Researchers Achieve Breakthrough in Understanding Complex Systems in Thermodynamics

Researchers Achieve Breakthrough in Understanding Complex Systems in Thermodynamics

Sept. 18 2024 | By Dave DeFusco

Researchers from the UNC-Chapel Hill have made a significant breakthrough in understanding complex systems in thermodynamics, chemistry and biophysics by introducing a new way to distinguish different systems based on their behaviors, or trajectories, according to their study, “Stochastic Distinguishability of Markovian Trajectories,” published in The Journal of Chemical Physics.

In fields like thermodynamics, chemistry and biophysics, it’s crucial to understand how different systems behave over time by analyzing the processes and mechanisms that govern them. Traditionally, researchers have used a tool called the Kullback-Leibler (KL) divergence to measure the differences between sets of trajectories; however, calculating this has been difficult due to sampling issues and the lack of clarity over which aspects of the systems’ behaviors are responsible for their distinguishability.

The researchers address these challenges by developing a formula that simplifies the calculation of the KL divergence by breaking it down into space and time components. This formula works for any Markov process—which describe how systems change over time—whether they are stable or not.

“The study also establishes a direct link between the KL divergence and specific events in the trajectories, such as individual transitions and the times between them,” said Lu, senior author of the paper and an assistant professor of chemistry. “This connection provides a deeper understanding of what makes the systems different and could lead to innovative designs for biological sensors and improvements in signal transmission within cells.”

Markov models are essential for describing how systems behave over time in physics, chemistry and biology. They are particularly valuable for studying complex systems that are not in equilibrium, like random thermodynamic processes, chemical reactions and molecular movements in biology.

“Distinguishing between different Markov systems is vital for solving practical problems,” said Asawary Pagare, a co-author of the paper and a Ph.D. student in chemistry. “For example, it helps in identifying mutations in proteins by comparing the behavior of mutant and normal proteins and recognizing signal patterns using sensors.”

The researchers use a mathematical framework, called the master equation, to describe Markov processes. This equation helps understand the probability of a system being in a certain state at a certain time and how it transitions between states.

The new approach splits the KL divergence into initial state differences and differences accumulated over time, making it clearer what distinguishes two Markov systems. Observing the trajectories of Markov processes allows for better differentiation between them. This is crucial for designing better biological sensors and understanding system responses to different signals. The study links KL divergence to thermodynamic entropy, a measure of disorder in a system. For certain Markov processes, KL divergence can represent changes in the system’s environment.

The theory aids in designing sensors that detect different input signals by observing changes in internal states over time, improving pattern recognition capabilities. To distinguish between two similar systems, using specific control protocols, like changing temperature or applying an electric field, can highlight their differences when they are not in a steady state.

“This breakthrough not only enhances our understanding of complex systems but also opens up new possibilities for technological advancements in multiple scientific fields,” said Zhongmin Zhang, a co-author of the paper and Ph.D. student in chemistry.

Measuring differences between long trajectories can be challenging, but breaking paths into shorter segments simplifies the calculation of overall differences. Another method is counting transition events and using a formula to calculate differences based on these counts.

“This study provides a new, efficient way to compare evolving systems with wide-ranging applications in various scientific fields, from thermodynamics to sensor technology,” said Jiming Zheng, a co-author of the paper and Ph.D. student in chemistry. “The new formula and theoretical framework offer deeper insights and practical tools for distinguishing complex stochastic systems, promising advancements in biological sensor design, signal processing and beyond.