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Integrability properties and stochastic McKean-Vlasov dynamics with singular Lennard-Jones drift: a mesoscale regularization

Ernesto M. Greco, Daniela Morale

TL;DR

The paper addresses the mean-field limit and well-posedness of particle systems with a singular Lennard-Jones drift across macro, meso, and micro scales. It develops a mesoscale regularization mechanism $K_N=K\ast V_N$ and proves existence/uniqueness for the corresponding MKV-SDE and the mild solution of the associated Fokker–Planck equation, plus convergence of the empirical density to the PDE solution with explicit rates depending on the mesoscale parameter $\alpha$ and integrability exponents. The results rely on detailed convolution estimates for $K$ and $\nabla K$ in various function spaces, including Hölder and Bessel potential spaces, and on a cut-off strategy to control singular drift in the particle system. A key contribution is a rigorous multi-scale framework that links microscopic dynamics to a macroscopic density evolution through a mesoscopic regularization, yielding strong convergence (and rates) for the empirical density, as well as weak convergence results in the mesoscale setting. This provides the first comprehensive, rigorous treatment of diffusion processes driven by Lennard–Jones–type kernels in a McKean–Vlasov context with quantified regularity and convergence across scales, with potential implications for mean-field limits in systems with singular interparticle forces.

Abstract

We study the convergence of the empirical measure of moderately interacting particle systems subject to singular forces derived by Lennard-Jones potential. Although the classical Lennard-Jones force is widely used in molecular dynamics, analytical results are not available. We consider a Lennard-Jones potential with free parameters in the McKean-Vlasov framework and proceed with a regularization at the mesoscale letting the particles interact moderately. We prove the well-posedness of the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards the solution of the McKean-Vlasov Fokker-Planck PDE, by means of a semigroup approach. We derive both the range of parameters characterizing the aggregation and repulsive force and the mesoscale order for which the convergence is achieved, by obtaining the right integrability regularity of the drift.

Integrability properties and stochastic McKean-Vlasov dynamics with singular Lennard-Jones drift: a mesoscale regularization

TL;DR

The paper addresses the mean-field limit and well-posedness of particle systems with a singular Lennard-Jones drift across macro, meso, and micro scales. It develops a mesoscale regularization mechanism and proves existence/uniqueness for the corresponding MKV-SDE and the mild solution of the associated Fokker–Planck equation, plus convergence of the empirical density to the PDE solution with explicit rates depending on the mesoscale parameter and integrability exponents. The results rely on detailed convolution estimates for and in various function spaces, including Hölder and Bessel potential spaces, and on a cut-off strategy to control singular drift in the particle system. A key contribution is a rigorous multi-scale framework that links microscopic dynamics to a macroscopic density evolution through a mesoscopic regularization, yielding strong convergence (and rates) for the empirical density, as well as weak convergence results in the mesoscale setting. This provides the first comprehensive, rigorous treatment of diffusion processes driven by Lennard–Jones–type kernels in a McKean–Vlasov context with quantified regularity and convergence across scales, with potential implications for mean-field limits in systems with singular interparticle forces.

Abstract

We study the convergence of the empirical measure of moderately interacting particle systems subject to singular forces derived by Lennard-Jones potential. Although the classical Lennard-Jones force is widely used in molecular dynamics, analytical results are not available. We consider a Lennard-Jones potential with free parameters in the McKean-Vlasov framework and proceed with a regularization at the mesoscale letting the particles interact moderately. We prove the well-posedness of the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards the solution of the McKean-Vlasov Fokker-Planck PDE, by means of a semigroup approach. We derive both the range of parameters characterizing the aggregation and repulsive force and the mesoscale order for which the convergence is achieved, by obtaining the right integrability regularity of the drift.
Paper Structure (19 sections, 29 theorems, 165 equations, 1 figure)

This paper contains 19 sections, 29 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convolution operators via the general Lennard-Jones force: integrability and continuity properties.
  3. Local integrability properties of the Lennard-Jones force
  4. Boundedness of convolution operators
  5. Convolution operator defined by the Lennard-Jones force K
  6. Convolution operators defined by nabla K. The sub-singular case d ge 3, a le d-2.
  7. Boundedness of convolution operators: the super-singular case a in (d-2, d-1)
  8. A McKean-Vlasov dynamics at the macro and microscale
  9. Brownian particles moderately interacting via a Lennard-Jones force at a mesoscale
  10. The mesoscale.
  11. A stochastic PDE for the empirical density
  12. The stochastic convolution integral: a priori estimates.
  13. Particle system and PDE with a uniformly regularized drift
  14. Meso-Macroscale: convergence for the empirical density
  15. The case of the regularized SDE: moment convergence
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $d \ge 2$. Let $K$ be the general Lennard-Jones force eq:JL_potential_force with parameters $d-1>a>b>0$. Then, $K \in L^1_\text{loc}(\mathbb{R}^d)$.

Figures (1)

  • Figure 1: (a) Classical Lennard-Jones potential $\widetilde{\Phi}$ (solid line) and the related force $\widetilde{K}$ (dashed line) as in \ref{['caption:eq:lennard_jones_classical']}. (b) Lennard-Jones potential with free parameters \ref{['eq:JL_potential_force']}, for different set $(a,b)$. Parameters: $\epsilon=\tilde{\epsilon}=2, R_0=1$.

Theorems & Definitions (79)

  • Proposition 2.1
  • proof
  • Proposition 2.2: LJ force local integrability
  • proof
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3: LJ force convolution operator
  • proof
  • Remark 2.3
  • Proposition 2.4
  • ...and 69 more