Sampling the Liquid-Gas Critical Point with Boltzmann Generators

Abstract

Generative models based on invertible transformations provide a physics-aware route to sample equilibrium configurations directly from the Boltzmann distribution, enabling efficient exploration of complex thermodynamic landscapes. Here, we evaluate their applicability in regions where conventional simulations suffer from severe dynamical bottlenecks, focusing on the liquid-gas critical point of a Lennard-Jones fluid. We show that Boltzmann Generators capture essential signatures of critical behavior, retain reliable performance when trained at or near criticality, and extrapolate across neighboring states of the phase diagram. An intriguing observation is that the model's efficiency metric closely traces the underlying phase boundaries, hinting at a connection between generative performance and thermodynamics. However, the approach remains limited by the small system sizes currently accessible, which suppress the large fluctuations that characterize critical phenomena. Our results delineate the current capabilities and boundaries of Boltzmann Generators in challenging regions of phase space, while pointing toward future applications in problems dominated by slow dynamics, such as glass formation and nucleation.

Figures (7)

• Figure 1: (a) Loss function, (b) Wasserstein distance, and (c) Effective Sample Size as a function of epochs for the liquid state point training.
• Figure 2: Energy density probability both for reference and generated samples. Inset: corresponding radial distribution function $g(r)$, reference results are shown as a shaded area.
• Figure 3: Efficiency maps of the liquid training as relative Wasserstein distances (a) and Effective Sample Size (b) between reference and generated configurations. The yellow point represent the training state point. As a conditional grid for efficiency evaluation we used a $5$ by $7$ mesh over $T^*=\left[1.1,1.5\right]$ and $P^*=\left[4,10\right]$. The black dashed curve, representing the liquid-solid coexistence, is adapted from Ref. schebek2024ccbylicense. The yellow full curve represent the system isomorph.
• Figure 4: (a) Loss function, (b) Wasserstein distance, and (c) ESS as a function of epochs for the critical state point training.
• Figure 5: (a) Efficiency map for training at the critical point, shown in terms of relative Wasserstein distances between the reference and generated configurations. The yellow point denotes the training state corresponding to the critical point . For the conditional evaluation of the efficiency, we combine a $4 \times 4$ grid spanning $T^* \in [1.12,1.15]$ and $P^* \in [0.10,0.13]$ with a $5 \times 5$ grid spanning $T^* \in [1.133,1.141]$ and $P^* \in [0.114,0.118]$. Grey points indicate the coexistence line as determined from the Maxwell construction, while the yellow line denotes the system isomorph. (b) An analogous representation of the ESS map.