Continuum models of kinetic many-particle systems with short-range interactions and clustering

Duration: 52 mins 50 secs

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Description:	Calum Braham (University of Oxford) 22/03/2022 Programme: FKT SemId: 35080


Created:	2022-04-05 02:43
Collection:	Frontiers in analysis of kinetic equations
Publisher:	Calum Braham
Copyright:	Isaac Newton Institute
Language:	eng (English)


Abstract:	Models of physical systems as sets of particles interacting through generalised `forces' are common to a wide variety of sciences. While individual-based (microscopic) models of such systems are generally conceptually simple, they are often intractable analytically and computationally as most systems to be modelled contain an infeasibly large number of particles. It is common in such circumstances to derive a macroscopic continuum model, which tracks the evolution of population-averaged probability densities over an individual particle's phase space. In this talk we will develop a general method for deriving continuum models of second-order (kinetic) systems with short-ranged interaction forces using matched asymptotic expansion in the small interaction-length parameter. As an archetypical example, we first show that this approach allows us to derive the Boltzmann equation as a continuum description of a low-density potential-force system. We then extend our method to apply to `clustering' systems, where particles' positions and velocities may be highly correlated post-interaction, a behaviour that would invalidate many of the typical assumptions made when deriving such continuum models. We use the Cucker-Smale individual-based model of collective behaviour as a test case to evaluate our matched asymptotic method against particle simulations and compare the accuracy to a standard mean-field approach.

Abstract:

Models of physical systems as sets of particles interacting through generalised `forces' are common to a wide variety of sciences. While individual-based (microscopic) models of such systems are generally conceptually simple, they are often intractable analytically and computationally as most systems to be modelled contain an infeasibly large number of particles. It is common in such circumstances to derive a macroscopic continuum model, which tracks the evolution of population-averaged probability densities over an individual particle's phase space. In this talk we will develop a general method for deriving continuum models of second-order (kinetic) systems with short-ranged interaction forces using matched asymptotic expansion in the small interaction-length parameter. As an archetypical example, we first show that this approach allows us to derive the Boltzmann equation as a continuum description of a low-density potential-force system. We then extend our method to apply to `clustering' systems, where particles' positions and velocities may be highly correlated post-interaction, a behaviour that would invalidate many of the typical assumptions made when deriving such continuum models. We use the Cucker-Smale individual-based model of collective behaviour as a test case to evaluate our matched asymptotic method against particle simulations and compare the accuracy to a standard mean-field approach.

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