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 ξ_{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 ξ_{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[η] where η is the viscosity of the fluid.