Similarity of relaxation in supercooled liquids and interacting arrays of oscillators

Dielectric relaxation and dynamic light scattering of small molecule glass-forming liquids invariably show that the fractional exponent ${\beta }_{\alpha }$ of the Kohlrausch-Williams-Watts correlation function, $\exp \left[-\right(t/{\tau }_{\alpha }{)}^{{\beta }_{\alpha }}],$ used to fit the α-relaxation data is temperature dependent, increasing towards the value of unity as temperature is raised and the relaxation time τ decreased. Comparing different glass formers, another property is the existence of a correlation between the value of $\left(1\mathrm{}-{\beta }_{\alpha }\right)$ at the glass temperature, $T_g,$ and the $T_g$-scaled temperature dependence of ${\tau }_{\alpha }.$ We analyze a system of interacting arrays of globally coupled nonlinear oscillators. Each array has its oscillators coupled among themselves with a coupling strength K. The coupling between arrays is characterized by the interarray coupling strength $K^{\prime }.$ The decay of the phase coherence r for each array is slowed down by the interarray coupling and its time dependence is well approximated at sufficiently long times by $\exp \left[-\right(t/\tau {)}^{\beta }].$ For a fixed $K^{\prime },$ on increasing K the results exhibit a decrease of τ and a concomitant increase of β, similar to the properties of dielectric relaxation and dynamic light scattering of glass-forming liquids on increasing temperature. For each $K^{\prime }$ we define $K_g$ to be the value of K at which τ is equal to an arbitrarily chosen long time. We find that $\beta {(K}_g)$ is correlated with the $K_g$-scaled K dependence of τ. The results obtained in this manner at various fixed values of $K^{\prime }$ reproduce the relaxation properties and temperature dependencies of strong, intermediate, and fragile glass-forming liquids.