(A) study on the effects of shapes and interactions of objects in the stochastic dynamical systems

Abstract

Shapes and interactions of objects play a nontrivial role in physics of the system they constitute, as evident from emergent phenomena and interactions observed from systems in equilibrium. Nevertheless, fundamental understanding of their effects on dynamical systems remains elusive. The central theme of this thesis is exploring how shapes of objects and interactions between them interplay with stochastic dynamics in a variety of systems. The objects with different shapes exhibit stochastic dynamics distinctive from each other, which has an enormous potential for applications. This aspect is studied in the context of separation of enantiomers by flows. A number of recent experimental and numerical studies observed chiral separation by flows, yet general understanding is lacking, and common physical properties of flows that induce the separation have not been clarified. I introduce a theoretical framework to understand the stochastic motion of rigid chiral objects and specify the conditions required for a flow field to bring about the chiral separation. From parity symmetry argument, the essential role of rate-of-strain field is revealed. Equally crucial requirement that the flow field should be quasi-two-dimensional for effective separation is drawn based on the eigenmode analysis. I demonstrate these claims by Langevin dynamics simulations while fully implementing hydrodynamic interactions. The Brownian trajectory is another example in which object shape plays a definitive role in its statistical properties. Statistics of the displacements measured on the Brownian trajectories deviate from the Gaussian distribution as the object becomes anisotropic. Thus, the shape information of the object can be retrieved from measures containing moments of fourth or higher orders, which, unfortunately, usually show poor statistical convergence. I suggest a measure derived from a gyration tensor and demonstrate its superior convergence in comparison to the fourth cumulant with exact analytic expressions and numerical confirmations. I further discuss how to derive the quartic measure with the least relative error. Lastly, the effect of interactions between objects on the first-passage dynamics is studied by comparing random target searching of independent and interacting searchers. Despite typical examples of random searchings involve multiple interacting searchers, relevant studies have been restricted to the searchings in one-dimensional systems or a few limiting cases of higher dimensions by noninteracting searchers. Employing the order statistics and eigenmode analysis, I clarify how the searching time depends on the number of searchers in one- to three-dimensional domains while modulating the initial distribution of the searchers as well. The target searching with interacting searchers are analyzed by employing fluctuating hydrodynamics. Repulsive interactions are found to enhance the searching speed while attractions between the searchers impede the searching process. These theoretical expectations are verified with Brownian dynamics simulations.