A mean-field approach to self-interacting networks, convergence and regularity

2021 ◽  
pp. 1-45
Author(s):  
Rémi Catellier ◽  
Yves D’Angelo ◽  
Cristiano Ricci
Keyword(s):  
Particle System ◽  
Mean Field ◽  
Spatial Evolution ◽  
Propagation Of Chaos ◽  
Interacting Particles ◽  
Field Limit ◽  
Mean Field Limit ◽  
Field Approach ◽  
Empirical Density ◽  
Interacting Networks

The propagation of chaos property for a system of interacting particles, describing the spatial evolution of a network of interacting filaments is studied. The creation of a network of mycelium is analyzed as representative case, and the generality of the modeling choices are discussed. Convergence of the empirical density for the particle system to its mean-field limit is proved, and a result of regularity for the solution is presented.

scholarly journals The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Hui Huang ◽  
Jinniao Qiu
Keyword(s):  
Existence Of Solutions ◽  
Particle System ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Unique Existence ◽  
Interacting Particle ◽  
Microscopic Derivation ◽  
The Mean ◽  
Limit Result

AbstractIn this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions $$d=2,3$$ d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

scholarly journals Mean Field Limit and Propagation of Chaos for a Pedestrian Flow Model

2016 ◽  
Vol 166 (2) ◽  
pp. 211-229 ◽  
Author(s):  
Li Chen ◽  
Simone Göttlich ◽  
Qitao Yin
Keyword(s):  
Flow Model ◽  
Mean Field ◽  
Propagation Of Chaos ◽  
Pedestrian Flow ◽  
Field Limit ◽  
Mean Field Limit

scholarly journals On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck System

2020 ◽  
Vol 181 (5) ◽  
pp. 1915-1965
Author(s):  
Hui Huang ◽  
Jian-Guo Liu ◽  
Peter Pickl
Keyword(s):  
Initial Data ◽  
Particle System ◽  
Mean Field ◽  
Newtonian Potential ◽  
Empirical Measure ◽  
Field Limit ◽  
Mean Field Limit ◽  
Brownian Motions ◽  
The Mean ◽  
Fokker Planck

AbstractWe rigorously justify the mean-field limit of an N-particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $$N^{-\frac{1}{3}+\varepsilon }$$ N - 1 3 + ε ($$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{-\delta }$$ N - δ ($$\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of $$1-N^{-\alpha }$$ 1 - N - α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.

scholarly journals Mean Field Derivation of DNLS from the Bose–Hubbard Model

2021 ◽  
Author(s):  
E. Picari ◽  
A. Ponno ◽  
L. Zanelli
Keyword(s):  
Hubbard Model ◽  
Coherent States ◽  
Quantum Dynamics ◽  
Mean Field ◽  
Interacting Particles ◽  
Field Limit ◽  
Mean Field Limit ◽  
Discrete Nonlinear Schrödinger Equation ◽  
The Mean ◽  
Wick Symbol

AbstractWe prove that the flow of the discrete nonlinear Schrödinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose–Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative $$L^p$$ L p -estimate, for any $$p \ge 1$$ p ≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.

scholarly journals STOCHASTIC MEAN-FIELD LIMIT: NON-LIPSCHITZ FORCES AND SWARMING

2011 ◽  
Vol 21 (11) ◽  
pp. 2179-2210 ◽  
Author(s):  
FRANÇOIS BOLLEY ◽  
JOSÉ A. CAÑIZO ◽  
JOSÉ A. CARRILLO
Keyword(s):  
Collective Behavior ◽  
Stochastic Systems ◽  
Lipschitz Constant ◽  
Mean Field ◽  
Classical Case ◽  
Interacting Particles ◽  
Field Limit ◽  
Mean Field Limit ◽  
Locally Lipschitz ◽  
The Mean

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker–Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.

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