longer measuring times, the temperature $T_{\rm drop}$ at which the generalized linear compressibility drops is roughly at the mode coupling temperature $T_C=0.303$. Let us define $T_{\rm drop}$ as the temperature at which $\chi_{l}$ has dropped halfway down. For $10^6$ md steps, $T_{\rm drop}\approx 0.33$; for $3\times 10^6$ md steps, $T_{\rm drop}\approx 0.30$; and for $10^7$ md steps, $T_{\rm drop}\approx 0.27$. Thus we are able to stay in equilibrium down to the mode coupling temperature for our longer runs. This is what we would expect when we compare these run times, which are longer than 1 million time steps, to the $\alpha$ relaxation time $\tau$ which is about 1 million time steps at $T=0.29$ which is just below $T_C$. Thus the fact that the drop in the linear generalized compressibility occurs at or very close to the mode coupling $T_C$ is a result of the relaxation times (see Fig. \[fig:mct\_tau\]) becoming comparable to and exceeding the simulation run times as the temperature drops below $T_C$. When this happens, the system falls out of equilibrium and undergoes a kinetic glass transition.

The behavior exhibited by $\chi_l$ can be quantified using a scaling ansatz: $\chi_{l}(t_M,T)=g(\mu=t_M/\tau (T))$, where the characteristic time has the Volgel–Fulcher 