Hydrodynamic behaviour of active and multi-species exclusion processes

This thesis investigates the effects of volume exclusion on large-scale collective behaviour in active and multi-species systems. We first focus on the effects of volume exclusion using the method of matched asymptotics on a multi-species exclusion process. This results in a cross-diffusion system of partial differential equations (PDEs) that captures non-trivial interaction terms, including a non-diagonal mobility matrix with explicit density dependence. This extends previous work to handle species with unequal jump rates, and provides a cubic polynomial approximation of the self-diffusion coefficient that remains numerically accurate across all densities. We also apply the method of matched asymptotics to a general exclusion process to yield a PDE for particle density. We factorise this into a gradient-flow structure, within which particle interactions were captured order by order in the mobility. We verify these results by working directly with mobilities's variational representation. We develop an exact hydrodynamic description for a lattice model of active matter with exclusion. The macroscopic limit is an integro-differential equation for the particles' density in phase space (positions and orientations) and includes nonlinearities in both the diffusive and advective components. Systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation. We explore the onset of such a transition in the parameter space of occupied volume fraction and self-propulsion speed via a linear stability analysis and numerical simulations. We also analyse the collective behaviour in a mixture of active and passive particles, deriving exact hydrodynamic equations for the particle densities. The analysis covers the stability of homogeneous states, phase coexistence, and pattern formation in dynamic steady states. At high activity levels, the spinodal curve for active phase separation intersects the binodal curve, leading to dynamic steady states, including anharmonic travelling and counterpropagating waves. We examine how these dynamical states behave in the thermodynamic limit of large system size, showing that sharp interfaces may travel at finite velocity but travelling phase-separated states are forbidden. The findings offer insights into the behaviour of active systems with non-reciprocal interactions and provide a foundation for future research in active matter.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Mason, James

Advisors dc:contributor.advisor

• Jack, Robert
• Bruna, Maria

• Active Matter
• Hydrodynamic Limit
• Matched Asymptotics
• Stochastic lattice gases
• Simple exclusion process
• Cross-diffusion system

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a0266766-37ed-4ab1-8fda-17d27b86580e/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.115705

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/379736