Dynamical non-linear susceptibility of the quenched and annealed frustrated lattice gas models

In this paper we study the 3D frustrated lattice gas model in the quenched and annealed versions. In the first case, the dynamical non-linear susceptibility grows monotonically as a function of time, until reaching a plateau that corresponds to the static value. The static non-linear susceptibility diverges at some density, signaling the presence of a thermodynamical transition. In the annealed version, where the disorder is allowed to evolve in time with a suitable kinetic constraint, the thermodynamics of the model is trivial, and the static non-linear susceptibility does not show any singularity. Nevertheless, the model shows a maximum in the dynamical non-linear susceptibility at a characteristic value of the time. Approaching the density corresponding to the singularity of the quenched model, both the maximum and the characteristic time diverge. We conclude that the critical behavior of the dynamical susceptibility in the annealed model is related to the divergence of the static susceptibility in the quenched case. This suggests a similar mechanism also in supercooled glass-forming liquids, where an analogous behavior in the dynamical non-linear susceptibility is observed.