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Supercritical Mass and Condensation in Fokker--Planck Equations for Consensus Formation

Monica Caloi, Mattia Zanella

TL;DR

The paper addresses consensus formation dynamics modeled by a one-dimensional nonlinear Fokker–Planck equation on $I=[-1,1]$ with diffusion that vanishes at the boundaries and a superlinear drift, demonstrating a finite-time loss of $L^2$ regularity when the initial mass is supercritical. It develops a general framework for both linear and superlinear drifts, deriving stationary states and identifying a Beta-type equilibrium in the linear case, and establishing a finite critical mass for condensation in the superlinear case via a family of steady states $f^ abla_C$ that blow up as $C\to\bar{C}$ for $\alpha>2$. By introducing diffusion weights $H(w)=(1-w^2)^\

Abstract

Inspired by recently developed Fokker--Planck models for Bose--Einstein statistics, we study a consensus formation model with condensation effects driven by a polynomial diffusion coefficient vanishing at the domain boundaries. For the underlying kinetic model, given by a nonlinear Fokker--Planck equation with superlinear drift, it was shown that if the initial mass exceeds a critical threshold, the solution may exhibit finite-time concentration in certain parameter regimes. Here, we show that this supercritical mass phenomenon persists for a broader class of diffusion functions and provide estimates of the critical mass required to induce finite-time loss of regularity.

Supercritical Mass and Condensation in Fokker--Planck Equations for Consensus Formation

TL;DR

The paper addresses consensus formation dynamics modeled by a one-dimensional nonlinear Fokker–Planck equation on with diffusion that vanishes at the boundaries and a superlinear drift, demonstrating a finite-time loss of regularity when the initial mass is supercritical. It develops a general framework for both linear and superlinear drifts, deriving stationary states and identifying a Beta-type equilibrium in the linear case, and establishing a finite critical mass for condensation in the superlinear case via a family of steady states that blow up as for . By introducing diffusion weights $H(w)=(1-w^2)^\

Abstract

Inspired by recently developed Fokker--Planck models for Bose--Einstein statistics, we study a consensus formation model with condensation effects driven by a polynomial diffusion coefficient vanishing at the domain boundaries. For the underlying kinetic model, given by a nonlinear Fokker--Planck equation with superlinear drift, it was shown that if the initial mass exceeds a critical threshold, the solution may exhibit finite-time concentration in certain parameter regimes. Here, we show that this supercritical mass phenomenon persists for a broader class of diffusion functions and provide estimates of the critical mass required to induce finite-time loss of regularity.
Paper Structure (3 sections, 2 theorems, 65 equations, 3 figures)

This paper contains 3 sections, 2 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Superlinear Fokker-Planck Equations
  3. Loss of regularity and supercritical mass

Key Result

Lemma 1

Let $f(w,t)$, $w \in I$, $t\ge0$, be the solution to the linear Fokker-Planck for consensus formation eq:generalFP with $J(f) = f$ and nonconstant diffusion weight eq:HBeta complemented with no-flux boundary condition. We assume that the initial distribution satisfies $f_0(w) \in L^1(I)\cap L^2(I)$. where $C_N = \frac{3^3}{2^5}$.

Figures (3)

  • Figure 1: Shape of $H(w) = (1-w^2)^{\gamma}$ for different values of $\gamma$.
  • Figure 2: Asymptotic distribution of \ref{['eq:generalFP']} with drift $J(f) = f(1+\beta H^\alpha(w)f^\alpha)$ as in \ref{['eq:asymptotic_m0']}. We fixed $\alpha=3$, $\beta=1$, $\sigma^2=0.025$, $m=0$ and different values of $\gamma\in \{1,10,50,100\}$. Left: subcritical case $C<\bar{C}$. Right: critical case $C=\bar{C}$.
  • Figure 3: Critical initial mass required for finite-time condensation in superlinear Fokker--Planck dynamics, showing its dependence on initial energy $E(0)$, nonlinearity exponent $\alpha$, diffusion shape parameter $\gamma$ and diffusion coefficient $\sigma^2$.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof