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Highly supercooled liquids with soft-core potentials are studied viamolecular dynamics simulations in two and three dimensions in quiescent andsheared conditions.We may define bonds between neighboring particle pairsunambiguously owing to the sharpness of the first peak of the pair correlationfunctions. Upon structural rearrangements, they break collectively in the formof clusters whose sizes grow with lowering the temperature $T$. The bond lifetime $\tau_b$, which depends on $T$ and the shear rate $\gdot$, is on the orderof the usual structural or $\alpha$ relaxation time $\tau_{\alpha}$ in weakshear $\gdot \tau_{\alpha} \ll 1$, while it decreases as $1/\gdot$ in strongshear $\gdot\tau_{\alpha} \gg 1$ due to shear-induced cage breakage.Accumulated broken bonds in a time interval ($\sim 0.05\tau_b$) closelyresemble the critical fluctuations of Ising spin systems. For example, theirstructure factor is well fitted to the Ornstein-Zernike form, which yields thecorrelation length $\xi$ representing the maximum size of the clusters composedof broken bonds. We also find a dynamical scaling relation, $\tau_b \sim\xi^{z}$, valid for any $T$ and $\gdot$ with $z=4$ in two dimensions and $z=2$in three dimensions. The viscosity is of order $\tau_b$ for any $T$ and$\gdot$, so marked shear-thinning behavior emerges. The shear stress is closeto a limiting stress in a wide shear region. We also examine motion of taggedparticles in shear in three dimensions. The diffusion constant is found to beof order $\tau_b^{-\nu}$ with $\nu=0.75 \sim 0.8$ for any $T$ and $\gdot$, soit is much enhanced in strong shear compared with its value at zero shear. Thisindicates breakdown of the Einstein-Stokes relation in accord with experiments.Some possible experiments are also proposed.Comment: 20pages (including figures

Dynamics of highly supercooled liquids: Heterogeneity, rheology, and diffusion