Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid
cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G in mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.
Self-propelled particle (SPP) systems are intrinsically out of equilibrium systems, where each individual particle converts energy into work to move in a dissipative medium. When interacting through a velocity alignment mechanism, and the medium acts
as a momentum sink, even momentum is not conserved. In this scenario, a mapping into an equilibrium system seems unlikely. Here, we show that an entropy functional can be derived for SPPs with velocity alignment and density-dependent speed, at least in the (orientationally) disordered phase. This non-trivial result has important physical consequences. The study of the entropy functional reveals that the system can undergo phase separation before the orientational-order phase transition known to occur in SPP systems with velocity alignment.Moreover, we indicate that the spinodal line is a function of the alignment sensitivity and show that density fluctuations as well as the critical spatial diffusion, that leads to phase separation, dramatically increase as the orientational-order transition is approached.
In self-gravitating stars, two dimensional or geophysical flows and in plasmas, long range interactions imply a lack of additivity for the energy; as a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many clai
ms, the equilibrium statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical approach to equilibrium and non equilibrium statistical mechanics for these systems, starting from first principles. We emphasize recent and new results, mainly a classification of equilibrium phase transitions, new unobserved equilibrium phase transition, and out of equilibrium phase transitions. We briefly discuss what we consider as challenges in this field.
