Collectivity in diffusion of colloidal particles: from effective interactions to spatially correlated noise

Abstract

The collectivity in the simultaneous diffusion of many particles, i.e. the interdependence of stochastic forces affecting different particles in the same solution, is a largely overlooked phenomenon with no well-established theory. Recently, we have proposed a novel type of thermodynamically consistent Langevin dynamics driven by the Spatially Correlated Noise (SCN) that can contribute to the understanding of this problem. This model draws a link between the theory of effective interactions in binary colloidal mixtures and the properties of SCN. In the current article we review this model from the perspective of collective diffusion and generalize it to the case of multiple (N > 2) particles. Since our theory of SCN-driven Langevin dynamics has certain issues that could not be resolved within its framework, in this article we also provide another approach to the problem of collectivity. We discuss the multi-particle Mori-Zwanzig model, which is fully microscopically consistent. Indeed, we show that this model supplies many information, complementary to the SCN-based approach, e.g. it predicts the deterministic dynamics of the relative distance between the particles, it provides the approximation for non-equilibrium effective interactions and predicts the collective subdiffusion of tracers in group. These results provide the short-range, inertial limit of the earlier model and agree with its predictions under some general conditions. In this article we also review the origin of SCN and its consequences for the variety of physical systems, with emphasis on the colloids.

Preface: Marian Smoluchowski’s 1916 paper—a century of inspiration

“…Majka and Góra span the range from effective interactions to spatially correlated noise for the collective motion of colloidal particles [41].…”

Frictionless mobility of submicron particles in model viscid fluid

“…Thus, the movement of the particle during the jump is nearly frictionless. Quite recently, such varying friction fluctuations have been rigidly formally related to the spatially correlated noise, which in some cases may lead to the formation of frictionless regions [31].…”

Effective one-component model of binary mixture: molecular arrest induced by the spatially correlated stochastic dynamics

“…hydrodynamic interactions 8,9,13,14 , stirring by active particles 15,16 , local density fluctuations 17 and phase transitions. Recently, the current authors have also shown that SCN might arise as the dynamic counterpart of entropic interactions 10,11 in binary mixtures (also named excluded volume or effective interactions 3,18,19 ). In this work, we build upon these results and investigate the SCN-driven Langevin equations as the possible one-component model for the effective dynamics in binary mixtures.…”

“…We will use r as a placeholder for relative distance throughout the paper. Using the projection operator method, the dynamics of the observed particles can be formally reduced to a set of Generalized Langevin Equations 4,5,8,11 , which have no explicit dependence on , i.e. :…”

“…here, , where is the Boltzmann constant, T is temperature and denotes the average. The formal definitions of , and as functions of , and other microscopic parameters can be found in many standard textbooks nowadays 4,5,8,11 . However, these exact expressions are rarely applied in practice as they require full microscopic information about the system, which is usually unavailable.…”