Phase Separation of Active Particles on a Lattice

Mukherjee, Indranil (2024) Phase Separation of Active Particles on a Lattice. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Indranil Mukherjee (20RS148)) 20RS148.pdf - Submitted Version Restricted to Repository staff only Download (4MB)

Abstract

In statistical mechanics, the steady-state measure of the equilibrium state is characterized by the well-known Gibbs-Boltzmann distribution for many-particle systems. This distribution, expressed as P(C) ∝ e⁻βH(C), relies on a prescribed Hamiltonian H(C) and the inverse temperature β. However, many processes exhibit a directional flow of certain observables in the natural world, called the current, indicating the onset of nonequilibrium phenomena. In these non-equilibrium systems, the assurance of the state functions of the system, like the Hamiltonian, is not always guaranteed. Nonequilibrium systems are modeled by stochastic dynamical rules that allow the system to evolve in time, and to determine the steady-state weights; one needs to solve the Master Equation under stationary conditions. The intricate dynamics of interacting stochastic systems exhibit various intriguing features, with a particular focus on phase separation transitions in many studies. In the equilibrium scenario, the non-existence theorem rules out the prospect of phase separation transition in one dimension, however, phase transition can occur in various nonequilibrium models, even in the one dimensional context. This thesis explores phase separation transitions in active particles across diverse nonequilibrium systems, specifically in the active matter systems (AMS) where individual constituents are self-propelled. A common phenomenon unique to active matter systems is the motility-induced phase separation (MIPS), transitioning from a homogeneous to an inhomogeneous phase with increased motility. Formation of such phase separated state without any explicit attractive interaction raises questions about its stability in one and two dimensions. We introduce a one-dimensional model of hardcore run-and-tumble particles (RTPs), where particles run in either +ve or -ve x-direction with an effective speed v and tumble with a constant rate ω when assisted by another particle from the right. We initially obtain its exact steady state using the multibalance scheme, a new flux cancellation scheme we introduced recently for nonequilibrium systems. Nevertheless, the steady state attained following this approach implies that all configurations occur with equal probability. Subsequently, we proceed to develop the coarse-grained dynamics of the system which can be mapped to a beads-in-urn model called the misanthrope process, where particles are identified as urns and vacancies as beads that hop to a neighboring urn situated in the direction opposite to the current. The hop rate is the same as the magnitude of the particle current; we calculate it analytically for a two-particle system and show that it does not satisfy the criteria required for a phase separation transition. The nonexistence of phase separation in this restricted tumbling model implies that motility-induced phase separation transition can not occur in other models in one dimension with unconditional tumbling. Expanding our analysis to two dimensions, we study hardcore RTPs in two dimensions with attractive interaction on a lattice and in continuum. These RTPs move exclusively along their internal orientation, tumbling with rate ω (ω⁻¹ quantifies motility). We find that various models of interacting RTPs, including conserved lattice gas, driven lattice gas, and interacting hard-disc models, exhibit phase separation transition with motility. However, irrespective of motility value, ordering vanishes when attractive interaction strength falls below a finite threshold. In the absence of any attraction, stable phase separation is precluded. We explain why, in the presence of attractive interaction, increased motility hinders cluster formation. This thesis also delves into exactly solvable models in nonequilibrium stochastic processes. Several flux cancellation schemes have been used to obtain the exact steady-state weight of nonequilibrium systems. We introduce a new flux cancellation scheme, multibalance (MB). Application of multibalance conditions to various models underscores its potential in solving exact steady states for nonequilibrium systems.

Actions (login required)

View Item