Diverging length scale of the inhomogeneous mode-coupling theory: a numerical investigation

Biroli 's extension of the standard mode-coupling theory to inhomogeneous equilibrium states [G. Biroli, J. P. Bouchaud, K. Miyazaki, and D. R. Reichman, Phys. Rev. Lett. 97, 195701 (2006)] allowed them to identify a characteristic length scale that diverges upon approaching the mode-coupling transition. We present a numerical investigation of this length scale. To this end we derive and numerically solve equations of motion for coefficients in the small q expansion of the dynamic susceptibility chiq(k;t) that describes the change of the system's dynamics due to an external inhomogeneous potential. We study the dependence of the characteristic length scale on time, wave vector, and on the distance from the mode-coupling transition. We verify scaling predictions of Biroli In addition, we find that the numerical value of the diverging length scale qualitatively agrees with lengths obtained from four-point correlation functions. We show that the diverging length scale has very weak k dependence, which contrasts with very strong k dependence of the q-->0 limit of the susceptibility, chiq=0(k;t) . Finally, we compare the diverging length obtained from the small q expansion to that resulting from an isotropic approximation applied to the equation of motion for the dynamic susceptibility chiq(k;t) .