# Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons

## TL;DR

This paper numerically investigates the Kaniadakis-Quarati model for bosons, demonstrating finite-time condensate formation, convergence to equilibrium, and universal blow-up profiles, thus providing insights into the PDE's solution behavior.

## Contribution

It introduces a validated numerical scheme to simulate the model's solutions over large times, revealing finite-time condensation and universal blow-up characteristics.

## Key findings

- Condensates form in finite time for supercritical mass
- Solutions converge exponentially to entropy minimizers
- Universal power-law blow-up profile near the origin

## Abstract

Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.