Rapid Turn Regime in Complex Systems

• Rapid Turn Regime is a phase in complex systems marked by sudden shifts in dynamics when critical turning parameters are reached.
• It manifests in contexts such as rotating convection-driven dynamos and multi-field inflation, altering magnetic scaling laws and inflationary trajectories.
• The regime enhances our understanding of system transitions, offering insights for predicting astrophysical phenomena and early universe behavior.

The rapid turn regime refers to dynamical transitions and sustained behaviors in complex systems, particularly in the contexts of rotating convection-driven dynamos and multi-field inflationary cosmology, where key system parameters cause trajectories, flows, or field-space orbits to change direction or character abruptly and/or for significant durations. The defining features of a rapid turn are a marked increase in the "turn rate"—a measure of non-geodesicity or non-collinearity in the relevant configuration space—that fundamentally alters the system's physical properties, mechanisms, or observable signatures. The concept finds technical realization in several domains, from magnetohydrodynamics (MHD) to early universe inflation, with each context setting its own criteria for "rapid" but invariably involving parameters (such as Ekman or Reynolds numbers, dimensionless turning rates, or field-space metric projections) reaching critical values that trigger substantial regime transitions.

1. Rapid Turn Regime in Rotating Convection-Driven Dynamos

A canonical example appears in the paper of plane layer Rayleigh-Bénard convection driven dynamos under rapid rotation (Tilgner, 2012). In this context, the rapid turn regime demarcates a transition zone within a hydrodynamically uniform (rapidly rotating) state, where the magnetic field generation mechanism switches character as flow driving increases (parameterized by Rayleigh number, Ra, at fixed Ekman number, Ek). Two distinct scaling regimes for the magnetic field amplitude, $E_B$, are observed:

• Helicity-driven, magnetostrophic regime (below transition):
  • The Lorentz and Coriolis forces balance (magnetostrophic equilibrium).
  • Magnetic energy satisfies:

  $$\frac{E_B\,\mathrm{Ek}^{5/6}\,\mathrm{Pm}}{\mathrm{Rm}^{1/2}\,\mathrm{Pr}} \approx 0.11$$

  where $E_B$ is mean magnetic energy, Ek is the Ekman number, Pm is magnetic Prandtl number, Rm is magnetic Reynolds number, and Pr is Prandtl number. - The convective flow is columnar, with lateral scale $\ell \simeq 1.9\, \mathrm{Ek}^{1/3}$.

• Chaotic stretching regime (above transition):

  • Dynamo action becomes controlled not by helicity but by chaotic advection and stretching.
  • Magnetic energy scales as:

  $E_B \propto \mathrm{Rm}^2$ - Helicity is suppressed but dynamo action remains robust via non-helical mechanisms.

The transition between these regimes occurs when:

$\mathrm{Rm} \mathrm{Ek}^{1/3} \approx 13.5$

even though the gross hydrodynamic properties (as measured by $\mathrm{Re} \mathrm{Pr} \mathrm{Ek}^{1/2}$; flows with $\mathrm{Re} \mathrm{Pr} \mathrm{Ek}^{1/2}<2$ remain rotationally dominated) remain unchanged. The rapid turn regime thus labels the interval where, despite apparent uniformity in flow character, the dynamo mechanism and its scaling laws change sharply. This is relevant for interpreting magnetic behaviors in planetary and stellar interiors, where abrupt changes in field properties may arise as such internal parameters are tuned through critical thresholds.

2. Rapid Turn Regime in Multi-Field Inflation

In multi-field inflationary cosmology, the rapid turn regime describes scenarios where the trajectory in field space departs from geodesicity with a large and, typically, slowly varying turn rate, defined as:

$\eta_\perp \equiv \frac{\Omega}{H}$

where $\Omega$ is the rate at which the tangent vector to the field trajectory rotates in field space, and $H$ is the Hubble parameter.

Two principal formulations are recognized:

• Strong rapid turn: $\eta_\perp^2 \gg 1$

• General rapid turn: $$\eta_\perp^2 \gg \max(\epsilon, |\eta_\parallel|, |\xi|)$$

where $\epsilon$, $\eta_\parallel$, and $\xi$ are standard slow-roll parameters.

Key features:

• Background trajectory: The field velocities decompose along the potential gradient (v) and its orthogonal direction (w). In the attractor regime (Bjorkmo, 2019):

$$\dot{\phi}_v/\dot{\phi} = -3/\sqrt{9+\omega^2}, \quad \dot{\phi}_w/\dot{\phi} = \omega/\sqrt{9+\omega^2}$$

with $\omega = \Omega/H$ the (dimensionless) turn rate.

• Stability and perturbations: The stability of rapid-turn attractors is guaranteed by conditions on the mass matrix constructed from second covariant derivatives of the potential, such that the entropic (isocurvature) mass parameter satisfies $\mu_s^2 > -3 \omega^2$.

