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Failure of LMC statistical complexity in identifying structural order in the XY model

Dario Javier Zamora

TL;DR

The study probes whether the LMC statistical complexity $C=H\cdot D$ faithfully identifies structural order in nonequilibrium, using a 2D XY model simulated by Monte Carlo. While $C$ vanishes for perfect order and disorder, and can peak at intermediate states, it does not consistently align with visible structural features like vortices, sometimes peaking when the system is near equilibrium. The analysis shows that the time derivative $dC/dt = (dH/dt)\cdot D + (dD/dt)\cdot H$ carries more meaningful dynamical information about self-organization, suggesting a shift from static to dynamical interpretation of complexity. Overall, the work highlights the limitations of static scalar complexity measures and advocates for directionality and dynamical sensitivity in future metrics within a broader complexity toolkit.

Abstract

Quantifying complexity in physical systems remains a fundamental challenge, and many proposed measures fail to capture the structural features that intuitive or theoretical considerations would demand. Among them, the Lopez-Ruiz-Mancini-Calbet (LMC) statistical complexity has been widely cited due to its simplicity and analytic tractability. Here, we examine the performance and limitations of the LMC measure in a controlled physical setting: a two-dimensional XY model studied through Monte Carlo simulations. By computing LMC complexity at each step of the system's relaxation dynamics, and directly comparing these values with the evolving dipole configurations, we show that LMC complexity systematically fails to identify states of high structural complexity. In particular, the measure often assigns maximal complexity to nearly equilibrated configurations while underestimating vortex-rich intermediate states. We further show that the time derivative of LMC complexity contains more meaningful dynamical information. We propose that future measures incorporate directionality and dynamical sensitivity to better reflect the emergence and decay of organization in nonequilibrium systems.

Failure of LMC statistical complexity in identifying structural order in the XY model

TL;DR

The study probes whether the LMC statistical complexity faithfully identifies structural order in nonequilibrium, using a 2D XY model simulated by Monte Carlo. While vanishes for perfect order and disorder, and can peak at intermediate states, it does not consistently align with visible structural features like vortices, sometimes peaking when the system is near equilibrium. The analysis shows that the time derivative carries more meaningful dynamical information about self-organization, suggesting a shift from static to dynamical interpretation of complexity. Overall, the work highlights the limitations of static scalar complexity measures and advocates for directionality and dynamical sensitivity in future metrics within a broader complexity toolkit.

Abstract

Quantifying complexity in physical systems remains a fundamental challenge, and many proposed measures fail to capture the structural features that intuitive or theoretical considerations would demand. Among them, the Lopez-Ruiz-Mancini-Calbet (LMC) statistical complexity has been widely cited due to its simplicity and analytic tractability. Here, we examine the performance and limitations of the LMC measure in a controlled physical setting: a two-dimensional XY model studied through Monte Carlo simulations. By computing LMC complexity at each step of the system's relaxation dynamics, and directly comparing these values with the evolving dipole configurations, we show that LMC complexity systematically fails to identify states of high structural complexity. In particular, the measure often assigns maximal complexity to nearly equilibrated configurations while underestimating vortex-rich intermediate states. We further show that the time derivative of LMC complexity contains more meaningful dynamical information. We propose that future measures incorporate directionality and dynamical sensitivity to better reflect the emergence and decay of organization in nonequilibrium systems.

Paper Structure

This paper contains 5 sections, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Results
  4. Discussion
  5. Conclusions

Figures (2)

  • Figure 1: LMC complexity vs entropy.
  • Figure 2: LMC complexity as a function of simulation step, together with the corresponding dipole configurations at six different times: steps 0, 200, 4300, 4400, 7500, and 10000. Each configuration is shown with a matching color label, and the corresponding point in the LMC-complexity curve is marked with the same color.