Low-temperature bath induces order in mixed-chaotic Hamiltonian dynamics

Prove that Hamiltonian systems exhibiting mixed chaos, when weakly coupled to a thermal bath at sufficiently low temperature, converge to islands of regularity in phase space and thereby exhibit emergent dynamical order.

Background

The paper bridges thermodynamic and dynamical-systems perspectives by noting that damping and temperature are linked via the Einstein relation, so adding weak damping corresponds to coupling a Hamiltonian system to a thermal bath. In mixed-chaotic systems, islands of regularity coexist with chaotic regions; weak damping breaks Liouville’s theorem and can turn these regular islands into attractors.

The authors provide numerical evidence across five systems (including a many-body example) supporting the conjecture that low-temperature bath coupling selects regular trajectories. They explicitly leave formal proofs of this conjecture to future work, framing it as a central open problem.

References

In this paper we combine thermodynamic and dynamical systems perspectives by developing numerical evidence and physical intuition for the following conjecture: Hamiltonian dynamics exhibiting mixed chaos will settle into islands of order upon weakly coupling to a thermal bath at sufficiently low temperature. Any formal proofs of this conjecture are left to future work.

— Emergent order from mixed chaos at low temperature  (2509.11583 - Chvykov et al., 15 Sep 2025) in Conjecture 1, Introduction

Hamiltonian chaos is conjectured to be generally robust with respect to the addition of (weak, classical) noise. This gives some confidence that including noise (seismic, Brownian, laser) should not alter significantly the picture.

— Radiation-pressure induced chaos in multipendular Fabry-Perot resonators  (2609.25872 - Pierro et al., 22 Sep 2026) in Discussion following Fig. 7, comment (iii)