Inferring Phase Transitions and Critical Exponents from Limited Observations with Thermodynamic Maps

Academic Background

Phase transitions are ubiquitous phenomena in nature, ranging from the boiling of water to the ferromagnetic-paramagnetic transition in magnetic materials, and even to conformational changes in biomolecules such as proteins and nucleic acids. Despite their prevalence, accurately quantifying phase transitions and their temperature-dependent characteristics remains a significant challenge, especially in cases where data is sparse or complex. Traditional statistical mechanics methods provide a theoretical framework for studying phase transitions, but in practice, computing phase transition features (such as critical temperature, heat capacity, and critical exponents) often requires substantial computational resources due to the difficulty of sampling the transition region.

To address this issue, researchers Lukas Herron, Kinjal Mondal, John S. Schneekloth Jr., and Pratyush Tiwary proposed a novel method called “Thermodynamic Maps” ™. This approach combines statistical mechanics, molecular simulations, and score-based generative models to infer phase transition characteristics from limited observational data, particularly from stable phases far from the transition region. This research not only provides a new tool for quantifying phase transitions but also opens new avenues for studying complex systems.

Source of the Paper

The paper, co-authored by Lukas Herron, Kinjal Mondal, John S. Schneekloth Jr., and Pratyush Tiwary, was published on December 16, 2024, in the Proceedings of the National Academy of Sciences (PNAS) under the title Inferring Phase Transitions and Critical Exponents from Limited Observations with Thermodynamic Maps. The authors are affiliated with the Biophysics Program and Institute for Physical Science and Technology at the University of Maryland, the Chemical Biology Laboratory at the National Cancer Institute, and the Department of Chemistry and Biochemistry at the University of Maryland.

Research Process and Results

1. Proposal and Design of Thermodynamic Maps ™

The core idea of Thermodynamic Maps is to map the temperature dependence of a complex system onto a simple, idealized system, thereby efficiently generating samples with correct Boltzmann weights. Specifically, TM combines free energy perturbation theory with score-based generative models to learn the temperature dependence of the partition function and, consequently, the free energy.

1.1 Free Energy Perturbation Theory

Free Energy Perturbation (FEP) is a classical method for calculating free energy differences. TM builds on this by introducing an invertible mapping that increases the overlap between different states, thereby improving the efficiency of free energy estimation. Specifically, TM uses neural networks to represent this mapping and learns the score (the gradient of the probability density) by optimizing a score-matching objective function.

1.2 Application of Non-Equilibrium Thermodynamics

TM also leverages the properties of non-equilibrium thermodynamics, particularly diffusion processes. By modeling the diffusion process as a Fokker-Planck equation, TM can map any initial distribution to a Gaussian distribution, thereby increasing the overlap between different states. The reversibility of the diffusion process ensures the existence of an inverse mapping, allowing TM to generate samples of the complex system from a simple system.

2. Application of TM to the Ising Model

To validate the effectiveness of TM, the researchers first applied it to the two-dimensional Ising model, a classic model for studying phase transitions. The Ising model exhibits a well-defined ferromagnetic-paramagnetic phase transition with a critical temperature (Tc) and critical exponents. The researchers generated configurations of the Ising model using Monte Carlo (MC) sampling and trained TM using data from only two temperatures. The results showed that TM could accurately infer the critical temperature and generate samples with correct critical behavior, even though the training data did not include samples from the transition region.

Specifically, the magnetization and heat capacity predicted by TM exhibited divergence near the critical temperature, consistent with MC sampling. Although the critical exponents predicted by TM deviated slightly from the theoretical values due to finite-size effects, the inference capability of TM remained remarkable.

3. Application of TM to RNA Systems

To further demonstrate the broad applicability of TM, the researchers applied it to two RNA systems: the GCAA tetraloop and HIV-TAR RNA. The conformational transitions of these RNA systems are difficult to sample due to their glass-like energy landscapes. By combining bioinformatics approaches and multi-ensemble molecular dynamics simulations, the researchers used TM to efficiently describe the conformational distributions of RNA and compute their melting curves.

3.1 GCAA Tetraloop

The GCAA tetraloop is a highly stable RNA sequence whose conformational diversity primarily arises from the arrangement of nucleotides in the loop region. Using TM-accelerated Molecular Dynamics (TM-AMD), the researchers generated the conformational distribution of the GCAA tetraloop and predicted its temperature-dependent free energy changes. The results showed that the conformational distribution generated by TM was consistent with experimental and molecular dynamics simulation data, although the current force field still exhibited some deviations in temperature dependence.

3.2 HIV-TAR RNA

HIV-TAR RNA is an RNA hairpin structure with rich conformational diversity, where the loop and bulge regions play key roles in interactions with proteins and small molecules. Using TM-AMD, the researchers inferred the global equilibrium distribution of HIV-TAR RNA and computed its melting curve. The results showed that the melting temperature predicted by TM agreed with experimental data, indicating the potential of TM in describing conformational transitions in complex RNA systems.

Conclusions and Significance

The Thermodynamic Maps ™ method proposed in this study provides an efficient and general tool for quantifying phase transitions. By combining statistical mechanics, molecular simulations, and generative artificial intelligence, TM can infer phase transition characteristics from limited observational data, particularly in cases where data is sparse or complex. The results demonstrate that TM can not only accurately predict the critical behavior of the Ising model but also efficiently describe the conformational transitions and melting curves of RNA systems.

Research Highlights

1. Innovative Method: TM combines free energy perturbation theory with score-based generative models, offering a novel approach to quantifying phase transitions.
2. Broad Applicability: TM is applicable not only to the classic Ising model but also to complex biomolecular systems such as RNA.
3. Computational Efficiency: TM can efficiently infer phase transition characteristics without requiring samples from the global equilibrium distribution, significantly reducing computational costs.

Application Value

The introduction of TM provides a new tool for studying complex systems, especially in cases where data is scarce or computational resources are limited. In the future, TM is expected to find widespread applications in materials science, biophysics, and chemistry, helping researchers gain deeper insights into phase transitions and their manifestations in complex systems.

Additional Valuable Information

The researchers also explored the potential applications of TM in spin glasses and dynamics studies and proposed further optimizations for the TM-AMD method. Additionally, the Python implementation of TM has been open-sourced for other researchers to use and validate.

This research not only provides a new theoretical framework for quantifying phase transitions but also opens new directions for studying complex systems, holding significant scientific and application value.