# Renormalization Group for non-equilibrium

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For equilibrium/ground state systems, a (Wilson) renormalization group transformation produces a series of systems (flow of Hamiltonians/couplings $H_{\Lambda}$ where $\Lambda$ is the cut-off) such that long-wave/asymptotic behaviour of $H_{\Lambda}$ is the same as of $H_{\Lambda'}$ after rescaling by $\Lambda/\Lambda'$. The idea of this definition implies an exact starting point for RG formalisms, with technical details varying between the fields and approximation methods. (For examples, see arXiv:1012.5604 and Wikipedia article).

Now, for non-equilibrium condensed-matter systems there is research direction aiming at generalization of the RG approach to a steady state, e.g., a voltage-biased strongly interacting quatum dot (or Kondo impuryity). For examples, see arXiv:0902.1446 and related references.

I would like to understand the conceptional foundations for the non-equlibrium RG.

What is the definition of an RG transofrmation in a non-equilibirum, steady state ensemble?

I see a problem in the fact that the non-equilibirum desnity matrix which is used to define the problem is not a function of the Hamiltonian alone, thus it is not clear to me how is the effect of the change in the cut-off is split between the Hamiltonian (running couplings) and the density matrix (renormaltizaton of the boundary/external conditions?)

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asked Sep 17, 2011
retagged Mar 7, 2014
Something I know about! I'm on a phone right now, but I'll just leave a reference and expand on an answer later: http://m.iopscience.iop.org/1751-8121/40/9/002/

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Looks refreshingly interesting, haven't seen this applied to non-equilibrium quantum transport problems.

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@genneth: Please, answer before the bounty falls away...

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## 1 Answer

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This is less ambitious than your question (general non-equilibrium states): Near equilibrium correlation functions are described by hydrodynamic theories with stochastic forces, for example the famous models A-J of Hohenberg and Halperin (Reviews of Modern Physics 49, 435 (1977)). In these models I can use the standard RG technology of integrating out short range modes and get running coupling constants. This is known as the "dynamic RG" or sometimes "mode-coupling" theory. The most important result is the critical scaling of transport coefficients (thermal conductivity, sound attenuation, etc.) near second order phase transitions. There have also been attempts to write down RG equations for the CTP (a.k.a. Schwinger-Keldysh) effective action, see for example Dalvit, Mazzitelli, "Exact CTP renormalization group equation for the coarse grained effective action", Phys. Rev. D54, 6338 (1996), arXiv:hep-th/9605024.

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answered Nov 9, 2011 by (705 points)

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