# Is there a connection between the fluctuation-dissipation theorem and the Green–Kubo relations?

Is there a connection between the fluctuation-dissipation theorem and the Green–Kubo relations? I have a hard time finding out if there is a relation and what it is, because the fluctuation-dissipation theorem always seems to be stated in another way and for specific cases. Both formula seem to be a statement about how a macroscopic property is determined as an integral over Greens function'esque objects.

What is the exact relation and the hierarchy of the two concepts?

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Why not let ol' Kubo tell you himself? iopscience.iop.org/0034-4885/29/1/306 –  wsc Jan 31 '12 at 14:04

The fluctuation-dissipation theorem is a general principle that comes in many concrete forms. It expresses in each case a way how the spectrum of equilibrium fluctuations can be probed by applying weak external fields.

The Kubo relations are just one specific instance of it.

For example, Reichl derives in her book on statistical physics the F/D theorem in Section 15D from general principles of nonequilibrium thermodynamics, and gives several quantitative applications, before he gives in Section 15H the Kubo relation as the microscopic expression for it.

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her book :-) –  Siva Oct 18 at 19:02
corrected, thanks. –  Arnold Neumaier Oct 26 at 12:07

A Kubo formula is a relation between a transport coefficient (diffusion constants, shear viscosity, etc) and the zero momentum, zero frequency limit of a retarded correlation function. Kubo formulas are derived by matching hydrodynamics to linear response theory. Fluctuation-dissipation (FD) relations connect the correlation function (Wightman function) to the imaginary part of the retarded correlator. These relations are valid for all frequencies, not just $\omega\to 0$. FD relations follow from analyticity and the KMS condition. The FD relations imply that Kubo formulas can be stated in terms of the zero frequency limit of the correlation function.

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In my understanding, these two concepts are essentially the same. In the l.h.s. of a Kubo formula, you ask for a dynamical quantity (e.g., current) in response to an external perturbation (e.g., potential gradient). So you are asking for a coefficient characterizing dissipation. In the r.h.s, you have an equilibrium average of a correlation function, that is, fluctuations. So to use a Kubo formula means to use information about fluctuations to get an answer about dissipation.

In a more technical language, FD theorem is an extra relation between lesser ("fluctuations") and retared ("dissipation/dynamic") components of a Keldysh (non-equilibrium) Green function.

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