Can classical systems exhibit "strong coupling"? Does the concept of strong coupling mean anything in a classical setting? If strong coupling means just an inability to apply perturbative methods to the Hamiltonian, then obviously yes, we can provide examples of "classical" systems that cannot be handled perturbatively, such as granular materials, dusty plasmas, etc.
I have only rudimentary QFT knowledge (and strong coupling is originally a QFT concept), but my intuition from my background in spectroscopy tells me that there is more to the term strong coupling than just the need for nonperturbative methods. The example I am most deeply familiar with is Forster energy transfer or FRET in biophysics. You're in weak coupling if the non-radiative rate is small compared to the radiative rate. But you're in strong coupling if the non-radiative rate is so large that you can get energy migration, where multiple energy transfer processes occur in series. In that case, the excitation is delocalized and you have to treat the network of interacting atoms as a single entity.  I was under the understanding that delocalization was an essential feature of the term "strong coupling".
Now, back to the original question, to what degree is it appropriate to use the term strong-coupling in many-body classical systems, such as granular materials, dusty plasmas, and dense colloidal glasses? These systems are classical in the sense that there is complete decoherence, and hard sphere potentials are the rule. In one sense, I want to say yes, the kinetic energy terms carry little information and most of the energy is stored in potential energy terms, but I am still hung up on the issue of delocalization. Can anyone help set me straight?
 A: In my experience (complex, spatially extended systems) we talk about coupling of oscillators.  People usually talk about weak coupling because it allows you to treat many oscillators more simply:
http://www.scholarpedia.org/article/Phase_model#Weakly_coupled_oscillators
Above, a system of coupled two-dimensional oscillators has been transformed into a system of single-dimensional oscillators (using their phase angle instead of their "x,y" position).  This essentially assumes that amplitude of the individuals in the ensembles is insignificant.
If the coupling were "strong", the amplitude of a member would be an important factor in the state of its coupling neighbors.
In classical systems, usually you do neighborly coupling (which is often treated by a diffusion term).  The closest you come to something like "decoherence" is that you can have "anomalous diffusion".  And to do that you, now have fractional derivatives (usually diffusion is a second derivative across the spatial aspect of the network... or in discrete systems a second difference across neighbors... but with anomalous diffusion, you now have the 2.4 derivative or whatnot).  And this causes non-local phase coupling, which "might" be comparable to decoherence if you stretch your imagination (not sure if it's formally a mathematical analog).
A: Not knowing the theory of granular materials, dusty plasmas or colloidal glasses, I am going to stick my neck out and say that what you describe as hard-potentials sound like you are referring to effective theories which are valid only within certain parameters.
Generally, strong coupling occurs along with de-localized excitation phenomena, but not always so. The only test of strong coupling is for the non-perturbatively-renormalized coupling to be large. I have not come across any classical theories with such phenomena - let me know if you know one.
edit: Actually I have come across some classical strongly coupled systems in Kogut's "An introduction to lattice gauge theory"
A: It seems like every field has a slightly different definition of strong coupling. I can imagine this comes from different quantities being experimentally accessible.
I'm from the field of quantum optics. If people talk about strong coupling between light and matter they mean that the energy exchange between light and matter is faster than all dissipation in the system. As an example, in Mark Fox – Quantum Optics: An Introduction chapter 10.2 the condition for strong coupling between the light in a cavity and an atom is fulfilled when the coupling constant $g_0$ (which is half the energy exchange rate between the atom and a photon in the cavity) is much larger than the energy loss rate of the cavity $\kappa$ and the energy loss rate of the atom $\gamma$.
One can define similar conditions for other systems, too. As you said, in FRET it would mean that energy is transferred faster between the molecules than it is dissipated in the environment.
Nothing stops us from using the same definition on classical systems as well. One typical example students usually have to solve is two pendulums connected by a spring. The spring constant $k$ determines the rate at which energy is transfered between the two pendulums. And if this process is faster than the damping they are strongly coupled.

