Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

A. Garroni, P. van Meurs, M. A. Peletier, L. Scardia

Research output: Contribution to journalArticle

Abstract

We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale δn> 0 , which converges to zero as the number of particles n tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma–Balogh equations. In the proof we rely on both the theory of λ-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δn-regularised version of the Groma–Balogh equations, and a priori estimates for the Groma–Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n→ ∞ we require δn to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δn converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma–Balogh equations. These results show how the validity of the Groma–Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

Original languageEnglish
Pages (from-to)1-47
Number of pages47
JournalArchive for Rational Mechanics and Analysis
Early online date19 Aug 2019
DOIs
Publication statusE-pub ahead of print - 19 Aug 2019

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Empirical Measures
Dislocations (crystals)
Dislocation
Converge
Zero
Crystals
Singularity
Continuum
Interaction
Infinity
Tend
Many-particle System
Interacting Particle Systems
Gradient Flow
Orlicz Spaces
A Priori Estimates
Length Scale
Convergence Results
Blow-up
Counterexample

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

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abstract = "We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale δn> 0 , which converges to zero as the number of particles n tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma–Balogh equations. In the proof we rely on both the theory of λ-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δn-regularised version of the Groma–Balogh equations, and a priori estimates for the Groma–Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n→ ∞ we require δn to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δn converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma–Balogh equations. These results show how the validity of the Groma–Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.",
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Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs. / Garroni, A.; van Meurs, P.; Peletier, M. A.; Scardia, L.

In: Archive for Rational Mechanics and Analysis, 19.08.2019, p. 1-47.

Research output: Contribution to journalArticle

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