# Nonlinear response in complex matter /1

Taking notes from this workshop, in a quite different area than my own…

I. Procaccia, Mechanical Yield in Amorphous Solids is a First Order Phase Transition. Charbonneau, Kurchan, Parisi, Urbani & Zamponi (CKPUZ) recently provided a theory of “jamming” of hard spheres in finite dimensions is available, mean field. It predicts exponents in infinite dimensions. Amazingly, this same exponents appear in d=2,3, to very high accuracy. Why?! Procaccia starts from a complete different perspective. [Majumdar and Behringer, Contact force measurements and stress-induced anisotropy in granular materials]: determining the profile of the F forces among disks with C contacts (which give constraints): $\sum_j f_{i,j} + F_i^{ext.}$. Underconstrained, so how do you solve this problem? They introduce an incidence matrix $M$ and then this looks like a Kirchhoff’s Current Law. Matrix $M$ might contain tangential an normal forces, so it is a $2N \times 2C$ matrix. What about the frictionless matrix? Jamming: number of unknowns = number of constraint, matrix $M$ is square (and is invertible, not always the case… what do these cases correspond to?). He now addresses the pressed case $2C > 2M$. He wants to add some unknowns. He takes the actual law for the forces: $f_{ij} = \sum_{k} a_k (r_{ij} - r_0)^k$ and now he has a lot of extra unknowns that describe the interactions (because now we are not talking about contact forces, but forces-at-a-distance… don’t know how this is actually relevant to jamming at all). Next he addresses thermal systems, giving some intuitions and plots as about why emergent forces emerge and one cannot describe the system by binary interactions, but entropic effects involve higher-order interactions (could this be related somehow to the appearance of long-range interactions when applying renormalisation group methods in statistical mechanics?). No idea how this relates to the work mentioned in the beginning of the talk. I’m not convinced by everything, but the calculations in the first part are simple and I might start taking a look at this kind of problems with one question in mind: could there be a role analogous to Kirchhoff’s Loop Law have something to say?

T. Truskett, On the formation of equilibrium gels via a macroscopic bond limitation. Gels have the most diverse “non equilibrium” behaviour, because it’s difficult to obtain gels in equilibrium conditions. From an engineering perspective, non-equilibrium conditions does not grant the same reproducibility, stability and design. So he will talk about equilibrium stuff (uff…). Despite the american style of presentation, I couldn’t go beyond the second slide.

G. Biroli, Breakdown of elasticity in amorphous solids. Second time the paper by Charbonneau et al. is mentioned. Infinite-dimensional hard spheres when compressed go through two phase transitions, one into a glass, and one into something else. The second transition was described by Gardner (1985) and Gross, Kanter, Sompolinsky into a spin-glass model. It’s interesting that the plots in CKPUZ have a nice hysteresis cycle, and as usual those might be used to produce a (glassy?) engine.

T. Voigtmann, Nonlinear response in crystallization. Finally a nonequilibrium talk. A strong shear flow. Work with Sven Dorosz and Tanja Schilling. Hard spheres, being compressed with some pressure rate, and use Brownian dynamics simulations. If you compress very quickly you don’t end with a perfect crystal because you also freeze defects. You can consider the work distribution which satisfies the Fluctuation Theorems. He shows that the work is typically much higher than the equilibrium free energies. It’s interesting this fact that long-time = perfect crystal, short-time=imperfect crystal, again might be related to the “efficiency” of a thermodynamic cycle (I could talk about efficiency and hysteresis cycles as a final slide… e.g. what is the role of the  imperfections in lowering efficiency?). Next they think systems under a constant shear from the outside, which is a truly nonequilibrium situation – and I wonder whether it could be relevant to apply two different independent thermodynamic forces on top of such systems. Unfortunately most of these talks are inventory of effects that one can only appreciate having an inside-view on the field. Someone asked during the question about the fact that the Jarzynski relation needs a final equilibrium state – clarify this (clarify the fact that the Fluctuation Theorems hold for quite general protocols and for nonequilibrium steady states).

R. Jack, Absence of dissipation in trajectory of biased ensembles – consequences for nonlinear response. Using MaxEnt methods to predict non-linear response. According to Jaynes, Maximum Caliber was meant to predict properties of non-equilibrium states by relating them to fluctuations of corresponding equilibrium systems (like Onsager’s regression hypothesis, mention in my own talk). He relies on the work of Pressè et al. The conclusion of this talk is that this is not right. Let $J(X)$ be the accumulated shear strain over a whole trajectory $X$. Least unlikely ensemble of trajectories with mean shear rate is

$P_{bias} = P_{eq.}(X) e^{-\nu J(X)} / Z$

Now let $\mathbb{T}$ be time reversal and $\mathbb{P}$ be plane reflection (we are considering a sheared fluid). Then

$P_{pias}(\mathbb{PT}X) = P(X)$

This biasing then is probably not the same as the tilting. And now he claims that the heat flow (which he did not define) $F(X) = - F(\mathbb{PT}X)$. And therefore there must be absence of dissipation. Despite the fact that he had a force acting on the system in his figures, and that it does not appear that he also reversed the force (being the dissipation a product of forces by currents, this might be the missing “$\mathbb{C}$” transformation that makes everything well-preserved and consistent). In other words, also the protocol, and not only the trajectory, has to change sign.