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 root 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 root 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 root t growth is proportional to root eta where eta is the viscosity of the fluid.