Sentence examples for mean field equation from inspiring English sources
To make use of the analytic tractability of the mean field equation it was necessary to consider a fully connected network.
We start with the law of large numbers, which establishes the connection to the deterministic mean field equation, and then proceed to central limit theorems which provide the basis for a Langevin approximation.
In keeping also the second-order terms, a 'stochastic' version of the mean field equation is also presented in the sense of coupling the first moment equation to an equation for the second moments.
In this article we consider the following fourth order mean field equation on smooth domain Ω⋐R4:Δ2u="ϱKeu∫ΩKeuin Ω,u= Δu= 0on ∂Ω, where ϱ∈R and 0<K∈C2.
In particular, the authors in [34] present a detailed existence and uniqueness result for the activity based Amari mean field equation and state that an analogous result hold for the Wilson Cowan equation (A.1) for spatial dimensions d ≤ 3, which covers all physical relevant domains.
As F ¯ n is essentially a piecewise constant approximation to F, the resulting equations for the mean correspond to a spatial discretisation of the Wilson Cowan equation, cf. the continuum limit in the derivation of the mean field equation in [5].
The interest in deriving mean field equations from stochastic microscopic model has been revived recently as it contains the possibility to derive deterministic 'corrections' to the mean field equations, also called second-order approximations.
These tools do not resolve turbulent fluctuations responsible for the bulk of cross-field transport in the Scrape-off Layer (SOL), and solve mean field equations instead.
As is thoroughly discussed in [5] establishing the connection between master equation models and mean field equations involves two limit procedures.
Failures owing to activity explosion (the lower side of the wedge) are almost independent of the skewness, due to the instantaneous feedback inhibition in the mean field equations.
We derive the averaged mean field equations and show that there are changes in the stability as the homeostatic time constant changes.
