Ruppeiner geometry
Encyclopedia
Ruppeiner geometry is thermodynamic geometry (a type of information geometry
) using the language of Riemannian geometry
to study thermodynamics
. George Ruppeiner proposed it in 1979. He claimed that thermodynamic system
s 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
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
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
and it is similar to the Fisher-Rao metric used in mathematical statistics . The Ruppeiner metric is conformally related to the Weinhold metric via
where T is the temperature of the system under consideration. Proof of the conformal relation can be easily done when one writes down the first law of thermodynamics in differential forms with a few manipulations. 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.
where Na refer to extensive parameters of the system. It has long been observed that the Ruppeiner metric is flat for systems with noninteracting underlying statistical mechanics such as the ideal gas. Curvature singularities signal critical behaviors. In addition, it has been applied to a number of statistical systems including Van de Waals gas. Recently the anyon gas has been studied using this approach.
Black hole's entropy is given by the well-known Bekenstein-Hawking formula
where
is Boltzmann's constant, 
the speed of light, 
the Newton's constant and 
is the area of the event horizon of the black hole. Calculating the Ruppeiner geometry of the black hole's entropy is, in principle, straightforward but important the entropy should be written in terms of extensive parameters, 
where 
is ADM mass of the black hole and 
conserved charges and 
runs from 1 to n. The signature of the metric reflects the sign of the hole's specific heat. For Reissner-Nordström black hole, the Ruppeiner metric has a Lorentzian signature which corresponds to the negative heat capacity it possess, while for the BTZ
black hole we have a Euclidean
signature. This cannot be done for the Schwarzschild black hole because its entropy is
which renders the metric degenerate.
Information geometry
Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. It derives its name from the fact that the Fisher information is used as the Riemannian metric when considering the geometry of probability distribution families...
) using the language of Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
to study thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
. George Ruppeiner proposed it in 1979. He claimed that thermodynamic system
Thermodynamic system
A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
s 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
Equilibrium thermodynamics
Equilibrium Thermodynamics is the systematic study of transformations of matter and energy in systems as they approach equilibrium. The word equilibrium implies a state of balance. Equilibrium thermodynamics, in origins, derives from analysis of the Carnot cycle. Here, typically a system, as...
, 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
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
gij in the distance formula (line element) between the two equilibrium states
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
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
Information geometry
Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. It derives its name from the fact that the Fisher information is used as the Riemannian metric when considering the geometry of probability distribution families...
and it is similar to the Fisher-Rao metric used in mathematical statistics . The Ruppeiner metric is conformally related to the Weinhold metric via
where T is the temperature of the system under consideration. Proof of the conformal relation can be easily done when one writes down the first law of thermodynamics in differential forms with a few manipulations. 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.
where Na refer to extensive parameters of the system. It has long been observed that the Ruppeiner metric is flat for systems with noninteracting underlying statistical mechanics such as the ideal gas. Curvature singularities signal critical behaviors. In addition, it has been applied to a number of statistical systems including Van de Waals gas. Recently the anyon gas has been studied using this approach.
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.Black hole's entropy is given by the well-known Bekenstein-Hawking formula
where
is Boltzmann's constant, the speed of light, the Newton's constant and is the area of the event horizon of the black hole. Calculating the Ruppeiner geometry of the black hole's entropy is, in principle, straightforward but important the entropy should be written in terms of extensive parameters, where is ADM mass of the black hole and conserved charges and runs from 1 to n. The signature of the metric reflects the sign of the hole's specific heat. For Reissner-Nordström black hole, the Ruppeiner metric has a Lorentzian signature which corresponds to the negative heat capacity it possess, while for the BTZBTZ black hole
The BTZ black hole, named after Maximo Banados, Claudio Teitelboim, and Jorge Zanelli, is a black hole solution for -dimensional gravity with a negative cosmological constant....
black hole we have a Euclidean
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
signature. This cannot be done for the Schwarzschild black hole because its entropy is
which renders the metric degenerate.