PDE/statistical mechanics duality: relation between Guerra's interpolated $p$-spin ferromagnets and the Burgers hierarchy

Abstract

We examine the duality relating the equilibrium dynamics of the mean-field $p$-spin ferromagnets at finite size in the Guerra's interpolation scheme and the Burgers hierarchy. In particular, we prove that - for fixed $p$ - the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the $p-1$-th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also obtain the solutions for the $p$-spin ferromagnets at fixed $N$, allowing us to easily generate specific solutions of the corresponding equation in the Burgers hierarchy. Finally, we obtain an effective description of the finite $N$ equilibrium dynamics of the $p=2$ model with some standard tools in PDE side.