Long-range spatial correlations of particle displacements and the emergence of elasticity

We examine correlations of transverse particle displacements and their relationship to the shear modulus of a glass and the viscosity of a fluid. To this end we use computer simulations to calculate a correlation function of the displacements, $S_4(q;t)$, which is similar to functions used to study heterogeneous dynamics in glass-forming fluids. We show that in the glass the shear modulus can be obtained from the long-time, small-q limit of $S_4(q;t)$. By using scaling arguments, we argue that a four-point correlation length $\xi_4(t)$ grows linearly in time in a glass and grows as $\sqrt{t}$ at long times in a fluid, and we verify these results by analyzing $S_4(q;t)$ obtained from simulations. For a viscoelastic fluid, the simulation results suggest that the crossover to the long-time $\sqrt{t}$ growth of $\xi_4(t)$ occurs at a characteristic decay time of the shear stress autocorrelation function. Using this observation, we show that the amplitude of the long-time $\sqrt{t}$ growth is proportional to $\sqrt{\eta}$ where $\eta$ is the viscosity of the fluid.