Application of Riemannian geometry to the thermodynamics of a simple fluctuating magnetic system

It has been suggested previously by the author that the correlation length $\xi$ can be calculated from the Gaussian curvature in a Riemannian geometric model of thermodynamics. In the present paper, this hypothesis is tested for the one-dimensional Ising model. It is found that for ferromagnetic interactions ($J>\mathrm{}0\mathrm{}$) the thermodynamic ${\xi }_G$ is in excellent agreement with the correlation length $\xi$ which gives the range of the exponentially decaying spin-spin correlation function $G\left(r\right)\mathrm{}$; ${\xi }_G$ and $\xi$ are never found to deviate by more than one lattice site. If $J=0\mathrm{}$, there is no curvature, in accordance with expectations that curvature be a measure of interactions. If $J<\mathrm{}0\mathrm{}$, ${\xi }_G$ does not give the range of the envelope of the oscillatory $G\left(r\right)\mathrm{}$, and hence, does not agree with usual definitions of the correlation length; however, ${\xi }_G$ can be interpreted satisfactorily as giving the average length, due to interactions, of clusters of aligned spins. The results in this paper support the contention that thermodynamics in general contains more information than had previously been thought.