Predicting phase transitions across temperature, pressure, and chemical potential using exponentially tilted thermodynamic maps

The authors propose expTM, which inserts an exponential tilting factor e^{βαx} into the Gaussian prior of a diffusion model, turning the prior into a tilted Gaussian with mean α and variance β^{-1}, where α corresponds to pressure (α=−P) or chemical potential (α=μ) and β^{-1} to temperature. The original thermodynamic maps (TM) framework encoded only temperature fluctuations through the canonical ensemble, with a prior N(0, β^{-1}), and could not treat pressure or chemical potential; expTM maps control parameters directly onto the prior mean, generalizing the framework to the isothermal-isobaric (NPT) and grand canonical (μVT) ensembles. The method section provides a full stochastic-differential-equation derivation (forward and reverse SDEs, Eqs. 20 and 21) and learns a joint score in the (x, η) ∈ R^{3D} space rather than treating η as a fixed label; the derivation and the two demonstrations support each other.

In the grand canonical lattice-gas model, training on only two chemical potentials (μ=−16 and μ=0) at a single temperature T=7 lets expTM generate particle densities over a 625-point grid of μ ∈ [−20, 4] and T ∈ [1, 13], reproducing the critical boundary at μc=−8. Whereas the original TM used temperature as the sole control variable, this work controls temperature and chemical potential simultaneously; an ablation replacing the tilted prior with a thermodynamic-parameter-independent unit Gaussian fails to recover the sharp phase boundaries seen in MCMC. Benchmarked against Monte Carlo Metropolis-Hastings simulations, the density difference Δρ = |ρ_MC − ρ_expTM| stays within 0.05 except near the critical chemical potential μc=−8; the ablation appears in Supplementary Note 4 and Supplementary Figs. 3–5.

In the isothermal-isobaric ensemble, expTM trained on configurations at P=1 GPa and P=8 GPa predicts CO2 phase behavior across pressure and temperature, identifying an intermediate state between Phase I and Phase III at roughly P=4.5–5.4 GPa. This demonstration moves expTM from a lattice model to a real molecular crystal, using atom-wise collective variables λ1 and λ3 (two 256-dimensional feature vectors) to describe phase states and showing the ability to capture structural arrangements in high- and low-pressure regions beyond MD simulation capabilities. Training data come from enhanced-sampling MD (well-tempered metadynamics) at 350 K and 1, 3, 5, and 8 GPa, with 10^5 configurations selected per pressure according to the Boltzmann distribution; the phase boundary uses a λ1 threshold of 0.58 and a 0-to-1 closeness-to-Phase-III metric, and the free-energy difference ΔG_{I-III} is reported with error bars from three block averages.

expTM offers a computational advantage over MCMC when sampling across multiple thermodynamic conditions, and its prior mean and variance correspond to pressure/chemical potential and temperature while retaining the tractability and simplicity of a Gaussian prior. The original TM mapped only temperature fluctuations onto the prior variance; expTM adds a fluctuation variable acting on the prior mean, making the prior more expressive without sacrificing Gaussian tractability. A wall-clock time benchmark comparison is given in Supplementary Note 3; the theory yields the analytic results ⟨x⟩=α and Var(x)=β^{-1}.

The framework targets settings where limited observations are available at a few thermodynamic conditions far from the phase boundary and one wishes to infer equilibrium behavior at other temperatures, pressures, or chemical potentials; the two demonstrations are a lattice-gas model in the grand canonical ensemble and CO2 crystal in the isothermal-isobaric ensemble, trained on only two data points and on configurations at two pressures, respectively. For researchers who want to extrapolate phase diagrams from a small number of simulations or experiments and need thermodynamically consistent samples, the approach offers a reusable strategy; it presumes the system has well-defined phase boundaries and that the chosen features (such as λ1 and λ3) can distinguish the target phases.

A careful reader may still watch: near the critical chemical potential μc=−8 the density difference exceeds 0.05, so accuracy near critical points is a direction for further examination; the CO2 phase-boundary determination relies on the adjusted λ1 threshold of 0.58, and the robustness of the intermediate-state interval under different thresholds is worth observing; training data at each of the two pressures cover only a single phase, so extension to more phases or more extreme conditions remains to be verified; the suggested applications to spin glasses, crystal nucleation, intrinsically disordered proteins, and RNA are outlooks without results in this paper; and this is an Article in Press version, so the ablation, timing benchmark, and MD setup details in the supplementary material were not loaded with the main text and would need to be consulted in the supplementary information to assess those details.