Continuous-time random-walk approach to supercooled liquids: Self-part of the van Hove function and related quantities

From equilibrium molecular dynamics (MD) simulations of a bead-spring model for short-chain glass-forming polymer melts we calculate several quantities characterizing the single-monomer dynamics near the (extrapolated) critical temperature T-c of mode-coupling theory: the mean-square displacement g(0)(t), the non-Gaussian parameter alpha(2)(t) and the self-part of the van Hove function G(s)(r, t) which measures the distribution of monomer displacements r in time t. We also determine these quantities from a continuous-time random walk (CTRW) approach. The CTRW is defined in terms of various probability distributions which we know from previous analysis. Utilizing these distributions the CTRW can be solved numerically and compared to the MD data with no adjustable parameter. The MD results reveal the heterogeneous and non-Gaussian single-particle dynamics of the supercooled melt near T-c. In the time window of the early a relaxation alpha(2)(t) is large and G(s)(r, t) is broad, reflecting the coexistence of monomer displacements that are much smaller ("slow particles") and much larger ("fast particles") than the average at time t, i.e. than r = g(0)(t)(1/2). For large r the tail of G(s)(r, t) is compatible with an exponential decay, as found for many glassy systems. The CTRW can reproduce the spatiotemporal dependence of G(s)(r, t) at a qualitative to semiquantitative level. However, it is not quantitatively accurate in the studied temperature regime, although the agreement with the MD data improves upon cooling. In the early a regime we also analyze the MD results for G(s)(r, t) via the space-time factorization theorem predicted by ideal mode-coupling theory. While we find the factorization to be well satisfied for small r, both above and below T-c, deviations occur for larger r comprising the tail of G(s)(r, t). The CTRW analysis suggests that single-particle "hops" are a contributing factor for these deviations.