# The variational reduction for low-dimensional Fermi gases and Bose - Fermi mixtures: A brief review

## TL;DR

This paper reviews recent theoretical advances in low-dimensional Fermi and Bose-Fermi gases, focusing on dimensional reduction techniques and their accuracy in modeling ground states and dynamics within a quasi-mean-field framework.

## Contribution

It introduces a variational reduction method for low-dimensional Fermi and Bose-Fermi gases and compares reduced models with full 3D equations, demonstrating high accuracy.

## Key findings

- Good agreement between reduced and full models for ground states
- Effective 1D and 2D equations capture complex dynamics
- Dimensional reduction simplifies analysis of degenerate gases

## Abstract

We present a summary of some recent theoretical results for matter-wave patterns in Fermi and Bose-Fermi degenerate gases, obtained in the framework of the quasi-mean-field approximation. We perform a dimensional reduction from the three-dimensional (3D) equations of motion to 2D and 1D effective equations. In both cases, comparison of the low-dimensional reductions to the full model is performed, showing very good agreement for ground-state solutions. Some complex dynamical regimes are reported too for the corresponding 1D systems.