Penn Engineers have developed a mathematical "Rosetta Stone" that translates atomic and molecular movements into predictions of larger-scale effects, like proteins unfolding, crystals forming and ice melting, without the need for costly, time-consuming simulations or experiments. That could make it easier to design smarter medicines, semiconductors and more.
In a recent paper in Journal of the Mechanics and Physics of Solids , the Penn researchers used their framework, Stochastic Thermodynamics with Internal Variables (STIV), to solve a 40-year problem in phase-field modeling, a widely used tool for studying the shifting frontier between two states of matter, like the boundary between water and ice or where the folded and unfolded parts of a protein join.
"Phase-field modeling is about predicting what happens at the thin frontier between phases of matter, whether it's proteins folding, crystals forming or ice melting," says Prashant Purohit , Professor in Mechanical Engineering and Applied Mechanics (MEAM) and one of the paper's co-authors. "STIV gives us the mathematical machinery to describe how that frontier evolves directly from first principles, without needing to fit data from experiments."
In a third, related paper, in the Journal of Non-Equilibrium Thermodynamics , the researchers generalize the framework, giving it broader mathematical power. "Just as the Rosetta Stone unlocked countless ancient texts, the STIV framework can translate microscopic movements into larger-scale behavior across non-equilibrium systems," says Celia Reina , Associate Professor in MEAM and the papers' senior author.
"STIV could potentially help us design new materials," adds Reina. "In the same way the Rosetta Stone allowed scholars to compose in hieroglyphs, this framework could let us start with the property we want and work backward to the molecular movements that create it."
How STIV Works
In the 20th century, French physicist Paul Langevin pioneered mathematics to describe the activity of atoms and molecules embedded in fluctuating environments. "STIV captures the average evolution of such systems by introducing 'internal' variables, extra quantities that capture the non-equilibrium features of a system," says Travis Leadbetter (Gr'25), the papers' first author and a recent Applied Mathematics and Computational Scienc e (AMCS) doctoral graduate.
Choosing the right variables matters: like the Rosetta Stone, whose alignment of hieroglyphs with Greek and Demotic text made translation possible, STIV depends on selecting the variables that best predict the system's large-scale behavior. "You need to have some sense of the context," adds Leadbetter. "But once those variables are chosen, STIV gives you their evolution, without having to adjust the mathematics to fit experimental data each time."
However, the groups' first efforts only showed that STIV worked in a narrow subset of contexts. "We needed to generalize the mathematics," says Leadbetter. That resulted in the groups' most recent paper, which presents three methods to account for virtually any situation. "Two are quicker and cover most systems, while the other takes longer to calculate but handles rare cases," says Leadbetter. "Together they make the framework both practical and universal."
The Power of STIV
For centuries, scientists have strived to mathematically describe the world as generally as possible. The better math can describe a system, the easier that system is to analyze and ultimately control.
But for complex systems outside equilibrium, achieving that level of rigor is usually slow and costly. "If you want a rigorous model, typically it takes a long time to compute, and if you want results fast, you have to simplify and lose accuracy," says Purohit. STIV promises to overcome that tradeoff, although the upside depends on the problem to which the framework is applied.
In addition to the applications explored by the authors, researchers in the United States and Italy recently used STIV to derive new insights into how biological cells move. "STIV gives us a common language for problems that used to be treated in isolation," says Reina. "That means researchers studying subjects as varied as proteins, crystals and cells can draw on the same framework. That kind of universality points to enormous potential for future discoveries."
These studies were conducted at the University of Pennsylvania School of Engineering and Applied Science and supported by the National Science Foundation (CAREER award CMMI-2047506, DMR 2212162); the National Defense Science and Engineering Graduate Fellowship Program, sponsored by the Air Force Research Laboratory, the Office of Naval Research and the Army Research Office (FA9550-21-F-0003); the National Institutes of Health (R01 HL-148227); the American Chemical Society (PRF-61793ND10); and the University of Pennsylvania Applied Mathematics and Computational Science graduate group.
