# Exact solution for the non-equilibrium attractor in number-conserving relaxation time approximation

## TL;DR

This paper derives an exact numerical solution for the non-equilibrium attractor in a conformal kinetic theory with number conservation, showing faster convergence to the attractor when chemical potential dynamics are included.

## Contribution

It introduces a dynamical fugacity to enforce number conservation and derives coupled integral equations for temperature and fugacity, providing the first exact solution for this system.

## Key findings

- Solutions converge to a unique non-equilibrium attractor.
- Number conservation accelerates the convergence of moments.
- Comparison with hydrodynamic models shows good agreement.

## Abstract

We extend previous studies of the conformal 0+1d kinetic non-equilibrium attractor in relaxation time approximation by enforcing number conservation through the introduction of a dynamical fugacity (chemical potential). We derive two coupled integral equations for the effective temperature and fugacity which are then solved numerically to obtain the exact solution. The resulting solutions exhibit convergence to a unique non-equilibrium attractor when the scaled moments of the distribution function are plotted as a function of the rescaled time w = tau/tau_eq. This occurs even though the system is out of chemical equilibrium at late times. In addition, compared to the case where number conservation was not imposed, we find that the moments converge to their respective attractors more quickly, particularly for moments with m=0. Finally, we compare the resulting attractor moments with predictions from different hydrodynamic frameworks.