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Information theoretic aspects of the two-dimensional Ising model.
Hon Wai Lau1, Peter Grassberger
1Complexity Science Group, University of Calgary, Calgary, Canada T2N 1N4.
Summary
We studied information theoretic properties of the Ising model. The excess entropy of a cylinder and a line of spins diverge logarithmically at critical points, suggesting a universal behavior in phase transitions.
Area of Science:
- Statistical Mechanics
- Information Theory
- Condensed Matter Physics
Background:
- The Ising model is a fundamental tool for understanding phase transitions.
- Information theoretic properties offer new insights into critical phenomena.
Purpose of the Study:
- To numerically investigate information theoretic properties of the square lattice Ising model.
- To quantify finite length corrections to entropy and analyze excess entropy.
- To explore the behavior of mutual information and excess entropy at critical points.
Main Methods:
- Bond propagation algorithm for entropy calculations.
- Transfer matrix method for precise estimations.
- Monte Carlo simulations for excess entropy analysis.
Main Results:
- Quantified finite length corrections to entropy scaling with cylinder circumference.
- Calculated mutual information between cylinder halves, confirming previous results with higher precision.
- Observed logarithmic divergence of mutual information and excess entropy at the critical point.
- Conjectured generic logarithmic divergence for 1D subsets at 2D second-order phase transitions.
Conclusions:
- The study provides precise numerical results for information theoretic properties of the Ising model.
- Logarithmic divergences in mutual information and excess entropy are observed at criticality.
- The findings suggest universal behavior for one-dimensional subsets at second-order phase transitions.
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