scholarly journals Mean-Field Limit of a Stochastic Particle System Smoothly Interacting Through Threshold Hitting-Times and Applications to Neural Networks with Dendritic Component

2015 ◽  
Vol 47 (5) ◽  
pp. 3884-3916 ◽  
Author(s):  
J. Inglis ◽  
D. Talay
Keyword(s):  
Neural Networks ◽  
Particle System ◽  
Mean Field ◽  
Hitting Times ◽  
Field Limit ◽  
Mean Field Limit ◽  
Stochastic Particle System ◽  
Stochastic Particle

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 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 The Pair-Replica-Mean-Field Limit for Intensity-based Neural Networks

2021 ◽  
Vol 20 (1) ◽  
pp. 165-207
Author(s):  
François Baccelli ◽  
Thibaud Taillefumier
Keyword(s):  
Neural Networks ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit

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 Derivation of the Landau–Pekar Equations in a Many-Body Mean-Field Limit

2021 ◽  
Vol 240 (1) ◽  
pp. 383-417
Author(s):  
Nikolai Leopold ◽  
David Mitrouskas ◽  
Robert Seiringer
Keyword(s):  
Mean Field ◽  
Einstein Condensate ◽  
Back Reaction ◽  
Many Body ◽  
Field Limit ◽  
Mean Field Limit ◽  
Phonon Field ◽  
Bose Einstein Condensate ◽  
Weakly Couple ◽  
Bose Einstein

AbstractWe consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.

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
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