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 c 1⁄4 1=2 as the critical temperature Tc is approached from below. When Tc 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 Tc 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. 2002 Elsevier Science B.V. All rights reserved. PACS: 64.70.Pf; 05.20.)y