I just finished reading an impressive article from 2003 entitled "Observing Brownian motion in vibration-fluidized granular matter". In the article's beginning, the authors established a simple question: can linear response theory describe a nonequilibrium thermodynamic system? This question is important because systems that are not in thermodynamic equilibrium are both difficult to analyze and serve as appropriate models for most natural phenomena. However, very powerful mathematical tools exist for systems that are in equilibrium, so it would be very convenient if their mathematical formalism could be extended to nonequilibrium cases.

In particular, the authors explore whether a torsion oscillator driven by a "heat bath" of randomly vibrating glass beads can be described by the fluctuation-dissipation theorem (FDT). The FDT is arguably the hallmark of linear response theory and describes the return to equilibrium of a many-body system subjected to a small perturbation. (A small perturbation means that the response is linearly proportional to the perturbation.)

A canonical example where the FDT finds use is in describing the motion of ions in a fluid between two plates of a capacitor after a voltage difference has been applied to the plates. Prior to the voltage being applied, the motion of the ions is erratic and Brownian. A long time after the constant voltage is applied, they move with an average velocity that is proportional to the electric field between the plates, the proportionality constant being called the mobility. The FDT describes the very short times immediately after the field is applied. It also links the noise (the random movement of the ions in equilibrium) to the mobility of the ions.

Returning to the article, the authors find that the motion of the oscillator is described by the FDT so long as an "effective" temperature is adopted. Effective temperatures are very appealing as analytical tools for describing nonequilibrium systems because they are very, very simple modifications to the FDT. Simply replace T with T_eff and you're done.

As Cugliandolo points out, to be a good thermodynamic descriptor, an effective temperature should be measurable by a thermometer. I'm not sure what the thermometer is in this system, but I suspect that it's the torsion oscillator itself. Furthermore, she stresses that not all nonequilibrium phenomena are describable by effective temperatures. It seems that one requires coupling between fast processes and slower observables, among other requirements. The beauty of the Nature article is that the authors not only confirmed this point (which seems to currently be an area of contention), but did so convincingly by measuring the relevant quantities directly and under a number of different conditions.

I'm not sure whether the effective temperature is a universal property of nonequilibrium systems; I'm inclined to say it is not. Hopefully more experiments like this one will be done that may further elucidate the current maze of theoretical papers on the topic.