2D Regime Map: Principles & Applications

• 2D regime maps are graphical tools that partition a two-parameter space into distinct regimes characterized by different physical or mathematical behaviors.
• They are widely used across materials discovery, nonlinear dynamics, reaction-diffusion systems, phase separation, and granular flow to guide both experimental and theoretical studies.
• The methodology involves scanning control parameters and using diagnostics like DFT, bifurcation analysis, and Lyapunov exponents to predict and classify stability and phase transitions.

A two-dimensional (2D) regime map is a graphical or computational representation partitioning a two-parameter space into distinct regions (regimes) characterized by qualitatively different behaviors, phases, or structural prototypes of a physical or mathematical system. In contemporary research, 2D regime maps are used to analyze stability, dynamics, transport, coarsening laws, or phase transitions by systematically varying two control parameters and diagnosing regular, singular, or chaotic structure across this grid. Such maps are essential in condensed matter physics, nonlinear dynamics, materials science, and computational modeling for both interpreting and predicting parameter-dependent behavior.

1. Structural Regime Maps in 2D Materials Discovery

The construction of the structure map for AB$_2$-type 2D monolayers exemplifies a high-throughput application of the 2D regime map paradigm, where both axes enumerate chemical groups (A and B) from the periodic table to classify possible structural ground states of 3844 candidate compounds (Fukuda et al., 2019). Three core elements define this map:

• Axes: Rows represent the group of element A; columns represent group of element B.
• Cell content: Each cell contains the outcome of DFT-based structure prediction, with color or symbol denoting the lowest-energy structural prototype (e.g., 1T, 1H, planar, distorted-planar, memory, amorphous, unknown).
• Screening and classification: Starting from prototype geometries (octahedral, trigonal prismatic, planar), full relaxation is performed. The stability is quantified via formation energies, with states within 0.03 Hartree/cell retained as metastable candidates.

This regime map provides direct structural guidance for experimental synthesis and theory, identifying "territories" in compositional space that reliably yield known 2D motifs (e.g., transition metal dichalcogenides as 1T/1H, MXenes as 1T). It delineates new, sparsely populated regions with prospects for unconventional planar, distorted planar, or "memory" structures (bistable for data storage), systematically classifying families by their lowest-energy motifs and thus serving as a predictive atlas for 2D materials (Fukuda et al., 2019).

Map domain	Axes	Prototypes classified
AB$_2$ monolayers	(A-group, B-group), periodic table	1T, 1H, planar, distorted, memory, amorphous, unknown

2. Regime Maps in Nonlinear Dynamics and Bifurcation Theory

In nonlinear discrete dynamics, 2D regime maps are foundational for classifying behaviors distinguished by stability, periodicity, and chaos. The two-parameter regime diagram for area-preserving symplectic maps (e.g., Hénon map) or coupled bistable lattices (e.g., parametric maps with double-well potentials) is constructed by scanning control parameters—such as potential amplitude ($E$) and deformability ($\mu$), or Hénon map parameters ($a, b$)—and then algorithmically categorizing the asymptotic dynamics at each grid point (Dikande, 2021, Zolkin et al., 7 Dec 2024).

Classification is established via analytic and numerical diagnostics:

• Fixed point and periodic orbit stability: Linearization yields Jacobian eigenvalues, whose loci of bifurcation are computed analytically (e.g., $a = 2\cos(2\pi m/n)$ for isochronous curves).
• Bifurcation detection: Period-doubling and pitchfork loci are traced by monitoring monodromy traces along symmetry lines; higher-order multiplicity is found by Floquet analysis or direct numerical mapping.
• Chaoticity criteria: Lyapunov exponents, Reversibility Error Method (REM), and Generalized Alignment Index (GALI) distinguish between regular and chaotic regimes.

Representative diagrams reveal rich topologies—cascades of period-doubling tongues, shrinkage of bifurcation intervals (non-universality of the Feigenbaum constant with parameter $\mu$), and the appearance of twistless tori (Dikande, 2021, Zolkin et al., 7 Dec 2024).

Application	Axes	Regimes identified
Symplectic maps	$(a, b)$ or $(a, q)$	Isochrones, period-doubling curves, chaos, twistless, KAM
Bistable lattices	$(E, \mu)$	Periodic (various period), quasi-periodic, chaotic

3. Regime Maps in Stochastic Reaction-Diffusion and Multiscale Coupling

For hybrid simulations combining molecular (Brownian) and compartment-based (mesoscopic) reaction-diffusion models, the 2D regime map in the $(h, \Delta t)$-plane prescribes optimal interface-coupling rules on a square mesh (Flegg et al., 2013):

• The axes are mesh size $h$ and Brownian time step $\Delta t$.
• Two sampling regimes are identified: the “triangle” law (recommended when $D\,\Delta t \sim h^2$ for optimal $O(h^2)$ tangential accuracy), and the “step” law (when $D\,\Delta t \ll h^2$ for better stability under stiff kinetics).
• The dividing curve is $D\,\Delta t = h^2$, above which the triangle rule is used and below which the step rule is preferred.
• At right-angled corners, specific split and re-injection rules maintain consistency.

The map provides explicit guidelines for algorithm selection as a function of resolution and timestep, balancing accuracy and efficiency in multi-regime simulations (Flegg et al., 2013).

