# On the Dynamics of Large Particle Systems in the Mean Field Limit

@article{Golse2013OnTD, title={On the Dynamics of Large Particle Systems in the Mean Field Limit}, author={Franccois Golse}, journal={arXiv: Analysis of PDEs}, year={2013}, pages={1-144} }

This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics—such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics—can be rigorously derived from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these…

## 180 Citations

### Topics in Analysis of Many Particle Systems

- Computer Science
- 2020

Depending on time and interest it will include part or all of the following topics: the Liouville/Master equations of N -particle systems, the notion of empirical measures, the BBGKY hierarchy, the Hewitt-Savage theorem, the Dobrushin’s stability estimate, the coupling method, the concepts of chaos and entropic chaos.

### On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

- MathematicsCommunications in Mathematical Physics
- 2021

We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on…

### On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

- MathematicsCommunications in Mathematical Physics
- 2021

We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on…

### The mean-field approximation for higher-dimensional Coulomb flows in the scaling-critical L ∞ space

- Mathematics
- 2022

In the mean-field scaling regime, a first-order system of particles with binary interactions naturally gives rise to a scalar partial differential equation (PDE), which, depending on the nature of…

### Mean-field limit for particle systems with topological interactions

- Physics, MathematicsMathematics and Mechanics of Complex Systems
- 2021

Many interesting physical systems can be described at the microscopic level as particle dynamics and at the mesoscopic level with kinetic equations. In the wide field of two-body interactions, the…

### On the mean field limit of Random Batch Method for interacting particle systems

- Physics, Computer Science
- 2020

The Random Batch Method can be viewed as a model of particle system in which particles interact, at discrete time, with randomly selected mini-batch of particles, and the mean-field limit of this model as the number of particles N → ∞ is investigated.

### Beyond Bogoliubov dynamics

- Mathematics, PhysicsPure and Applied Analysis
- 2021

We consider a quantum system of N interacting bosons in the mean field scaling regime. We construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to…

### On the mean field limit of the Random Batch Method for interacting particle systems

- Physics, Computer ScienceScience China Mathematics
- 2021

The Random Batch Method can be viewed as a model of the particle system in which particles interact, at discrete time, with randomly selected mini-batch of particles, and the mean-field limit of this model as the number of particles N→∞ is investigated.

### From Quantum Many-Body Systems to Ideal Fluids

- Mathematics, Physics
- 2021

Abstract. We give a rigorous, quantitative derivation of the incompressible Euler equation from the many-body problem for N bosons on T with binary Coulomb interactions in the semiclassical regime.…

### Mean-Field Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity

- Mathematics
- 2020

We consider first-order conservative systems of particles with binary Coulomb interactions in the mean-field scaling regime in dimensions $d\geq 3$. We show that if at some time, the associated…

## References

SHOWING 1-10 OF 114 REFERENCES

### On the Mean Field and Classical Limits of Quantum Mechanics

- Physics
- 2015

The main result in this paper is a new inequality bearing on solutions of the N-body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field…

### Statistical mechanics of classical particles with logarithmic interactions

- Mathematics
- 1993

The inhomogeneous mean-field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic…

### The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

- Mathematics
- 2011

The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Commun Math Phys…

### Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

- Mathematics
- 2004

We consider the dynamics of N boson systems interacting through a pair potential N−1Va(xi−xj) where Va(x)=a−3V(x/a). We denote the solution to the N-particle Schrödinger equation by ΨN, t. Recall…

### The mean-field limit for the dynamics of large particle systems

- Physics
- 2003

This short course explains how the usual mean-field evolution PDEs in Statistical Physics — such as the Vlasov-Poisson, Schrodinger-Poisson or time-dependent Hartree-Fock equations — are rigorously…

### Dynamical Systems, Theory and Applications

- Mathematics
- 1975

Time evolution of large classical systems.- Ergodic properties of infinite systems.- Time evolution and ergodic properties of harmonic systems.- The laser: A reversible quantum dynamical system with…

### Empirical Measures and Vlasov Hierarchies

- Mathematics
- 2013

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach…

### The Vlasov limit and its fluctuations for a system of particles which interact by means of a wave field

- Mathematics, Physics
- 2005

In two recent publications, [Commun. PDE 22, 307–335 (1997), Commun. Math. Phys. 203, 1–19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a…

### A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

- Mathematics
- 2015

This paper is devoted the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative…

### WASSERSTEIN DISTANCES FOR VORTICES APPROXIMATION OF EULER-TYPE EQUATIONS

- Mathematics
- 2009

We establish the convergence of a vortex system towards equations similar to the 2D Euler equation in vorticity formulation. The only but important difference is that we use singular kernel of the…