# Quantum Joule Expansion of One-Dimensional Systems

## TL;DR

This paper studies the quantum Joule expansion in one-dimensional nonintegrable systems, revealing thermalization, negative effective temperatures, and the potential to create negative temperature states dynamically, with implications for quantum thermodynamics.

## Contribution

It demonstrates how quantum Joule expansion leads to thermalization and negative temperatures in 1D systems, and proposes a method to generate negative temperature states dynamically.

## Key findings

- Diagonal and canonical ensemble weights match at high initial temperatures.
- Subsystem entropies equilibrate to thermal entropy at long times.
- Negative effective temperatures occur in finite systems and vanish in the thermodynamic limit for bosons.

## Abstract

We investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. A barrier initially confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time $t = 0$. We investigate the properties of the time-evolved density matrix, the diagonal ensemble density matrix and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation using exact diagonalization. The weights for the diagonal ensemble and the canonical ensemble match well for high initial temperatures that correspond to negative effective final temperatures after the expansion. At long times after the barrier is removed, the time-evolved R\'enyi entropy of subsystems bigger than half can equilibrate to the thermal entropy with exponentially small fluctuations. The time-evolved reduced density matrix at long times can be approximated by a thermal density matrix for small subsystems. Few-body observables, like the momentum distribution function, can be approximated by a thermal expectation of the canonical ensemble with strongly suppressed fluctuations. The negative effective temperatures for finite systems go to nonnegative temperatures in the thermodynamic limit for bosons, but is a true thermodynamic effect for fermions, which is confirmed by finite temperature density matrix renormalization group calculations. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions.