Dimensional dependence of the Stokes-Einstein relation and its violation

We generalize to higher spatial dimensions the Stokes–Einstein relation (SER) as well as the leading correction to diffusivity in finite systems with periodic boundary conditions, and validate these results with numerical simulations. We then investigate the evolution of the SER violation with dimension in simple hard sphere glass formers. The analysis suggests that the SER violation disappears around dimension du = 8, above which SER is not violated. The critical exponent associated with the violation appears to evolve linearly in 8 − d, below d = 8, as predicted by Biroli and Bouchaud [J. Phys.: Cond. Matt. 19, 205101 (2007)], but the linear coefficient is not consistent with the prediction. The SER violation with d establishes a new benchmark for theory, and its complete description remains an open problem.