School of Physics Thesis Dissertation Defense

Han Liang
Advisor: Dr. Predrag Cvitanović, School of Physics, Georgia Institute of Technology 

A Deterministic Lattice Field Theory of Spatiotemporal Chaos

Date: Monday, May 12, 2025
Time: 1:00 p.m.
Location: Howey N201/N202

Zoom link:  https://gatech.zoom.us/j/98851319184

Committee members:
Dr. Elisabetta Matsumoto,  School of Physics, Georgia Institute of Technology

Dr. Martin Mourigal,  School of Physics, Georgia Institute of Technology

Dr. Zeb Rocklin, School of Physics, Georgia Institute of Technology
Dr. Luca Dieci, School of Mathematics, Georgia Institute of Technology

Abstract:
Traditional periodic orbit theory enables the evaluation of statistical properties of finite-dimensional chaotic dynamical systems through the

hierarchy of their periodic orbits. However, this approach becomes impractical for spatiotemporally chaotic systems over large or infinite

spatial domains. As the spatial extents of these systems increase, the physical dimensions grow linearly, requiring exponentially more distinct

periodic orbits to describe the dynamics to the same accuracy. To address this challenge, we propose a novel approach, describing spatiotemporally

chaotic or turbulent systems using the chaotic field theories discretized over multi-dimensional spatiotemporal lattices. The `chaos theory' is

here recast in the language of statistical mechanics, field theory, and solid state physics, with traditional periodic orbit theory of

low-dimensional, temporally chaotic dynamics a special, one-dimensional case.

In this field-theoretical formulation, there is no time evolution. Instead, by treating the temporal and spatial directions on equal footing,

one determines the spatiotemporally periodic states that contribute to the theory's partition function, each a solution of the system's

deterministic defining equations, with sums over time-periodic orbits of dynamical systems theory now replaced by sums of d-periodic states over

d-dimensional spacetime geometries, weighted by their global orbit stabilities.

The orbit stability of each periodic state is evaluated using the determinant of its spatiotemporal orbit Jacobian matrix. We derive the

Hill's formula, which relates the global orbit stability to the conventional low-dimensional forward-in-time evolution stability, and

show that the field-theoretical formulation is equivalent to the temporal periodic orbit theory for systems with fixed finite spatial extent. By

summing the partition functions over different spacetime geometries, we extend the temporal periodic orbit theory to spatiotemporal systems. The

multiple periodicities of spatiotemporally periodic states are described in the language of crystallography using Bravais lattices. Applying the

Floquet-Bloch theorem to evaluate the spectrum of orbit Jacobian operators of periodic states, we compute their multiplicative weights, leading to

the spatiotemporal zeta function formulation of the theory in terms of prime orbits. Hyperbolic shadowing of periodic orbits by pseudo orbits

ensures that the predictions of the theory are dominated by the prime periodic orbits with shortest spatiotemporal periods.