Geometry-protected Probabilistic Structure Concentrates Measure in Hamiltonian Dynamics of Both Finite and Infinite-dimensional Systems

Understanding how complex systems evolve over time remains a central challenge in physics, and recent work by Yue Liu, Chushun Tian, and Dahai He sheds new light on this problem. The researchers investigate the dynamics of isolated systems, moving beyond traditional statistical mechanics which relies on averaging over many possibilities, to focus on the behaviour of a single, mechanically isolated system. Their findings reveal a surprisingly robust probabilistic structure, protected by the geometry of the system’s phase space, that governs the behaviour of both finite and infinite-dimensional systems. This structure explains how an isolated system, like the well-studied Fermi-Pasta-Ulam-Tsingou model, can behave like a thermal gas, and how another, the Gross-Pitaevskii equation, avoids a predicted energy crisis through a process of wave localisation, offering potential implications for fields like nonlinear optics and the study of cold atoms.

Addressing many-body dynamics remains a significant challenge, particularly deriving macroscopic properties from solutions to equations that describe the microscopic motion of isolated systems. This work advances our understanding of this long-standing problem, shifting the focus from statistical ensembles to the behavior of individual, isolated systems. Scientists uncovered a common probabilistic structure, known as concentration of measure, within the Hamiltonian dynamics of two distinct systems: the Fermi-Pasta-Ulam-Tsingou (FPUT) model and the Gross-Pitaevskii equation (GPE).

Geometry, Topology, and Thermal Equilibrium Dynamics

This research investigates how systems reach thermal equilibrium, a state of uniform energy distribution. It challenges traditional assumptions about ergodicity, the idea that a system explores all possible states, and proposes a new perspective based on the interplay of geometry, topology, and statistical mechanics. The team demonstrates that thermalization doesn’t necessarily require complete exploration of state space, but rather arises from a concentration of probability around a specific region dictated by the system’s underlying structure. They connect these theoretical ideas to observed phenomena in diverse physical systems, ranging from classical mechanics to quantum chaos and ultracold atoms.

The research demonstrates that ergodicity is not always a necessary condition for thermalization. Many systems may not fully explore all possible states, yet still reach a stable thermal equilibrium. The geometry and topology of the system’s phase space play a crucial role in determining how quickly and effectively thermalization occurs. Certain geometric structures can concentrate probability, leading to faster thermalization even without full ergodicity. This framework applies to a wide range of physical systems.

In the FPUT problem, it explains the surprising lack of equipartition of energy in nonlinear lattices. In quantum chaos, it helps understand the behavior of quantum systems exhibiting chaotic behavior. For ultracold atoms, it explains recent experiments on thermalization. It even provides insight into the Rayleigh-Jeans distribution observed in multimode optical fibers.

Phase Space Concentration Unifies Dynamics

This research establishes a common probabilistic structure within the seemingly disparate Hamiltonian dynamics of the FPUT model and the GPE. The team demonstrates that both systems, despite differing dimensionality and ergodic properties, exhibit a concentration of measure within their phase space, a structure protected by the geometry of that space and independent of ergodicity. This finding challenges conventional understanding of how macroscopic properties emerge from microscopic dynamics. Specifically, the researchers show that a single trajectory of the FPUT model behaves as an ideal gas, even with strong interactions between modes, and that thermalization occurs through a process of “self-repulsion” within the trajectory itself. For the GPE, which describes Bose-Einstein condensates and wave propagation, the analysis reveals nonlinear wave localization and the establishment of Rayleigh-Jeans thermal equilibrium within a defined volume, effectively resolving the ultraviolet catastrophe within a purely classical framework. These results suggest that established statistical mechanics may not fully capture the behaviour of isolated many-body systems.

The underlying principle driving these results is a geometric phenomenon called concentration of measure. Scientists proved that in high-dimensional spaces, subsets with sufficient measure concentrate on extremely small regions. This concentration manifests as a near-constancy of Lipschitz-continuous functions, meaning their values are almost uniform across the space. The team demonstrated that this concentration of measure is naturally embedded within the Hamiltonian dynamics of both the FPUT and GPE systems, providing a unifying framework for understanding their statistical behavior. These findings have implications for nonlinear optics and cold-atom dynamics, offering new insights into the behavior of complex systems. The authors acknowledge that their theory currently applies to the specific models investigated, and further work is needed to determine its generality. They also note the potential for applying these findings to areas such as nonlinear optics and cold-atom dynamics, suggesting avenues for future research into the statistical properties of isolated systems and the emergence of macroscopic behaviour from microscopic origins.