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. Samesign particles repel each other, and oppositesign 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 blowup in finite time. We remedy this situation by regularising the interaction potential at a lengthscale δ_{n}> 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 λconvex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δ_{n}regularised version of the Groma–Balogh equations, and a priori estimates for the Groma–Balogh equations to pass to the smallregularisation 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 δ_{n} to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δ_{n} 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 manyparticle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.
Original language  English 

Pages (fromto)  349 
Number of pages  47 
Journal  Archive for Rational Mechanics and Analysis 
Volume  235 
Early online date  19 Aug 2019 
DOIs  
Publication status  Epub ahead of print  19 Aug 2019 
ASJC Scopus subject areas
 Analysis
 Mathematics (miscellaneous)
 Mechanical Engineering
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Lucia Scardia
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)