[Paper Review] Statistical measure of complexity of hard-sphere gas: applications to nuclear matter

Ch. C. Moustakidis, K. Ch. Chatzisavvas+2 more|arXiv (Cornell University)|Feb 12, 2010

Statistical Mechanics and Entropy2 references3 citations

TL;DR

This paper applies the López-Ruiz-Mancini-Calbet (LMC) statistical complexity measure to a hard-sphere Fermi gas model to quantify complexity in momentum space, linking it to particle correlations and energy-like quantities. It finds a strong, nearly linear dependence between complexity and energy, validating the LMC measure as a robust indicator of correlated behavior in quantum many-body systems like nuclear matter.

ABSTRACT

We apply the statistical measure of complexity, introduced by López-Ruiz, Mancini and Calbet to a hard-sphere dilute Fermi gas whose particles interact via a repulsive hard-core potential. We employ the momentum distribution of this system to calculate the information entropy, the disequilibrium and the statistical complexity. We examine possible connections between the particle correlations and energy of the system with those information and complexity measures. The hard-sphere model serves as a test bed for concepts about complexity.

Motivation & Objective

• To investigate the connection between statistical complexity and particle correlations in a uniform Fermi system using a probabilistic, information-theoretic framework.
• To extend previous work on complexity in uniform Fermi systems by focusing on momentum-space probability distributions.
• To assess the validity and robustness of the LMC complexity measure in quantum many-body systems with strong correlations.
• To explore whether complexity increases intuitively with correlation strength, as parameterized by the wound parameter $k_{\text{dir}}$ and Fermi discontinuity $Z$.
• To establish a quantitative link between complexity and thermodynamically relevant quantities like specific heat, via empirical trends.

Proposed method

• Employs the LMC complexity measure $C = e^S D$, where $S$ is information entropy and $D$ is disequilibrium, derived from the momentum distribution of a hard-sphere Fermi gas.
• Calculates $S$ and $D$ from the momentum distribution $n(k)$, using the Fermi-Dirac-like form derived from the Low Order Approximation (LOA) for nuclear matter.
• Uses the LOA momentum distribution $n_{\text{LOA}}(k)$, which incorporates the correlation parameter $\beta$ and the wound parameter $k_{\text{dir}}$ to model inter-particle repulsion.
• Relies on the normalization condition $\int n_{\text{LOA}}(k) k^2 dk = \frac{1}{3}k_F^3$ to ensure physical consistency of the momentum distribution.
• Analyzes the dependence of $S_{\text{cor}}$, $D_{\text{cor}}$, and $C$ on the correlation parameter $k_F c$ and the Fermi discontinuity $Z = 1 - Z$, with $Z$ quantifying momentum-space depletion.
• Compares results from the LOA method with those from the hard-sphere model to assess consistency in complexity trends across different correlation models.

Experimental results

Research questions

• RQ1 How does the LMC complexity measure $C$ respond to increasing correlations in a hard-sphere Fermi gas, as parameterized by $k_{\text{dir}}$ and $Z$?
• RQ2 Is there a quantitative relationship between the statistical complexity $C$ and energy-like observables such as specific heat $C_V$?
• RQ3 Does the LMC complexity measure exhibit the expected asymptotic behavior—vanishing for both ideal gas (disorder) and perfect crystal (order)?
• RQ4 How do the information entropy $S$ and disequilibrium $D$ evolve with increasing correlation strength, and do they reflect physical intuition?
• RQ5 Can the LMC complexity measure serve as a reliable, physically meaningful indicator of correlated behavior in quantum many-body systems like nuclear matter?

Key findings

• The LMC complexity measure $C$ exhibits a strong, nearly linear dependence on the energy-like parameter $1 - Z$, indicating a direct link between complexity and the degree of momentum-space depletion due to correlations.
• Information entropy $S_{\text{cor}}$, disequilibrium $D_{\text{cor}}$, and complexity $C$ all show a consistent trend with increasing correlation strength, as quantified by the wound parameter $k_{\text{dir}}$ in the range $0.02 \leq k_{\text{dir}} \leq 0.3$.
• The results confirm a previously observed empirical connection between complexity and specific heat $C_V$, extending this link to the hard-sphere Fermi gas model.
• The LMC measure $C$ reaches a minimum value of 1 for uniform (equipartition) distributions, validating its theoretical consistency and robustness as a complexity indicator.
• Despite different physical origins, the LOA and hard-sphere models yield qualitatively similar trends in $S_{\text{cor}}$, $D_{\text{cor}}$, and $C$ as functions of $1 - Z$, supporting the reliability of the complexity measure.
• The study demonstrates that complexity, as defined by $C = e^S D$, increases with correlation strength, satisfying intuitive expectations and validating its use in quantum many-body systems.