# Nonlinear response in complex matter /2

Liesbeth Janssen, Activity-induced aging in a topologicaly constrained glass. Active matter can be glassy, and interesting things happen when activity is coupled with topology. She considers a simple model of rods with a repulsive potential and an active force in the direction of the rod’s direction, and look at the overdamped Brownian dynamics on a sphere (but then she says that the dynamics has no noise, I missed something here…(“Brownian dynamics at zero temperature”)). (Again, my fundamental doubt (FD) about active matter and Brownian dynamics stands: can this kind of system can be modelled by SDE’s with external forces, or is active behavior induced by some implicit coarse-graining? See here for more about this). Short rods in low density go alone. Long rods at low density flock together. At high density there’s a lot of alignment. Going back to short rods, at high density, the system locks into a glass but there’s an overall spinning of the spins across the sphere (so there’s a symmetry breaking phenomenon here, because a priori there’s no favourite direction of rotation). The rotation is very stable. We then have the interesting idea of “melting and freezing” the glass by expanding and contracting the sphere. If this “breathing” is not enough to completely melt the configuration, the system reaches a quasi-stable configuration that has the behavior of aging. “Activity” here stands for a nonconservative force, hence I can indeed make the connection to active response (despite the FD). Nice simulations, not sure what the message was.

A. Sharma, Green-Kubo approach to active Brownian particles. Spherically symmetric particles that are self- propelled in one favourite direction. For this reason these systems are out of equilibrium, but still he is looking for equilibrium concepts like an equation of state. He defines an average swim velocity (like a current). In his world, the Green-Kubo approach is to write a Langevin equation of this kind:

$\dot{r}_i = v_0(t) p_i + \gamma^{-1} F + \xi_i, \dot{p}_i = \eta_i \times p_i$

He writes the Fokker-Planck equation and he separates it into equilibrium and a nonequilibrium part. And then he has a term (what is the gradient of the momentum??!). He goes into autocorrelation functions etc. So this is the perfect talk to make contact with my own stuff. Autocorrelation with time-symmetric “active” terms. Funny behavior like getting less from pushing more.

(By the way, all this literature seems to relate to cellular automata, see works by Turchetti etc. the action-reaction principle does not hold. If my FD is resolved, this might open the way to thermodynamic analysis of a self-propelled active Brownian particle; the coarse-graining might be related to forgetting the angle).