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

TL;DR

This paper investigates a nonlinear Fokker--Planck equation modeling consensus formation, demonstrating that supercritical mass leads to finite-time concentration, extending previous results to broader diffusion functions.

Contribution

It extends the supercritical mass phenomenon to a wider class of diffusion functions and provides estimates for the critical mass needed for finite-time regularity loss.

Findings

Supercritical mass causes finite-time concentration in the model.

The phenomenon persists for a broader class of diffusion functions.

Critical mass estimates are provided for regularity loss.

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.