Ruppeiner geometry

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The Ruppeiner geometry is a way to study thermodynamics using the language of Riemannian geometry. George Ruppeiner proposed it in 1979. He claimed that thermodynamic systems can be represented by Riemannian geometry, and that statistical properties can be derived from the model.

This geometrical model is based on the inclusion of the theory of fluctuations into the axioms of equilibrium thermodynamics, namely there exist equilibrium states which can be represented by points on two-dimensional surface (manifold) and the distance between these equilibrium states is related to the fluctuation between them. This concept is associated to probabilities, i.e. the less probable a fluctuation between states, the further apart they are. This can be recognized if one considers the metric tensor gij in the distance formula (line element) between the two equilibrium states

ds^2 = g^R_{ij} dx^i dx^j, \,

where the matrix of coefficients gij is the symmetric metric tensor which is called a Ruppeiner metric, defined as a negative Hessian of the entropy function

ds^2_R = -\partial_i \partial_j S(M, N^a)

where M is the mass (internal energy) of the system and Na refer to extensive parameters of the system. Mathematically the Ruppeiner geometry is one particular type of information geometry. The basic idea is to take the Hessian matrix of the entropy function. The Ruppeiner metric is conformally related to the Weinhold metric via

ds^2_R = \frac{1}{T} ds^2_W \,

where T is the temperature of the system under consideration. The Weinhold geometry is also considered as a thermodynamic geometry. It is defined as a Hessian of mass (internal energy) with respect to entropy and other extensive parameters.

ds^2_W = \partial_i \partial_j M(S, N^a)

where Na refer to extensive parameters of the system. The Ruppeiner metric is flat for systems with noninteracting underlying statistical mechanics such as the ideal gas. Curvature singularities signal critical behaviors.

[edit] Application to black hole systems

In the last five years or so this geometry has been applied to black hole thermodynamics with some results physically relevant. The most physically significant case is in the Kerr black holes in higher dimensions where the curvature singularity signals thermodynamic instability as found earlier by conventional method.


[edit] References

  • Ruppeiner, George (1995), “Riemannian geometry in thermodynamic fluctuation theory”, Reviews of Modern Physics 67: 605–659, doi:10.1103/RevModPhys.67.605 .

[edit] External links