scholarly journals On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

2021 ◽  
Author(s):  
Matias G. Delgadino ◽  
Rishabh S. Gvalani ◽  
Grigorios A. Pavliotis
Keyword(s):  
Diffusion Processes ◽  
Mean Field ◽  
Rates Of Convergence ◽  
Kuramoto Model ◽  
Field Energy ◽  
Field Limit ◽  
Mean Field Limit ◽  
Weakly Interacting ◽  
The Mean ◽  
Field Plane

AbstractThe objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

scholarly journals Derivation of the Hartree equation for compound Bose gases in the mean field limit

2017 ◽  
Vol 29 (07) ◽  
pp. 1750022 ◽  
Author(s):  
Ioannis Anapolitanos ◽  
Michael Hott ◽  
Dirk Hundertmark
Keyword(s):  
Mean Field ◽  
Rates Of Convergence ◽  
Hartree Equation ◽  
Initial Condition ◽  
Field Limit ◽  
Bose Gases ◽  
Mean Field Limit ◽  
Relativistic Case ◽  
The Mean ◽  
Reduced Density

We consider mixtures of Bose gases of different species. We prove that in the mean field limit and under suitable conditions on the initial condition, a system composed of two Bose species can be effectively described by a system of coupled Hartree equations. Moreover, we derive quantitative bounds on the rates of convergence of the reduced density matrices in Sobolev trace norms. We treat both the non-relativistic case in the presence of an external magnetic field [Formula: see text] and the semi-relativistic case.

scholarly journals On the complete phase synchronization for the Kuramoto model in the mean-field limit

2015 ◽  
Vol 13 (7) ◽  
pp. 1775-1786 ◽  
Author(s):  
Dario Benedetto ◽  
Emanuele Caglioti ◽  
Umberto Montemagno
Keyword(s):  
Phase Synchronization ◽  
Mean Field ◽  
Kuramoto Model ◽  
Field Limit ◽  
Mean Field Limit ◽  
The Mean ◽  
The Kuramoto 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.

scholarly journals Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Carina Geldhauser ◽  
Marco Romito
Keyword(s):  
Euler Equations ◽  
Mean Field ◽  
Point Vortices ◽  
Positive Temperature ◽  
Field Limit ◽  
Mean Field Limit ◽  
Large Numbers ◽  
The Mean ◽  
Formal Solutions ◽  
2D Torus

AbstractWe prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

scholarly journals Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Pierre Roux ◽  
Delphine Salort
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Mean Field ◽  
Classical Solutions ◽  
Field Limit ◽  
Mean Field Limit ◽  
Diffusive Approximation ◽  
The Mean ◽  
Weak Connectivity ◽  
Time Existence

<p style='text-indent:20px;'>The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.</p>

Long Time Estimates in the Mean Field Limit

2008 ◽  
Vol 190 (3) ◽  
pp. 517-547 ◽  
Author(s):  
E. Caglioti ◽  
F. Rousset
Keyword(s):  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Time Estimates ◽  
Long Time ◽  
The Mean

scholarly journals Insensitivity of the mean field limit of loss systems under SQ(d) routeing

2019 ◽  
Vol 51 (4) ◽  
pp. 1027-1066
Author(s):  
Thirupathaiah Vasantam ◽  
Arpan Mukhopadhyay ◽  
Ravi R. Mazumdar
Keyword(s):  
Fixed Point ◽  
Service Time ◽  
Mean Field ◽  
Service Time Distribution ◽  
Field Limit ◽  
Mean Field Limit ◽  
Mean Field Equations ◽  
The Mean ◽  
Loss Systems ◽  
Exponential Case

AbstractIn this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

scholarly journals On the mean field limit for Brownian particles with Coulomb interaction in 3D

2019 ◽  
Vol 60 (11) ◽  
pp. 111501 ◽  
Author(s):  
Lei Li ◽  
Jian-Guo Liu ◽  
Pu Yu
Keyword(s):  
Coulomb Interaction ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Brownian Particles ◽  
The Mean
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