Theory of non-linear susceptibility and correlation length in glasses and liquids

Within the framework of the effective potential theory of the structural glass transition, we calculate for the p-spin model and for a hard sphere liquid in the hypernetted chain approximation a static non-linear susceptibility related to a four-point density correlation function, and show that it diverges in mean field with exponent γ = 1/2 as the critical temperature T_c is approached from below. When T_c is approached from above, we calculate for the p-spin model a time dependent non-linear susceptibility and show that there is a characteristic time where this susceptibility has a maximum, and that this time grows with decreasing T. This susceptibility diverges as T_c is approached from above, and has key features in common with a generalized susceptibility related to particle displacements, previously introduced to measure correlated particle motion in simulations of glass-forming liquids.