Information-Geometric Signatures from Nonextensivity in the $1$-D Blume-Capel Model

Abstract

We study the thermodynamic geometry of the one-dimensional Blume--Capel model within the Tsallis nonextensive framework to understand how generalized statistics modify correlation structure and pseudo-critical behaviour. Using the transfer matrix method, we construct the Tsallis entropy based thermodynamic metric as its negative Hessian on the parameter space $(\beta ,J)$, with the crystal-field anisotropy $D$ as a control parameter, and compute the associated scalar curvature $R(T)$ as a measure of correlations. Although no true phase transition occurs in one dimension, $R(T)$ exhibits finite peaks signaling pseudo-critical crossovers. We analyze both $D<J$ and $D>J$ regimes and show that deviations from the Boltzmann--Gibbs limit ($q=1$) systematically deform the curvature profile: for $q>1$ the peak shifts and correlations persist beyond the crossover, whereas for $q<1$ the peak is weakened or suppressed. Our results demonstrate that the Tsallis parameter $q$ geometrically reshapes the entropy surface, providing a clear information-geometric interpretation of nonextensive effects in spin-1 systems.

arXiv comment15 pages, 9 figures

References in corpus (15)

• Superstatistics
• Measuring thermodynamic length
• Geometry of black hole thermodynamics
• Thermodynamic curvature measures interactions
• Ruppeiner Geometry of RN Black Holes: Flat or Curved?
• First-order phase transition and tricritical scaling behavior of the Blume-Capel model: a Wang-Landau sampling approach
• Thermodynamic geometry of black holes in f(R) gravity
• Critical and tricritical singularities from small-scale Monte Carlo simulations: The Blume-Capel model in two dimensions
• Thermodynamic curvature for a two-parameter spin model with frustration
• Renormalization Group on hierarchical lattices in finite dimensional disordered Ising and Blume-Emery-Griffiths Models
• Spin dependence of the tricritical point in the mixed-spin Blume-Capel model on three-dimensional lattices: Metropolis and Wang-Landau sampling approaches
• Alternative description of the 2D Blume-Capel model using Grassmann algebra
• Geometrical mutual information at the tricritical point of the two-dimensional Blume-Capel model
• Blume-Capel model analysis with microcanonical population annealing method
• Existence of a tricritical point for the Blume-Capel model on ${\mathbb{Z}}^d$