Emergent order from mixed chaos at low temperature (2509.11583v1)

Abstract: This paper explores a novel connection between a thermodynamic and a dynamical systems perspective on emergent dynamical order. We provide evidence for a conjecture that Hamiltonian systems with mixed chaos spontaneously find regular behavior when minimally coupled to a thermal bath at sufficiently low temperature. Numerical evidence across a diverse set of five dynamical systems supports this conjecture, and allows us to quantify corollaries about the organization timescales and disruption of order at higher temperatures. Balancing the damping-induced phase-space contraction against thermal exploration, we are able to predict the transition temperatures in terms of the relaxation timescales, indicating a novel nonequilibrium fluctuation-dissipation relation, and formally connecting the thermodynamic and dynamical systems views. Our findings suggest that for a wide range of real-world systems, coupling to a cold thermal bath leads to emergence of robust, non-trivial dynamical order, rather than a mere reduction of motion as in equilibrium.

• The paper demonstrates that weak damping coupled with low-temperature effects drives chaotic trajectories into islands of regularity, as confirmed by extensive numerical simulations.
• It employs simulations across diverse dynamical systems to reveal a power-law scaling between damping and regularization timescales, highlighting predictable relaxation dynamics.
• The study establishes a nonequilibrium fluctuation-dissipation relation that connects thermal noise with emergent order, offering insights into self-organization in many-body systems.

Emergent Order in Mixed-Chaotic Hamiltonian Systems Coupled to Low-Temperature Baths

Introduction and Theoretical Context

This paper investigates the emergence of dynamical order in Hamiltonian systems exhibiting mixed chaos when weakly coupled to a thermal bath at low temperature. The central conjecture posits that such systems, regardless of their initial conditions, will spontaneously settle into islands of regularity—regions of phase space characterized by vanishing local Lyapunov exponents—when subjected to minimal damping and thermal noise. This work bridges thermodynamic and dynamical systems perspectives, highlighting a nonequilibrium mechanism for self-organization that is distinct from equilibrium self-assembly and extends to many-body systems.

The theoretical framework draws on the interplay between phase-space contraction induced by damping and stochastic exploration driven by thermal noise. The authors generalize the Boltzmann distribution to nonequilibrium steady states, emphasizing the role of dissipative work histories in selecting ordered trajectories. The analysis leverages KAM theory to explain the ubiquity of mixed chaos in nonlinear Hamiltonian systems and elucidates how weak damping transforms islands of regularity into global attractors by breaking Liouville's theorem and enabling phase-space contraction.

Numerical Evidence Across Dynamical Systems

The conjecture is substantiated through extensive numerical simulations across five distinct dynamical systems: the kicked rotor (Chirikov standard map), the Zaslavsky web map, the Duffing oscillator, the bouncing ball system, and a novel many-body extension termed the Kicked Harmonic Net (KHN). In each case, the addition of weak damping and low-amplitude thermal noise leads to the collapse of trajectories onto islands of regularity, as identified by local Lyapunov exponent analysis.

Figure 1: Numerical evidence for the conjecture across five dynamical systems, showing the collapse of trajectories onto islands of regularity under weak thermal bath coupling.

The simulations demonstrate that the emergence of order is robust to the choice of system, the degree of fine-tuning required, and the complexity of the regular attractors. Notably, in the high-dimensional KHN, partial order emerges, with attractors exhibiting nontrivial fractal dimensions, indicating the possibility of complex, partially ordered steady states in many-body systems.

Quantitative Analysis of Regularization Timescales

A key practical consideration is the timescale $\tau$ required for regularization, i.e., the time for trajectories to settle into ordered regions. The authors empirically establish a power-law relationship between $\tau$ and the damping coefficient $b$, $\tau \propto b^{-\eta}$, where the scaling exponent $\eta$ depends on system parameters and the phase-space volume fraction $\epsilon$ occupied by regular islands.

Figure 2: Regularization timescale $\tau$ as a function of damping $b$, with scaling exponent $\eta$ depending on the volume fraction $\epsilon$ of regular islands.

For systems where regular islands are macroscopic ($\epsilon \gtrsim 0.05$), $\eta \approx 1$, corresponding to trivial relaxation dynamics. In regimes of strong fine-tuning (small $\epsilon$), $\eta > 1$, indicating anomalously slow regularization due to the rarity of ordered states and the stickiness of chaotic trajectories near regular islands. In many-body systems, $\eta$ appears less sensitive to $\epsilon$, suggesting a form of universality dominated by collective modes and self-averaging.

Breakdown of Order at Finite Temperature and Nonequilibrium Fluctuation-Dissipation Relation

The stability of emergent order is further analyzed as a function of thermal noise. The transition temperature $T^*$, above which order is destroyed, is found to scale as a power law with $b$, $T^* \propto b^\gamma$, with the exponent $\gamma$ related to $\eta$ via $\gamma = \eta - 1$. This relation is interpreted as a nonequilibrium generalization of the Einstein relation, connecting relaxation dynamics and steady-state fluctuations.

Figure 3: Phase diagram of the kicked rotor showing the dependence of self-organization probability on temperature $T$ and kick strength $K$, and the scaling of transition temperature $T^*$ with damping $b$.

The analysis employs a two-state Markov model to estimate transition rates between chaotic and regular regions, yielding a generalized fluctuation-dissipation relation:

$D/\mu = b^{1-\eta} T$

where $D$ is the effective diffusivity and $\mu$ the motility. Deviations from this relation in many-body systems are attributed to noise amplification within partially ordered attractors, highlighting the need for further theoretical refinement.

Implications and Future Directions

The results have significant implications for the control and understanding of nonequilibrium self-organization in physical and engineered systems. The conjecture provides a predictive framework for the emergence and stability of dynamical order in systems with mixed chaos, with quantitative tools for estimating relaxation times and phase boundaries. The findings suggest that cooling or damping does not merely suppress motion but actively stabilizes nontrivial dynamical patterns, a phenomenon with potential applications in areas ranging from pattern formation to robust machine learning architectures inspired by the KHN.

The observed scaling laws and fluctuation-dissipation relations open avenues for developing a more general theory of nonequilibrium self-organization, particularly in high-dimensional and many-body contexts. The connection to recurrent neural network architectures and the possibility of physically realizable learning systems based on these dynamics are especially noteworthy.

Conclusion

This paper provides compelling numerical evidence and physical intuition for the conjecture that weak coupling to a low-temperature thermal bath induces robust dynamical order in Hamiltonian systems with mixed chaos. The work unifies thermodynamic and dynamical systems perspectives, introduces quantitative scaling laws for regularization timescales and transition temperatures, and proposes a nonequilibrium fluctuation-dissipation relation. The results underscore the generality and practical relevance of this mechanism for emergent order, while highlighting open questions regarding universality and the dynamics of partial order in many-body systems. Further exploration of these phenomena is likely to yield deeper insights into the principles governing nonequilibrium self-organization.