# [Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas](positive-temperature-hard-spheres.pdf)

**Author:** OpenAI

**Date:** October 5, 2026

## Citation

```bibtex
@misc{OAI:Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026,
  author = {{OpenAI}},
  title = {{Bose--Einstein condensation at positive temperature in the dilute hard-sphere gas}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf}{OAI:Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026}},
  year = {2026}
}
```