Configurational entropy of Wigner crystals

Abstract: We present a theoretical study of classical Wigner crystals in two- and three-dimensional isotropic parabolic traps aiming at understanding and quantifying the configurational uncertainty due to the presence of multiple stable configurations. Strongly interacting systems of classical charged particles confined in traps are known to form regular structures. The number of distinct arrangements grows very rapidly with the number of particles, many of these arrangements have quite low occurrence probabilities and often the lowest-energy structure is not the most probable one. We perform numerical simulations on systems containing up to 100 particles interacting through Coulomb and Yukawa forces, and show that the total number of metastable configurations is not a well defined and representative quantity. Instead, we propose to rely on the configurational entropy as a robust and objective measure of uncertainty. The configurational entropy can be understood as the logarithm of the effective number of states; it is insensitive to the presence of overlooked low-probability states and can be reliably determined even within a limited time of a simulation or an experiment.

• The paper demonstrates that configurational entropy robustly quantifies uncertainty in finite Wigner crystals by focusing on dominant metastable configurations.
• It employs extensive simulations using the Metropolis algorithm and energy quenching to uncover power-law statistics and super-exponential entropy scaling with system size.
• The study reveals that short-range interactions and enhanced screening intensify rare state occurrence and trigger distinct entropy peaks linked to structural transitions.

Configurational Entropy in Classical Wigner Crystals

Introduction

This paper provides a systematic theoretical investigation of the configurational entropy in two- and three-dimensional classical Wigner crystals confined within isotropic parabolic traps. The primary focus is the quantification of uncertainty associated with the proliferation of metastable configurations, as the number of stable arrangements grows rapidly with system size in these strongly interacting, confined charged particle systems. Unlike approaches that count all stationary states, this work proposes configurational entropy as a robust, objective measure of uncertainty, emphasizing its insensitivity to the abundance of low-probability states that are commonly missed in practical simulations and experiments.

Model and Numerical Methodology

The physical system analyzed comprises up to $N=100$ identical charged particles interacting via either pure Coulomb or Yukawa (screened Coulomb) potentials. The particles are confined by a symmetric parabolic trap, forming canonical models of finite Wigner crystals. The total system energy includes the sum of the external potential and pairwise interaction terms. In the presented simulations, particles are initialized with randomized positions according to a Boltzmann distribution at temperature $kT$ (well above potential barrier heights), ensuring exploration of the entire basin structure in configuration space.

Stationary states—including the ground state (global minimum) and all local energy minima—are found via iterated cycles of equilibrium sampling (using the Metropolis algorithm) followed by a steepest-descent (with occasional Newton's method refinement) quench to the nearest minimum. For each independent run, the system settles into some stable configuration; the occurrence probability of each configuration is then estimated across $2 \times 10^4$ to $3 \times 10^4$ simulation runs per system.

The configurational entropy $S=-\sum_{k=1}^M p_k \ln p_k$ is defined, with $p_k$ the measured probability of the $k$-th metastable configuration and $M$ the number found. Crucially, since new metastable configurations continue to be discovered with increasing runs in large systems, $M$ is not a reliable or convergent descriptor; however, $S$ converges rapidly because the overwhelming majority of configurations have extremely low probability and do not contribute to the entropy.

Analysis of Metastable States and Probability Distribution

Results show that the number of discovered metastable configurations grows nearly exponentially with $N$ (for both 2D and 3D systems), but without sign of saturation even at $10^4$ simulation runs for $N=100$. However, the configurational entropy saturates quickly, highlighting that only a few “dominant” configurations capture most of the probability mass. This is consistent with observed power-law (Zipf/Pareto) statistics for the probability distribution $f(p)\sim p^{-\alpha}$ over the set of metastable states: the exponent $\alpha$ varies with system parameters, increasing with screening strength and dimensionality. Thus, shorter-range interactions (larger screening) exacerbate the proliferation of rare configurations.

A significant empirical claim is that the frequently cited exponential growth of the number of metastable states with system size, while numerically corroborated, does not reflect any operational uncertainty for large $N$ due to the vanishing probabilities of almost all configurations.

Configurational Entropy: Scaling and Properties

The configurational entropy $S$ exhibits a nontrivial, strongly super-linear dependence on $N$. The effective number of states, $\Omega(N)=\exp[S(N)]$, grows faster than exponentially with $N$, exhibiting trends compatible with scaling of the form $S = \gamma \ln(N!)$ for both 2D and 3D systems. Fitted values for the proportionality constant $\gamma$ are higher for 3D and larger screening, indicating enhanced uncertainty in systems with more pronounced short-range correlations and greater geometric complexity.

Entropy landscapes are not smooth but show pronounced peaks and troughs as a function of $N$, corresponding to the emergence or dominance of competing shell- or lattice-like arrangements. Notably, the ground state is not always the most probable configuration: certain metastable states have larger basins of attraction and thus higher occurrence probabilities, a finding consistent with earlier experimental reports.

It is also shown that the positions and nature of entropy peaks vary as the system transitions between different shell structures (especially pronounced in 3D with screening), further substantiating the configurational entropy as a sensitive probe of structural transitions and associated uncertainties.

Implications and Future Perspectives

The use of configurational entropy as an uncertainty quantifier in the structural analysis of finite Wigner crystals is justified by (i) its rapid convergence in simulations, (ii) its robustness against undiscovered low-probability states, and (iii) its direct connection to the effective number of significant metastable configurations. The established super-exponential scaling with $N$ has implications for the thermodynamic behavior, accessible state spaces, and equilibration of strongly correlated mesoscopic systems.

Practically, these results inform the characterization of experimentally realized systems such as electrons on helium, quantum dots, and dusty plasmas, where observed metastable structure does not simply correspond to ground states. From a theoretical standpoint, the emergence of power-law statistics in the distribution of metastable state probabilities links these structural problems to a broad class of complex systems exhibiting heavy-tailed statistics and rich phase-space geometry.

Future investigation may target connections between configurational entropy and dynamical properties (e.g., relaxation kinetics, glassiness, or ergodicity breaking), extension to quantum regimes, and the development of more efficient algorithms for rare-event sampling in high-dimensional configuration spaces. Moreover, the interplay between shell structure transitions and entropy peaks merits further analytic and numerical scrutiny, particularly in regard to the applicability of these findings in contexts such as quantum computing architectures and precision control of mesoscopic crystalline assemblies.

Conclusion

The study establishes that configurational entropy, rather than the sheer number of metastable states, is the key operational measure of uncertainty in finite Wigner crystals. It robustly captures the proliferation of significant configurations with system size, interaction range, and dimensionality, and reveals subtle structure in the configurational landscape inaccessible to naive state counting. The findings contribute to both foundational understanding and practical analysis of classical many-body systems with complex energy landscapes.