# Scaling laws for harmonically trapped two-species mixtures at thermal equilibrium

## TL;DR

This paper investigates how interaction energy scales with particle number in a two-species mixture at thermal equilibrium, revealing how interaction strength, range, and system dimensionality influence the scaling laws.

## Contribution

It introduces a comprehensive analysis of scaling laws for interaction energy in two-species mixtures, bridging classical models and numerical simulations across different interaction regimes.

## Key findings

- Scaling laws depend on interaction strength, nature, and dimensionality.
- Numerical simulations confirm analytic predictions.
- A threshold between two distinct interaction regimes is identified.

## Abstract

We discuss the scaling of the interaction energy with particle numbers for a harmonically trapped two-species mixture at thermal equilibrium experiencing interactions of arbitrary strength and range. In the limit of long-range interactions and weak coupling, we recover known results for the integrable Caldeira-Leggett model in the classical limit. In the case of short-range interactions and for a balanced mixture, numerical simulations show scaling laws with exponents that depend on the interaction strength, its attractive or repulsive nature, and the dimensionality of the system. Simple analytic considerations based on equilibrium statistical mechanics and small interspecies coupling quantitatively recover the numerical results. The dependence of the scaling on interaction strength helps to identify a threshold between two distinct regimes. Our thermalization model covers both local and extended interactions allowing for interpolation between different systems such as fully ionized gases and neutral atoms, as well as parameters describing integrable and chaotic dynamics.