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Deterministic reversible model of non-equilibrium phase transitions and stochastic counterpart
Sapienza Universit`a di Roma, Italy.
Universit`a degli Studi dell’Aquila, Italy.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-1160-0007
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-2185-641x
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2020 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 53, no 30, article id 305001Article in journal (Refereed) Published
Abstract [en]

Npoint particles move within a billiard table made of two circular cavities connected by a straight channel. The usual billiard dynamics is modified so that it remains deterministic, phase space volumes preserving and time reversal invariant. Particles move in straight lines and are elastically reflected at the boundary of the table, as usual, but those in a channel that are moving away from a cavity invert their motion (rebound), if their number exceeds a given thresholdT. When the geometrical parameters of the billiard table are fixed, this mechanism gives rise to non-equilibrium phase transitions in the largeNlimit: lettingT/Ndecrease, the homogeneous particle distribution abruptly turns into a stationary inhomogeneous one. The equivalence with a modified Ehrenfest two urn model, motivated by the ergodicity of the billiard with no rebound, allows us to obtain analytical results that accurately describe the numerical billiard simulation results. Thus, a stochastic exactly solvable model that exhibits non-equilibrium phase transitions is also introduced.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2020. Vol. 53, no 30, article id 305001
Keywords [en]
billiards; Ehrenfest urn model; non-equilibrium states; phase transitions
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-77788DOI: 10.1088/1751-8121/ab94ecISI: 000553030100001Scopus ID: 2-s2.0-85088575439OAI: oai:DiVA.org:kau-77788DiVA, id: diva2:1430981
Available from: 2020-05-18 Created: 2020-05-18 Last updated: 2021-03-31Bibliographically approved
In thesis
1. Multiscale models and simulations for diffusion and interaction in heterogeneous domains
Open this publication in new window or tab >>Multiscale models and simulations for diffusion and interaction in heterogeneous domains
2021 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains, with a focus on multiscale diffusions. Although diffusion is often studied in terms of continuum observables, it is a consequence of the motion of individual particles. Incorporating interactions between constituents and geometry often runs into complications, since interactions typically act on multiple length scales. We address this issue by studying different types of multiscale models and by applying them to a variety of scenarios known for their inherent complexity.

Our contributions can be grouped in two parts. In the first part, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct finite element schemes with desirable approximation properties that resolve the microscopic domain variations and support parallel execution. In the second part of the thesis, we investigate certain interacting particle systems and their links to families of partial differential equations. In this spirit, we analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within an interacting particle system in an extension of the classical Ehrenfest model.

Our overall focus is two-fold. On the one hand, we increase the theoretical understanding of multiscale models by providing modeling, analysis and simulation of specific two-scale couplings. On the other hand, we design computational frameworks and tailored implementations to improve the application of multiscale modeling to complex scenarios and large-scale systems. In this way, our contributions aim to expand the capacity of mathematical modeling to numerically approximate the rich and complex physical world.

Abstract [en]

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains. Our contributions can be grouped in two parts. First, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct convergent and efficient finite element schemes that resolve the microscopic domain variations. Second, we investigate certain interacting particle systems and their links to a family of partial differential equations. We analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We also construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within a particle system extending the classical Ehrenfest model.

Our focus is two-fold: we increase the theoretical understanding of certain two-scale couplings, while on the other hand, we develop computational multiscale frameworks for a variety of scenarios known for their inherent complexity.

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2021. p. 212
Series
Karlstad University Studies, ISSN 1403-8099 ; 2021:10
Keywords
multiscale modeling, finite element methods, interacting particle sytems, population dynamics, non-equilibrium dynamics
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-83568 (URN)978-91-7867-195-3 (ISBN)978-91-7867-205-9 (ISBN)
Public defence
2021-05-19, Via Zoom, 15:00 (English)
Opponent
Supervisors
Available from: 2021-04-28 Created: 2021-03-31 Last updated: 2021-05-11Bibliographically approved

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Publisher's full textScopushttps://iopscience.iop.org/journal/1751-8121?gclid=CjwKCAjw5Ij2BRBdEiwA0Frc9U7wLSq0Dc-G9bD9rLh7iI18Ld5-G7L4r08-TzgRsfeQjVFutX25NBoCX-wQAvD_BwE

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Muntean, AdrianRichardson, Omar

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