# Fixed points and flow analysis on off-equilibrium dynamics in the boson Boltzmann equation

## TL;DR

This paper analyzes fixed points and flow directions in the off-equilibrium dynamics of the boson Boltzmann equation, revealing approximate fixed points with power-law spectra and their relation to thermalization processes.

## Contribution

It introduces a graphical flow analysis of the boson Boltzmann equation, identifying approximate fixed points and their connections, enhancing understanding of thermalization out of equilibrium.

## Key findings

- Existence of approximate fixed points with power-law spectra
- Flow diagrams illustrating fixed point connections and critical lines
- Insights into thermalization processes from flow analysis

## Abstract

We consider fixed points of steady solutions and flow directions using the boson Boltzmann equation that is a one-dimensionally reduced kinetic equation after the angular integration. With an elastic collision integral of the two-to-two scattering process, in the dense (dilute) regime where the distribution function is large (small), the boson Boltzmann equation has approximate fixed points with a power-law spectrum in addition to the thermal distribution function. We argue that the power-law fixed point can be exact in special cases. We elaborate a graphical presentation to display evolving flow directions similarly to the renormalization group flow, which explicitly exhibits how fixed points are connected and parameter space is separated by critical lines. We discuss that such a flow diagram contains useful information on thermalization processes out of equilibrium.