS->BEC, quark->hadron etc where neither small coupling perturbation nor large coupling effective theories apply). These regions are indirectly shown to exist through extrapolating RG flow equation derived for weak coupling ranges. I want to know if such phases have been known to exist in non-quantum systems.

PS: I had asked a similar question (which is still open/unaswered), but that was purely in the context of classical turbulence: Similarities between laminar-turbulence transition and others like BCS-BEC crossover, quark-hadron transition etc

edited Nov 11, 2014

Clarifying what I meant by classical theories: I am using 'classical theory' in its traditional form ie. a non-quantum theory.

Vladimir Kalitvianski wrote (which was deleted for some reason?!):

My first idea may be silly, but let us consider a case of two attracting particles. When the kinetic energy of their relative motion is everywhere higher than the local attractive potential, such particles can only scatter. We may consider it as a regime of weak coupling. We can solve the scattering problem by the perturbation theory and obtain some series in powers of effective coupling.

However, if the kinetic energy of their relative motion is lower than the local attractive potential, the particles get coupled in some sort of a limited relative motion (a bound state). We can consider it as a regime of strong coupling.

You want to know whether we can/cannot extract information about a bound state from a perturbative (or exact) solution of a scattering problem?

No, its not silly! This is the essence of what happens in the strongly coupled quantum systems. But the bound state limits (hadrons of partons->hadron, BEC of BCS->BEC etc) are impossible (as yet) to derive from their weak coupling limits, but can at least be studied in terms of an effective theory of bound states. But the intermediate coupling ranges can neither be studied by effective theories nor by weakly coupled exact theories.

Similarly, I am interested in the intermediate coupling ranges of classical theories. To be specific, I would like to know if there are well known classical systems where such intermediate couplings appear, and how are they handled.

PS: Perhaps I should not have mentioned 'intermediate and large', but in many (quantum) systems the intermediate coupling ranges over several orders of magnitude before a bound state transition can occur.

Dear crackjack, I can share my experience of successfully using perturbative series in the intermediate and strong coupling ranges, but not in the Q/A section. My experience may have a limited significance for you, because I do not know what exactly you are interested in.

I have cleared some off-topic meta comments between Vladimir and myself. @VladimirKalitvianski If you have anything else to say, please continue it on my wall or send me a private message.

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