• Consistency conditions: Derived in (Anguelova et al., 2023, Anguelova et al., 2022, Anguelova, 19 May 2024, Wolters et al., 19 May 2024), these relate directional projections of the potential's Hessian in the "potential basis" (n, τ) to V itself. For sustained rapid-turn slow-roll inflation (e.g. third-order slow roll, large and slowly varying turn rate), the condition is:

$3V V_{nn}^2 \simeq V_{n\tau}^2 V_{\tau\tau}$

where $V_{nn}$, $V_{n\tau}$, $V_{\tau\tau}$ are projections of the covariant Hessian onto n and τ. Generalized forms relate all projections and allow for arbitrary $\omega$.

• Longevity of the rapid turn regime: Contrary to earlier suggestions that rapid-turn phases are generically brief due to the evolution equation for the characteristic rotation angle $\vartheta$, recent work (Anguelova, 19 May 2024, Wolters et al., 19 May 2024) demonstrates that the above consistency conditions themselves enforce $\mathrm{d}\vartheta/\mathrm{d}N \simeq 0$, allowing rapid-turn regimes to persist for many e-folds, including at not-strictly-large turn rates.

• Special case—angular inflation: For models where $V_{nn} \simeq -V_{\tau\tau}$ along the trajectory, rapid-turn conditions are satisfied even for order-unity turn rates, relevant to "angular inflation" scenarios.

3. Physical and Theoretical Implications

• Phenomenology:

  • In multi-field inflation, rapid turn regimes enable inflation along steeper potentials, with implications for swampland constraints.
  • Rapid turn dynamics can amplify curvature perturbations transiently (exponential enhancement near horizon crossing), yet due to commutator structure, higher-point non-Gaussianities (e.g. $f_{\rm NL}$, $g_{\rm NL}$) remain mild, preserving perturbative consistency (Bjorkmo et al., 2019).
  • Order-one local or mixed-shape non-Gaussianity is a generic signature (Iarygina et al., 2023).
  • Robust gravitational production of massive dark matter is possible in rapid-turn monodromy models, consistent with relic abundance and isocurvature bounds (Kolb et al., 2022).
• In dynamo theory:
  • Abrupt regime transitions (e.g. quick switch from helicity-driven to chaotic-stretching dynamos at fixed flow morphology) exemplify non-trivial responses of complex systems in rapid turn regimes.
  • Sharp crossovers in scaling laws at critical dimensionless parameters provide theoretical pivots for interpreting astrophysical and experimental observations.

4. Broader Application and System-Theoretic Generalization

• The rapid turn regime bears analogy to more general critical transitions in multi-attractor systems, e.g., rapid regime shifts signaled in stochastic models by decreases (rather than increases) in variance and autocorrelation—dubbed "critical speeding up" (Titus et al., 2019)—as the basin of attraction contracts.
• In highly multi-dimensional optimization or sampling landscapes, analogous regime transitions arise when landscape modifications induce rapid mixing and tunneling, eliminating torpid dynamics by artificially flattening barriers, as in spin glass models (Choi, 2022).

5. Summary Table: Key Aspects of the Rapid Turn Regime

System/Context	Diagnostic Parameter(s)	Regime Transition/Indicator
Rotating Dynamos	$\mathrm{Rm} \mathrm{Ek}^{1/3}$	Transition at $\approx 13.5$
Multi-field Inflation	$\eta_\perp = \Omega/H$; Consistency conditions	Sustainment if Hessian constraints satisfied
Stochastic Systems (CSU)	Variance, autocorrelation	Drop signals imminent regime shift
Spin System Sampling	Effective barrier height (modified Hamiltonian)	O(1) barrier, rapid mixing/tunneling replaces slow dynamics

Alternative interpretations of "rapid turn regime" in other contexts converge toward the universal role of geometric, dynamical, or statistical criteria defining sharp transitions in system behavior, often controlled by the geometry (real or abstract) of the underlying configuration space and the tuning of system-specific parameters.

6. Future Directions and Open Problems

• Model engineering: Precise construction of potentials and field-space metrics in multi-field inflation to realize desired rapid-turn dynamics, especially those compatible with swampland conjectures and producing acceptable observational signals.
• Dynamical stability: Thorough investigation of stability properties, especially in high-dimensional field spaces or with nontrivial curvature, including potential for tachyonic modes and their control (Aragam et al., 2021).
• Computational tools: Automated identification of rapid-turn regimes via packages such as inflatox, which efficiently scan field space for satisfaction of the generalized consistency conditions (Wolters et al., 19 May 2024).
• Transients and observability: Quantitative predictions of transients (such as those leading to PBH formation (Anguelova, 2020)) and connections to observational probes (CMB non-Gaussianity, isocurvature perturbations, AGN/quasar state changes).
• Generalizability: Application of the rapid turn paradigm to complex, high-dimensional dynamical systems outside physics, e.g., ecological regime shifts, climate, and network dynamics.

The rapid turn regime thus encapsulates a multidisciplinary frontier, unified by the mathematical criteria for sharp, dynamic reorientations in the state or scaling properties of complex, high-dimensional systems.