Geometry-Dependent Crossover Speed

• Geometry-dependent crossover speed is a concept where physical or abstract system geometries define the critical thresholds between different dynamical response regimes.
• It appears in diverse areas such as granular impact, solar wind acceleration, thermal relaxation, optimization, and quantum speed limits, with geometry controlling scaling laws and transition points.
• The study integrates empirical experiments, analytical models, and simulation data to link geometric features decisively with regime-switching phenomena.

Geometry-dependent crossover speed denotes a phenomenon wherein the transition between different regimes of dynamical response occurs at a critical speed determined by the geometric features of the system or its governing structures. This construct appears in granular impact physics, solar wind acceleration, thermal relaxation in classical systems, linear programming crossovers, and quantum speed limits—each embodying specific mechanisms where geometry dictates not only the magnitude but also the regime-switching point of the relevant dynamical quantity.

1. Granular Impact: Geometry-Controlled Onset of Inertial Drag

In granular impact dynamics, the crossover speed $v_c$ separates quasi-static force transmission (localized chains) from an inertial regime (collective grain acceleration). Systematic experiments with conical intruders of varying half-angle $\phi$ show that sharp cones (small $\phi$) require substantially higher $v$ to manifest inertial scaling, while blunt cones (large $\phi$) exhibit inertial drag even at low $v$ (Williams, 28 Dec 2025). The governing force law

$$F(z,v) = F_{\rm qs}(z) + C\,\rho_g\,A_{\rm eff}\,v^2$$

enables the definition of a geometry-dependent crossover speed as the lowest $v$ above which the peak acceleration $a_{\rm peak}$ scales quadratically with $v$:

$$v_c(\phi) \approx \alpha \tan\phi, \quad \alpha \simeq 0.25\,\mathrm{m/s}$$

with $A_{\rm eff} \sim A_0 \tan\phi$ encoding the width of the jammed front. Above $v_c$, the inertial force magnitude collapses onto a universal curve when rescaled by $\tan\phi$:

$a_{\rm peak}(v,\phi) \approx \beta\,\tan\phi\,v^2$

where $\beta$ is determined empirically. This architecture reveals that cone geometry governs both the scaling of the inertial drag and the velocity threshold for collective grain entrainment.

2. Solar Wind Acceleration: Flux-Tube Geometry and Crossover Flows

Solar wind speed fluctuation at 1 AU is linked to geometric properties of open magnetic flux tubes (Pinto et al., 2016). Empirical fits to extensive MHD simulation data yield a composite scaling law:

$$v_\infty = C_0 + C_1\,f_{\text{exp}}^\alpha + C_2\,(\sin\theta)^\beta + C_3 B_0^\gamma$$

where $f_{\text{exp}}$ is the total expansion ratio, $\theta$ is the field-line inclination at the sonic point, and $B_0$ is the foot-point magnetic field. The geometry-dependent crossover speed $v_c$ is defined as the locus in parameter space where expansion and inclination effects exactly cancel:

$$C_1\,f_{\text{exp}}^\alpha + C_2\,(\sin\theta)^\beta = 0 \implies (\sin\theta)_c = \left(-C_1/C_2\right)^{1/\beta} f_{\text{exp}}^{\alpha/\beta}$$

At this geometric threshold, the terminal wind speed is dictated only by $B_0$:

$v_c(f_{\text{exp}}, B_0) = C_0 + C_3\,B_0^\gamma$

Geometry thus controls not only spatial distribution of wind speeds but also the critical regime-transition points between fast and slow solar wind sectors.

3. Relaxation Speed Crossover in Anharmonic Potentials

The rate of thermal equilibration for overdamped diffusion in single-well anharmonic potentials depends on the potential's geometric exponent $n$ and the initial temperature ratio $\tau = T_h/T_c$ (Meibohm et al., 2021). Relaxation-speed phase diagrams in $(n, \tau)$ space exhibit three distinct regimes:

• Regime A ($n < n_c^{\text{short}}(\tau)$): Heating is always faster than cooling.
• Regime B ($n > n_c^{\text{long}}(\tau)$): Cooling is always faster than heating.
• Crossover Region ($n_c^{\text{short}} < n < n_c^{\text{long}}$): Heating is initially slower but overtakes cooling asymptotically.

The crossover boundaries are computed analytically (short-time expansion, eigenmode analysis) and are robust against moderate perturbations (added harmonic terms or alternative free-energy measures). Here, geometric parameters of the potential modulate whether cooling or heating is dominant, and the crossover speed emerges from the critical lines in the phase diagram.

4. Algorithmic Crossovers in Optimization: Geometry-Guided Transitions

In linear programming, traditional crossover from a suboptimal interior-point solution to an optimal basic feasible solution represents a significant computational bottleneck. Geometry-aware approaches ("Smart Crossover") exploit polyhedral and network structures (Ge et al., 2021):

• Network-based crossover: Graph structure is leveraged; BFS corresponds to a spanning tree. The flow-ratio indicator identifies potential tree arcs.
• General LPs: Approximate optimal faces are identified via active-set partitioning; a small perturbed objective ensures transition to a unique extreme point, with geometry dictating both candidate selection and regime change.

Geometric information thus identifies structures (trees, faces) where rapid transition to an optimal vertex ("crossover") occurs, and empirical results show geometry-dependent speed-ups in practice.

5. Quantum Speed Limits: Geometric Crossover Between Bounds

Quantum evolution speed is fundamentally limited by two geometric bounds—the Mandelstam-Tamm (MT, set by energy uncertainty $\Delta E$) and Margolus-Levitin (ML, set by mean energy $E$) (Ness et al., 2021). The crossover time $t_c$ separating regimes dominated by MT or ML bounds is determined by their geometric intersection:

$$t_c = \frac{\pi \hbar E}{2 \Delta E^2} = \frac{\tau_{MT}^2}{\tau_{ML}}$$

where $\tau_{MT} = \pi\hbar/(2\Delta E)$, $\tau_{ML} = \pi\hbar/(2E)$. Experimental investigation with multi-level atomic systems reveals two regimes: large-level excitations remain MT-constrained, while small-level (qubit-like) excitations exhibit transition at $t_c$ to ML-constrained evolution. The crossover is intrinsically geometric, tied to Hilbert-space trajectories and energy spectra.

6. Cross-Disciplinary Interpretation and Physical Significance

Geometry-dependent crossover speed is a unifying concept highlighting how system geometry—whether physical (shape, spatial embedding), abstract (potential exponent, graph structure), or algebraic (polyhedral constraints, Hilbert-space path)—determines not only magnitudes but the thresholds for regime transitions. This construct encapsulates the critical speed, time, or parameter value at which collective or qualitatively distinct dynamics emerge, shaping physical processes, computational algorithms, and fundamental limits across disciplines. The robustness of geometric criteria for crossover speed, as evidenced by empirical collapse and analytical boundaries, underscores the universality and predictive power of geometry-driven regime transitions in complex systems.