Figure 5: “Dynamic scaling” relation between the number of dynamically correlated particles, evaluated by the peak height of ${\chi }_4$, and relaxation timescale, ${\tau }_{\alpha }$, for a number of glass-formers [36], determined using the bound provided by Eq. (7). For all systems, dynamic correlations increase when the glass transition is approached. The full line through the data [36] suggests a crossover from algebraic, ${\chi }_4\sim {\tau }_{\alpha }^z$, to logarithmic, ${\chi }_4\sim \text{exp}({\tau }_{\alpha }^{\psi }$), growth of dynamic correlations with increasing ${\tau }_{\alpha }$.