Although thermodynamic fluctuation theory originated from statistical mechanics, it may be put on a completely thermodynamic basis, in no essential need of any microscopic foundation. This review views the theory from the macroscopic perspective, emphasizing, in particular, notions of covariance and consistency, expressed naturally using the language of Riemannian geometry. Coupled with these concepts is an extension of the basic structure of thermodynamic fluctuation theory beyond the classical one of a subsystem in contact with an infinite uniform reservoir. Used here is a hierarchy of concentric subsystems, each of which samples only the thermodynamic state of the subsystem immediately larger than it. The result is a covariant thermodynamic fluctuation theory which is plausible beyond the standard second-order entropy expansion. It includes the conservation laws and is mathematically consistent when applied to fluctuations inside subsystems. Tests on known models show improvements. Perhaps most significantly, the covariant theory offers a qualitatively new tool for the study of fluctuation phenomena: the Riemannian thermodynamic curvature. The thermodynamic curvature gives, for any given thermodynamic state, a lower bound for the length scale where the classical thermodynamic fluctuation theory based on a uniform environment could conceivably hold. Straightforward computation near the critical point reveals that the curvature equals the correlation volume, a physically appealing finding. The combination of the interpretation of curvature with a well-known proportionality between the free energy and the inverse of the correlation volume yields a purely thermodynamic theory of the critical point. The scaled equation of state follows from the values of the critical exponents. The thermodynamic Riemannian metric may be put into the broader context of information theory.

Riemannian geometry in thermodynamic fluctuation theory

https://pure.rug.nl/ws/files/2807768/2007RepMathPhysEberard.pdf

An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes

We discuss a scheme based on Ehrenfest-like equations to exhibit and classify transitions between two phases (with ``smaller'' and ``larger'' masses) of Kerr-AdS black holes. We show that for fixed angular velocity this phase transition is of second order as both Ehrenfest's equations are satisfied. Finally, we make a close connection of the results found from this analysis with those obtained from the thermodynamic state space geometry approach.

Second order phase transition and thermodynamic geometry in Kerr-AdS black holes

It is shown that the stability of ideal quantum gases can be measured by means of the Riemann scalar curvature R of the parameter space. The components of the metric tensor were assumed to be the second moments of energy and the number of particle fluctuations. As a result, R is a function of the second and third moments of those quantities. For bosons R is positive and increases monotonically from zero at the classical limit to positive infinity in the condensation region. A system is less stable if R is bigger and vice versa. For fermions R is negative and this means that Fermi gases are more stable than the ideal Bose and ideal classical systems.

https://iopscience.iop.org/article/10.1088/0305-4470/23/4/016/pdf

Riemannian geometry and stability of ideal quantum gases

In this paper, we study various aspects of the equilibrium thermodynamic state space geometry of AdS black holes. We first examine the Reissner-Nordstrom-AdS (RN-AdS) and the Kerr-AdS black holes. In this context, the state space scalar curvature of these black holes is analysed in various regions of their thermodynamic parameter space. This provides important new insights into the structure and significance of the scalar curvature. We further investigate critical phenomena, and the behaviour of the scalar curvature near criticality, for KN-AdS black holes in two mixed ensembles, introduced and elucidated in our earlier work arXiv:1002.2538 [hep-th]. The critical exponents are identical to those in the RN-AdS and Kerr-AdS cases in the canonical ensemble. This suggests an universality in the scaling behaviour near critical points of AdS black holes. Our results further highlight qualitative differences in the thermodynamic state space geometry for electric charge and angular momentum fluctuations in these black holes.

R. Mrugała

Papers

Geometrical structure of the state space in classical statistical and phenomenological thermodynamics