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, , is in order from which the thermodynamic behavior can be observed by taking the thermodynamic limit () posteriorly. This article presents such a study. We find that for a trapped gas, as increases, the values of curvature decrease proportionally to a power of , 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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Many works on Ruppeiner geometry13,16,21,22 use macroscopic value densities—energy density and particle density , where is the volume accessible by the system, fixed here by —instead of and in (13) to calculate the metric. In this convention, the metric terms are divided by the volume accessible to the system, resulting in a curvature with units of volume. However, this does not change the main results of the present article, as the curvature is still limited and converges to zero as the number of particles increases. Additionally, the convention followed here is consistent with information geometry as an emerging geometric consequence of probability theory1,2,3,4 rather than phenomenological thermodynamics.
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2023
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