Domain	Axes	Coupling regime
Brownian-compartment	$(h, \Delta t)$	Triangle (diffusive), Step (stiff)

4. Flow-Pattern Regime Maps in Phase Separation

Flow-pattern maps in phase separation of binary fluids delineate regimes in the $(D, \nu)$-plane (dimensionless diffusion versus kinematic viscosity) (Naso et al., 2017), with Re (Reynolds), Pe (Péclet), and Cn (Cahn) numbers parameterizing the effective physics:

• Domains: Four universal regimes are separated by algebraic boundaries: diffusion-dominated ($\nu > D$), viscous-hydrodynamic, inertial-hydrodynamic, and “anomalous” (accelerated) coarsening. These cuts are defined by analytic crossover equations such as $D\,\text{Re} = 1$ and $D\,\text{Re}^3\,\text{Cn}^2 = 1$.
• Significance: Each regime is characterized by distinct growth laws for the coarsening length scale $L(t)$. For example, $L(t) \sim (D\,\text{Cn}\,t)^{1/3}$ in the diffusion-dominated regime and $L(t) \sim t^{2/3}$ in the inertial-hydrodynamic regime.

The regime map is predictive: it determines which physical process dominates the evolution and provides design principles for engineering or interpreting coarsening kinetics (Naso et al., 2017).

5. Statistical Fluctuation Regime Maps in Random Ensembles

Regime maps also describe transitions among statistical behaviors, as in the paper of extremal statistics of eigenvalue moduli for the complex Ginibre ensemble (interpreted as a 2D Coulomb gas) (Lacroix-A-Chez-Toine et al., 2017). Here, the regime map partitions the probability landscape of $r_{\max} = \max_i |z_i|$ into four shadowed windows:

• Left large deviations: Deep tail for $r < 1$, with probability $\sim \exp[-N^2 \Phi_-(r)]$.
• Intermediate deviation regime (IDR): Crossover window $r = 1 + s/\sqrt{2N}$, with universal CDF scaling as $\exp[-\sqrt{2N}\varphi_I(s)]$; $\varphi_I$ is independent of the confinement potential's fine details.
• Typical (Gumbel) regime: Center region scaled by $b_N$ and $a_N$, with Gumbel-type statistics.
• Right large deviations: Far tail $r>1$, with probability $1-\exp[-N\Phi_+(r)]$.

Crossover matching among these windows is analytic, and the existence and universality of the IDR are distinguished features for 2D random ensembles (Lacroix-A-Chez-Toine et al., 2017).

Regime	Window	Probability Scaling
Left large dev.	$1 - r = O(1)$	$\exp[-N^2\Phi_-(r)]$
Intermediate (IDR)	$r = 1 + s/\sqrt{2N}$	$\exp[-\sqrt{2N} \varphi_I(s)]$
Typical (Gumbel)	$r = b_N + x/a_N$	$G(x) = \exp[-e^{-x}]$
Right large dev.	$r-1 = O(1)$	$1-\exp[-N\Phi_+(r)]$

6. Flow and Collision Regime Maps in Granular Matter

In granular silo discharge, a regime map in the $(D^*, \Gamma^*)$-plane (normalized outlet size and collision frequency) organizes the transition from intermittent clogging to continuous flow (Arévalo, 2022):

• Clogging-dominated regime: $1\le D^* \lesssim 5$; slow rise in packing fraction and collision rate.
• Mixed/intermittent regime: $5\lesssim D^* \lesssim 10$; saturation sets in, flow and collision statistics are steady.
• Continuous regime: $D^* \gtrsim 10$; Beverloo law applies, distributions are collisionally equilibrated.

Empirically, all quantities vary smoothly without critical anomalies—arches causing clogs become exponentially rare but remain possible even for large $D^*$, confirming the absence of a hard transition (Arévalo, 2022).

7. Methodological Principles for Building 2D Regime Maps

In all contexts, the construction of a 2D regime map follows a general workflow:

1. Define two physically or mathematically relevant control parameters to scan.
2. At each grid point $(x,y)$, compute or evaluate classification metrics (e.g., energy minima, Lyapunov exponents, order parameter scaling, bifurcation markers).
3. Assign a discrete or continuous label by diagnostic (e.g., structural prototype, dynamical regime, coarsening law).
4. Visualize the resulting regime diagram as a color-coded or symbol-annotated 2D matrix, using further overlays for transition curves, bifurcation loci, or crossover lines.
5. Interpret the emergent phase or behavior boundaries; provide predictions or experimental strategies accordingly.

This methodology is essential for both theory-guided exploration (e.g., predicting material realizability, algorithm selection) and for empirical diagnostics (e.g., pinpointing nonstandard dynamical or statistical behavior). Numerical recipes, as detailed for nonlinear maps and stochastic methods, emphasize reproducible, grid-based grid search with post-classification (Dikande, 2021, Zolkin et al., 7 Dec 2024, Flegg et al., 2013).

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References:

• (Fukuda et al., 2019): Structure map of AB$_2$ type 2D materials by high-throughput DFT calculations
• (Dikande, 2021): Route to chaos in two-dimensional discrete parametric maps with bistable potentials
• (Zolkin et al., 7 Dec 2024): Isochronous and period-doubling diagrams for symplectic maps of the plane
• (Flegg et al., 2013): Analysis of the two-regime method on square meshes
• (Naso et al., 2017): A flow-pattern map for phase separation using the Navier-Stokes Cahn-Hilliard model
• (Lacroix-A-Chez-Toine et al., 2017): Extremes of $2d$ Coulomb gas: universal intermediate deviation regime
• (Arévalo, 2022): Collisional regime during the discharge of a 2D silo