Calculation of displacement correlation tensor indicating vortical cooperative motion in two-dimensional colloidal liquids

Abstract

As an indicator of cooperative motion in a system of Brownian particles that models two-dimensional colloidal liquids, a displacement correlation tensor is calculated analytically and compared with numerical results. The key idea for the analytical calculation is to relate the displacement correlation tensor, which is a kind of four-point space-time correlation, to the Lagrangian two-time correlation of the deformation gradient tensor. Tensorial treatment of the statistical quantities, including the displacement correlation itself, allows capturing the vortical structure of the cooperative motion. The calculated displacement correlation also implies a negative long-time tail in the velocity autocorrelation, which is a manifestation of the cage effect. Both the longitudinal and transverse components of the displacement correlation are found to be expressible in terms of a similarity variable, suggesting that the cages are nested to form a self-similar structure in the space-time.

Localization and diffusion of tracer particles in viscoelastic media with active force dipoles

“…Previously, such behavior was investigated at thermal equilibrium [20,21]. Experimental data for bacterial cells shows that subdiffusion characteristic for glassy systems disappears in the presence of metabolic activity inside a cell and that classical diffusion is then observed [1].…”

Collective Motion of Repulsive Brownian Particles in Single-File Diffusion with and without Overtaking

“…The calculated displacement correlation revealed collective motions behind the slow diffusion in SFD, in contrast to free diffusion in which R i R j vanishes (unless i = j). The formalism for analytical calculation of the displacement correlation can be extended to the case of two-dimensional (2D) colloidal liquids [10,21], which reproduces some numerical findings in 2D systems, such as vortical cooperative motion, with negative velocity autocorrelation being a manifestation of the cage effect. One of the delicate points in this extension is that the cage effect in 2D liquids cannot be infinitely strong, in the sense that eventually the particles can escape from the 2D cage.…”

“…The kinematics of overtaking were discussed in Subsection 2.5: as is illustrated in Figure 3, two particles exchange their labels according to Equation (21).…”

“…[21] and references therein). The density field, ρ(x, t), is governed by the Dean-Kawasaki equation [32][33][34][35][36][37], which can be presented as a set of equations of the following form:…”

“…Subsequently, by inverting the mapping (x, t) → ξ, we obtain the coordinate system with the independent variables (ξ, t) [4,10,19,21,30]. It should be emphasized that we take Equation (18), not Equation (17), as the definition of the mapping between ξ and x.…”