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  • Open Access

Deep Generative Modeling of the Canonical Ensemble with Differentiable Thermal Properties

Shuo-Hui Li1,*,†, Yao-Wen Zhang1,†, and Ding Pan1,2,‡

  • *Contact author: shuohuili@ust.hk
  • †These authors contributed equally to this work.
  • ‡Contact author: dingpan@ust.hk
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Phys. Rev. Lett. 135, 027301 – Published 8 July, 2025

DOI: https://doi.org/10.1103/8wx7-kyx8

Abstract

It is a long-standing challenge to accurately and efficiently compute thermodynamic quantities of many-body systems at thermal equilibrium. The conventional methods, e.g., Markov chain Monte Carlo, require many steps to equilibrate. The recently developed deep learning methods can perform direct sampling, but only work at a single trained temperature point and risk biased sampling. Here, we propose a variational method for canonical ensembles with differentiable temperatures, which gives thermodynamic quantities as continuous functions of temperature akin to an analytical solution. The proposed method is a general framework that works with any tractable density generative model. At optimal, the model is theoretically guaranteed to be the unbiased Boltzmann distribution. We validated our method by calculating phase transitions in the Ising and XY models, demonstrating that our direct-sampling simulations are as accurate as Markov chain Monte Carlo methods but more efficient. Moreover, our differentiable free energy aligns closely with the exact one to the second-order derivative, indicating that the variational model captures the subtle thermal transitions at the phase transitions. This functional dependence on external parameters is a fundamental advancement in combining the exceptional fitting ability of deep learning with rigorous physical analysis.

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Phys. Rev. Lett. 135, 027301– Published 8 July, 2025
  • Received 13 July 2024
  • Revised 15 March 2025
  • Accepted 10 June 2025
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