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 Decay of correlation rate in the mean field limit of point vortices ensembles

2020 ◽  
Vol 20 (06) ◽  
pp. 2040009
Author(s):  
Francesco Grotto ◽  
Marco Romito
Keyword(s):  
Classical Result ◽  
Mean Field ◽  
Coulomb Gas ◽  
Point Vortices ◽  
Field Limit ◽  
Mean Field Limit ◽  
Gaussian Integration ◽  
Euclidean Field Theory ◽  
Integration Techniques ◽  
The Mean

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional (2D) point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: We compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2D Coulomb gas and the Sine-Gordon Euclidean field theory.

scholarly journals The mean-field limit of quantum Bose gases at positive temperature

2021 ◽  
pp. 1
Author(s):  
Jürg Fröhlich ◽  
Antti Knowles ◽  
Benjamin Schlein ◽  
Vedran Sohinger
Keyword(s):  
Mean Field ◽  
Positive Temperature ◽  
Field Limit ◽  
Bose Gases ◽  
Mean Field Limit ◽  
The Mean

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 of Interacting Filaments for 3D Euler Equations

2018 ◽  
Vol 174 (3) ◽  
pp. 562-578 ◽  
Author(s):  
Hakima Bessaih ◽  
Michele Coghi ◽  
Franco Flandoli
Keyword(s):  
Euler Equations ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit

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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