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

Abstract

We address the question of 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 manyparticle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

Atomistic origins of continuum dislocation dynamics

“…Finally, the connection from (M F δ β ) to (M F β ) follows by the same argument as in the proof of (GB δ ) to (GB) in [GvMPS19].…”

“…We choose to rescale the total energy in such a way as to ensure that it remains bounded as the number of dislocations tends to infinity. Given a collection of dislocations, the rescaled interaction energy E n : T 2n × {±1} n → R is given by (see [GvMPS19])…”

“…with F as in (7). The system (28) is the starting point of [GvMPS19], where convergence to (GB) is proven in the joint limit where (n, δ) → (∞, 0). The proof of this joint limit is a combination of two results; the first is a quantitative estimate between the solution of (ODE δ n ) and that of the 'regularised Groma-Balogh equations' (see [GB99])…”

Discrete-To-Continuum Limits of Particles with an Annihilation Rule

“…The assumption that the dislocation lines are straight and parallel, such that their position can be identified by points in a 2D cross section [ADLGP14,Ber06,CL05,GvMPS20,GB99,GCZ03,GGK06,HO14,KHG15,MPS17].…”

Leader formation with mean-field birth and death models

“…Such approach has the advantage of reducing the computational complexity of the models (overcoming the curse of dimensionality [10]) and allows the so-called microfundation of macromodels, i.e., the validation of the macroscopic dynamics from the coherence with the behavior of individuals (a central issue in the field of macroeconomics). The mean-field limit of systems of interacting agents has been thoroughly studied also in conjunction with irregular interaction kernel [17,32], control problems [3,15,29,30,38] and multiple populations [4,5,12,21]. Also models where the total mass of the system is not preserved in time, due to the presence of source (or sink) terms, have been considered (see for instance [45,).…”