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arXiv:cond-mat/0401092 · Physica A Statistical Mechanics and its Applications

Information geometry and phase transitions

Wolfhard Janke E, Desmond A. Johnston, Ralph Kenna B

Published Feb 6, 2004• 84 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2004.01.023

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It is a long held conjecture in the connection between information geometry (IG) and thermodynamics that the curvature endowed by IG diverges at phase transitions. Recent work on the IG of Bose-Einstein (BE) gases challenged this conjecture by saying that in the limit of fugacity approaching unit-where BE condensation is expected-the curvature does not diverge; rather, it converges to zero. However, as the discontinuous behavior that identifies condensation is only observed at the thermodynamic limit, a study of the IG curvature at a finite number of particles, N, is in order from which the thermodynamic behavior can be observed by taking the thermodynamic limit ( N→∞) posteriorly. This article presents such a study. We find that for a trapped gas, as N increases, the values of curvature decrease proportionally to a power of N, while the temperature at which the maximum value of curvature occurs approaches the usually defined critical temperature. This means that, in the thermodynamic limit, the curvature has a limited value where a phase transition is observed, contradicting the forementioned conjecture.

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It is a long held conjecture in the connection between information geometry (IG) and thermodynamics that the curvature endowed by IG diverges at phase transitions. Recent work on the IG of Bose-Einstein (BE) gases challenged this conjecture by saying that in the limit of fugacity approaching unit-where BE condensation is expected-the curvature does not diverge; rather, it converges to zero. However, as the discontinuous behavior that identifies condensation is only observed at the thermodynamic limit, a study of the IG curvature at a finite number of particles, N, is in order from which the thermodynamic behavior can be observed by taking the thermodynamic limit ( N→∞) posteriorly. This article presents such a study. We find that for a trapped gas, as N increases, the values of curvature decrease proportionally to a power of N, while the temperature at which the maximum value of curvature occurs approaches the usually defined critical temperature. This means that, in the thermodynamic limit, the curvature has a limited value where a phase transition is observed, contradicting the forementioned conjecture.
We examine phase transition of the Husimi-Temperley model in terms of information geometry. For this purpose, we introduce the Fisher metric defined by the density matrix of the model. We find that the metric becomes hyperbolic at the critical point with respect to the energy scale. Then, the metric is invariant under the scale transformation. We also find that the equation of states is naturally derived from a necessary condition for the entropy operator that is a building block of the metric. Based on these findings, we conclude that the geometric quantities clearly detect the phase transition of the model.