Aubry-Mather Theory

Aubry-Mather theory is a branch of dynamical systems theory that studies the persistence of quasi-periodic motion in systems that are perturbations of integrable ones. Named after Serge Aubry and John Mather, who developed the theory independently in the 1980s, it provides a rigorous framework for understanding the transition from order to chaos in Hamiltonian systems. The theory is most famous for its application to the Frenkel-Kontorova model — a model of a chain of atoms in a periodic potential — and to the study of twist maps, a class of area-preserving maps that include the standard map and the Chirikov-Taylor map.

The central object of Aubry-Mather theory is the Mather set (or Aubry-Mather set): a closed, invariant set that is a generalization of a KAM torus. A KAM torus is a smooth, invariant torus on which the dynamics is quasi-periodic. The KAM theorem guarantees that such tori persist under small perturbations of integrable systems, provided the frequencies satisfy a Diophantine condition. But KAM tori do not persist for all perturbations: as the perturbation grows, the tori break down, and the system transitions from quasi-periodic motion to chaotic motion.

Aubry-Mather theory describes what happens after the breakdown. Even when the KAM tori are destroyed, the system retains invariant sets that are the ghosts of the tori: Mather sets that are not smooth but are still ordered, still invariant, and still carry quasi-periodic dynamics in a generalized sense. These sets bridge the gap between order and chaos, providing a continuous family of invariant structures that exist for all parameter values, not just small perturbations.

The Variational Principle

Aubry-Mather theory is built on a variational principle. The dynamics of a twist map is generated by a generating function, and the orbits of the map are the critical points of an action functional. The Mather sets are the orbits that minimize the action globally, not just locally. This variational structure is what makes the theory so powerful: it provides a selection principle that picks out the most important orbits from the infinite set of possible orbits.

The theory introduces the Mather's rotation number (or frequency), which generalizes the rotation number of circle homeomorphisms to higher dimensions. For each irrational rotation number, there exists a Mather set with that rotation number. The set is a graph of a function if the function is continuous — this is the case for KAM tori — but it can also be a Cantor set if the function is discontinuous. The Cantor set case is the cantorus: an invariant set that is a Cantor set, ordered by the rotation number, but not a smooth torus.

The existence of cantori is one of the most striking results of Aubry-Mather theory. It shows that the breakdown of a KAM torus is not a sudden disappearance but a gradual transformation: the torus becomes increasingly wrinkled, its graph becomes increasingly discontinuous, and eventually it shatters into a Cantor set. But the Cantor set is still invariant, still ordered, and still carries the same rotation number. The order persists even when the smoothness is lost.

The Connection to Chaos and Transport

Aubry-Mather theory has profound implications for the study of chaos in Hamiltonian systems. The cantori act as barriers to transport: trajectories cannot cross from one side of a cantorus to the other without passing through a gap in the Cantor set. But the gaps are narrow, and the transport through them is slow. This is the mechanism of slow chaos or weak chaos: the system is chaotic in the sense that nearby trajectories diverge, but the divergence is slow because the cantori act as partial barriers.

As the perturbation increases, the cantori become more fragmented, the gaps become larger, and the transport becomes faster. At some critical perturbation strength, the last cantorus breaks, and the system undergoes a global transition to strong chaos. This transition is a phase transition in the dynamical system: the transport properties change qualitatively, the diffusion coefficient jumps, and the system moves from a regime of slow, anomalous diffusion to a regime of fast, normal diffusion.

The theory also connects to the study of Arnold diffusion — the slow drift of action variables in nearly integrable Hamiltonian systems. The cantori are the barriers that contain the Arnold diffusion, and their structure determines the rate of diffusion. The gaps in the cantori are the channels through which the diffusion occurs, and the size of the gaps determines the diffusion coefficient. Aubry-Mather theory provides the tools for computing these gaps and predicting the diffusion rate.

The Frenkel-Kontorova Model

The Frenkel-Kontorova model is a one-dimensional chain of atoms connected by harmonic springs and subject to a periodic potential. It was introduced in 1938 to study the motion of dislocations in crystals, and it has since become a paradigm for the study of commensurate-incommensurate phase transitions, sliding friction, and the dynamics of driven systems.

In the Frenkel-Kontorova model, the Aubry-Mather theory describes the ground states of the chain as a function of the mismatch between the natural period of the springs and the period of the potential. When the mismatch is small (the commensurate case), the chain locks into the potential, and the ground state is periodic. When the mismatch is large (the incommensurate case), the chain slides freely, and the ground state is quasi-periodic. The transition between the two regimes is the Aubry transition, and it is described by the Aubry-Mather theory.

The Aubry transition is a phase transition in the ground state structure. Below the transition, the ground state is a smooth, periodic function. Above the transition, the ground state is a discontinuous, Cantor-like function — a cantorus. The transition is continuous in the sense that the ground state changes continuously as a function of the mismatch, but it is discontinuous in the sense that the smoothness is lost abruptly. The critical behavior at the transition is described by a universal exponent, analogous to the critical exponents in equilibrium phase transitions.

The Frenkel-Kontorova model also connects to the theory of sliding friction. The static friction force is the force required to depin the chain from the potential. Below the Aubry transition, the static friction is finite: the chain is pinned. Above the transition, the static friction is zero: the chain slides freely. The Aubry transition is thus the transition from stick to slip, and the critical behavior at the transition determines the scaling of the friction force with the system size.

The Philosophy of Order and Disorder

Aubry-Mather theory is a theory of the boundary between order and chaos. It does not ask whether a system is ordered or chaotic; it asks how the system moves from one to the other, and what structures persist during the transition. The Mather sets are the fossils of order: they are the structures that remain when the smoothness is gone, the order that persists when the chaos has taken over.

This has philosophical implications for the concept of emergence. The cantori are emergent structures in the sense that they are not present in the integrable system and they appear only when the perturbation is strong enough. But they are also remnants of the integrable structure, ghosts of the tori that once were. The emergence here is not the creation of something new but the persistence of something old in a transformed form. The cantorus is a torus that has been shattered but not destroyed.

The theory also has implications for the concept of predictability. In a system with KAM tori, the motion is predictable because the tori constrain the trajectories to regular, quasi-periodic orbits. In a system with cantori, the motion is partially predictable: the trajectories are constrained by the cantori, but they can leak through the gaps. The predictability is not all or nothing; it is a continuous function of the perturbation strength, and Aubry-Mather theory provides the quantitative framework for describing this continuity.

Aubry-Mather theory is the study of broken symmetries that refuse to die. The tori shatter, but the shards remember their shape. The order is lost, but the memory of order persists, and it is that memory — the cantorus, the ghost of the torus — that governs the transport, the diffusion, and the slow chaos that follows.