Universality and criticality of a second-order granular solid-liquid-like phase transition

Abstract

We experimentally study the critical properties of the non-equilibrium solid-liquid-like transition that takes place in vibrated granular matter. The critical dynamics is characterized by the coupling of the density field with the bond-orientational order parameter $Q_4$, which measures the degree of local crystallization. Two setups are compared, which present the transition at different critical accelerations as a a result of modifying the energy dissipation parameters. In both setups five independent critical exponents are measured, associated to different properties of $Q_4$: the correlation length, relaxation time, vanishing wavenumber limit (static susceptibility), the hydrodynamic regime of the pair correlation function, and the amplitude of the order parameter. The respective critical exponents agree in both setups and are given by $ν_{\perp} = 1$, $ν_{\parallel} = 2$, $γ= 1$, $η\approx 0.6 - 0.67$, and $β=1/2$, whereas the dynamical critical exponent is $z = ν_{\parallel}/ν_{\perp} = 2$. The agreement on five exponents is an exigent test for the universality of the transition. Thus, while dissipation is strictly necessary to form the crystal, the path the system undergoes towards the phase separation is part of a well defined universality class. In fact, the local order shows critical properties while density does not. Being the later conserved, the appropriate model that couples both is model C in the Hohenberg and Halperin classification. The measured exponents are in accord with the non-equilibrium extension to model C if we assume that $α$, the exponent associated in equilibrium to the specific heat divergence but with no counterpart in this non-equilibrium experiment, vanishes.