scholarly journals Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

2017 ◽  
Vol 28 (02) ◽  
pp. 223-258 ◽  
Author(s):  
Young-Pil Choi ◽  
Samir Salem
Keyword(s):  
Boundary Conditions ◽  
Particle System ◽  
Mean Field ◽  
Empirical Measure ◽  
Lipschitz Continuous ◽  
Stochastic Particle System ◽  
Flux Boundary Conditions ◽  
Geometrical Constraints ◽  
Discontinuous Kernels ◽  
Drift Terms

We consider an interacting [Formula: see text]-particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuity-type of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles [Formula: see text] goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak solution to the continuity equation. This extends previous results on an interacting particle system with bounded and Lipschitz continuous drift terms and normal reflecting boundary conditions by Sznitman [J. Funct. Anal. 56 (1984) 311–336] to that one with discontinuous kernels.

scholarly journals Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

2019 ◽  
Vol 21 (04) ◽  
pp. 1850039 ◽  
Author(s):  
José A. Carrillo ◽  
Young-Pil Choi ◽  
Samir Salem
Keyword(s):  
Particle System ◽  
Planck Equation ◽  
Newtonian Potential ◽  
Empirical Measure ◽  
Propagation Of Chaos ◽  
Interaction Forces ◽  
Fokker Planck Equation ◽  
Stochastic Particle System ◽  
Stochastic Particle ◽  
Fokker Planck

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.

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.

PHASE DIAGRAMS AND CORRELATION INEQUALITIES OF A THREE-STATE STOCHASTIC EPIDEMIC MODEL ON THE SQUARE LATTICE

2006 ◽  
Vol 16 (12) ◽  
pp. 3687-3693 ◽  
Author(s):  
KAZUMICHI OHTSUKA ◽  
NORIO KONNO ◽  
NAOKI MASUDA ◽  
KAZUYUKI AIHARA
Keyword(s):  
Particle System ◽  
Contact Process ◽  
Mean Field ◽  
Field Approximation ◽  
Correlation Inequalities ◽  
Square Lattice ◽  
Mean Field Approximation ◽  
Pair Approximation ◽  
Stochastic Particle System ◽  
Stochastic Epidemic

We study a three-state stochastic particle system on the square lattice, which extends the contact process. The phase diagram is analyzed by the mean-field approximation, the pair approximation, and numerics. The pair approximation turns out to be better than the meanfield ansatz. We also show that the Harris-FKG type correlation inequalities approximately hold in the present model.

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.

THE INNER CRUST OF NEUTRON STARS IN RELATIVISTIC MEAN FIELD APPROACH

2008 ◽  
Vol 17 (09) ◽  
pp. 1765-1773 ◽  
Author(s):  
JIGUANG CAO ◽  
ZHONGYU MA ◽  
NGUYEN VAN GIAI
Keyword(s):  
Boundary Conditions ◽  
Neutron Stars ◽  
Mean Field ◽  
Bcs Theory ◽  
Neutron Density ◽  
Field Approach ◽  
Hartree Fock ◽  
Relativistic Mean Field ◽  
Density Distributions ◽  
Rmf Model

The microscopic properties and superfluidity of the inner crust in neutron stars are investigated in the framework of the relativistic mean field(RMF) model and BCS theory. The Wigner-Seitz(W-S) cell of inner crust is composed of neutron-rich nuclei immersed in a sea of dilute, homogeneous neutron gas. The pairing properties of nucleons in the W-S cells are treated in BCS theory with Gogny interaction. In this work, we emphasize on the choice of the boundary conditions in the RMF approach and superfluidity of the inner crust. Three kinds of boundary conditions are suggested. The properties of the W-S cells with the three kinds of boundary conditions are investigated. The neutron density distributions in the RMF and Hartree-Fock-Bogoliubov(HFB) models are compared.

scholarly journals Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation

2018 ◽  
Vol 52 (2) ◽  
pp. 567-593 ◽  
Author(s):  
Li Chen ◽  
Simone Göttlich ◽  
Stephan Knapp
Keyword(s):  
Particle System ◽  
Rigorous Derivation ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Delta Potential ◽  
Intermediate Particle ◽  
Stochastic Particle System ◽  
Aggregation Equation ◽  
Discretization Schemes ◽  
Theoretical Results

In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a stochastic particle system while introducing an intermediate particle system with smooth interaction potential. The theoretical results are compared to numerical simulations relying on suitable discretization schemes for the microscopic and macroscopic level. In particular, the regime switch where the analytic theory fails is numerically analyzed very carefully and allows for a better understanding of the equation.

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