PornÑо¿Ëù

Kyoto, Japan -- Since Austrian physicist Ludwig Boltzmann founded statistical mechanics in the late 19th century, physicists have struggled to reconcile microscopic irreversibility embodied in the second law of thermodynamics, otherwise known as time-reversal symmetry.

Then in the mid-20th century, chaos theory emerged. In physics, chaos refers to the exponential sensitivity to initial conditions which hinders long-term prediction, and chaotic dynamics highlighted mixing -- the rapid loss of memory of initial conditions -- as a mechanism of irreversibility. Yet rigorous demonstrations of mixing in time-reversal-symmetric systems have been limited to billiard tables and geodesic flows on negatively curved manifolds.

This prompted a team of researchers at PornÑо¿Ëù to construct a concrete physical model that explains why systems that are microscopically reversible can still display macroscopic irreversibility: a paradox called the arrow of time.

Using a second-order symplectic integrator called the leap-frog method -- widely employed in AI algorithms -- the researchers employed discrete time while preserving time-reversal symmetry. As a result, they were able to prove that, given certain parameters, the resulting dynamical system is an Anosov system, a strong form of mixing: all trajectories converge to a single physical equilibrium measure.

This is the first proof of Anosov chaos in a system derived from the universal form of kinetic + potential energy, known as a Hamiltonian system. Such a result may open the door to mathematically analyzing the arrow of time problem in AI and to designing next-generation chaotic sampling methods for more efficient machine learning. It also has the potential for application in cryptography or in the predictability of complex systems.

"Proving irreversibility has been my goal since the late 1990s. Our next goal is to fuse AI and chaos into a new chaos-AI framework to tackle short-term earthquake prediction," says Ken Umeno, leader of the study.

"In effect, the same symplectic integrator underlying the sampling algorithm Hamiltonian Monte Carlo becomes a key to symplectic chaotic Monte Carlo, promising faster convergence in AI sampling," continues Umeno.

Ultimately, demonstrating that a Hamiltonian system with preserved time-reversal symmetry can still show Anosov irreversibility highlights the central role of chaotic mixing in the origin of the arrow of time. Quantum mechanics lacks an analogous system of classical mixing, suggesting the need for new concepts to address irreversibility in quantum systems.