# Local nonequilibrium fluctuation-dissipation relations

New paper uploaded to the arXiv:

B. Altaner, M. Polettini and M. Esposito, Local nonequilibrium fluctuation-dissipation relationsarXiv:1604.08832

Quite sadly, it’s my first paper as a second author. This is the beginning of the end I guess. No, seriously, I really think that with this work (and with some follow-ups in production, stay tuned…) we hit something quite important. As about myself, I see this as the point where all of my own expertize finally converges to one complete and consistent body of knowledge that is genuinely new and interesting.

This work started off after one intensive week of discussion in front of a blackboard and of Mathematica notebooks back in January 2015, when Bernhard visited Luxembourg on occasion of the conference Luxembourg Out-Of-Equilibrium, which says a lot on the importance of personal contact in scientific collaboration. We wanted to dig into a fact that some authors had observed in simple specific models, but that was somehow buried and certainly not fully understood.

The most important paradigm of statistical mechanics close to equilibrium is “Fluctuation-Dissipation” (FD), which roughly states that a system perturbed out of equilibrium by an external disturbance will behave as if it spontaneously fluctuated. By “equilibrium” it is meant that no observable current is flowing through the system. After a small conjugate force $x$ is exerted, some current starts flowing. The sensitivity of that current to the perturbation, on average, is identical to (one half) the variance of the current. In a formula:

$\frac{\partial}{\partial x} \langle \jmath \rangle_{\mathrm{eq.}} = \frac{1}{2} \langle \jmath^2 \rangle_{\mathrm{eq.}}$

Now, what if that particular current we are observing is just one of many currents that are flowing, at the same time, in that particular system? That is, instead of considering “global equilibrium”, we just consider “local equilibrium”, whereby some current vanishes in the whirling of other currents, and we perturb it and measure its spontaneous fluctuations. Will the above equation still hold, by replacing the equilibrium average $\langle \;\cdot\; \rangle_{\mathrm{eq.}}$ with a nonequilibrium average $\langle \;\cdot\; \rangle_{\mathrm{noneq.}}$?

And the answer is YES! Well, of course, given a lot of premises that I don’t really want to discuss here. What is most appealing to me, nevertheless, is this fact that we somehow managed to generalize an important “global” result into a “local” one, in a way that stirs my wildest fantasies about possible parallels with such global-to-local principles such as the equivalence principle of General Relativity, or the gauge principle in Quantum Field theory… So, there’s a lot to be excited about.