By including the theory of fluctuations in the axioms of thermodynamics it is shown that thermodynamic systems can be represented by Riemannian manifolds. Of special interest is the curvature of these manifolds which, for pure fluids, is associated with eA'ective interparticle interaction strength by means of a general thermodynamic "interaction hypothesis. " This interpretation of curvature appears to be consistent with hyperscaling and two-scale-factor universality. The Riemannian geometric model is a new attempt to extract information from the axioms of thermodynamics,
Singularities in the thermodynamics of Kerr-Newman black holes are commonly
associated with phase transitions. However, such interpretations are
complicated by a lack of stability and, more significantly, by a lack of
conclusive insight from microscopic models. Here, I focus on the later problem.
I use the thermodynamic Riemannian curvature scalar $R$ as a try to get
microscopic information from the known thermodynamics. The hope is that this
could facilitate matching black hole thermodynamics to known models of
statistical mechanics. For the Kerr-Newman black hole, the sign of $R$ is
mostly positive, in contrast to that for ordinary thermodynamic models, where
$R$ is mostly negative. Cases with negative $R$ include most of the simple
critical point models. An exception is the Fermi gas, which has positive $R$. I
demonstrate several exact correspondences between the two-dimensional Fermi gas
and the extremal Kerr-Newman black hole. $R$ diverges to $+\infty$ along curves
of diverging heat capacities $C_{J,\Phi}$ and $C_{\Omega,Q}$, but not along the
Davies curve of diverging $C_{J,Q}$. Finding statistical mechanical models with
like behavior might yield additional insight into the microscopic properties of
black holes. I also discuss a possible physical interpretation of $|R|$.Comment: 29 pages, 8 figure
Thermodynamic fluctuation theory originated with Einstein who inverted the relation S = k B ln Ω to express the number of states in terms of entropy: Ω = exp(S/k B ). The theory's Gaussian approximation is discussed in most statistical mechanics texts. I review work showing how to go beyond the Gaussian approximation by adding covariance, conservation, and consistency. This generalization leads to a fundamentally new object: the thermodynamic Riemannian curvature scalar R, a thermodynamic invariant. I argue that |R| is related to the correlation length and suggest that the sign of R corresponds to whether the interparticle interactions are effectively attractive or repulsive.
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