A mathematical framework originally developed to solve exactly solvable equations is being applied to chaotic systems that were thought to lie beyond its reach. Statistical mechanics, the theory that connects the behaviour of individual atoms to the properties of bulk materials like temperature and pressure, works well for systems that have reached equilibrium. But many real-world systems—from the flow of granular materials to the dynamics of biological molecules—never settle into equilibrium, and existing theory struggles to explain their behaviour. This project tackles that gap by testing whether tools from integrable systems theory, which describes systems with enough hidden constants to make them exactly solvable, can also describe systems that are chaotic and non-integrable. If successful, the work could provide a new mathematical foundation for statistical mechanics that bridges microscopic and macroscopic properties without requiring equilibrium. The project also aims to develop analytic methods for calculating Lyapunov exponents—a key measure of chaos—which are currently obtained only through numerical simulation. This is fundamental science with no immediate practical application, but deeper understanding of how order and chaos coexist in many-body systems could eventually inform models of heat transport, material stability, or energy dissipation in complex systems.
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This project aims to approach statistical mechanics using elements from integrable systems theory. This mathematical and computational approach intends to show that integrable systems theory is invaluable even beyond its applicability boundaries within a traditional perturbation theory framework. Integrable systems theory was mainly developed after the mid-20th century and describes dynamical systems that are ‘solvable’, meaning they contain a sufficient number of conserved quantities that constrain the dynamics of the system. Integrable systems became widely known after the discovery of special solutions called solitons in the Korteweg-De Vries equation. When a system is non-integrable, then these conserved quantities do not exist. Systems with chaotic and/or ergodic behaviour are typically non-integrable systems. In recent years, there has been a surge in investigating the enigmatic persistence of non-equilibrium phenomena in many-body Hamiltonian systems. Perturbation theory can partially explain, and within a limited scope, why non-integrable systems share many similarities with their integrable counterparts. Instead, there is a broad range of values and system parameters associated with non-equilibrium behaviour. This project aims to demonstrate that tools and methodologies derived from integrable systems theory can provide an ideal framework for studying Hamiltonian systems with many degrees of freedom, even at high energies or more generally under large perturbations, where stable integrable-like structures cease to exist. This research will investigate the role of adiabatic invariants, quantities that have the potential to reshape the framework of statistical mechanics and bring together microscopic and macroscopic properties of a many-body Hamiltonian system. Based on these findings, the project will conclude with a study on analytic methods for evaluating Lyapunov exponents, a well-established tool in dynamical systems theory that quantifies the extent of chaos in a system. Despite their significant applications, Lyapunov exponents are typically obtained numerically. This research aims to develop analytic methods for deriving these exponents in various classes of many-body Hamiltonian systems.
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