Thermodynamics from the differential geometry standpoint

Павлов, В. Ф.; Сергеев, Виктор Михайлович

doi:10.1007/s11232-008-0122-8

Cited by 10 publications

(10 citation statements)

References 4 publications

(6 reference statements)

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“…In statistical mechanics, a canonical ensemble [117][118][119][120][121] is the statistical ensemble that is used to represent the possible states of a mechanical system that is being maintained in thermodynamic equilibrium. Souriau has defined this Gibbs canonical ensemble on Symplectic manifold M for a Lie group action on M.…”

Section: Souriau-gibbs Canonical Ensemble Of Dynamical Group and Lie mentioning

confidence: 99%

Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics

Barbaresco

2014

Entropy

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Section: Souriau-gibbs Canonical Ensemble Of Dynamical Group and Lie mentioning

confidence: 99%

Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics

Barbaresco

2014

Entropy

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“…Pavlov and Sergeev (2008), among others, have discussed the interpretation of Gibbs' thermodynamics in terms of differential geometry. In the fully developed context of 'local' thermodynamics the dynamical variables are fields over an infinite dimensional configuration space.…”

Section: The Phenomenological Approachmentioning

confidence: 99%

Relativistic thermodynamics, a Lagrangian field theory for general flows including rotation

Frønsdal

2017

Int. J. Geom. Methods Mod. Phys.

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Any theory that is based on an action principle has a much greater predictive power than one that does not have such a formulation. The formulation of a dynamical theory of General Relativity, including matter, is here viewed as a problem of coupling Einstein's theory of pure gravity to an independently chosen and well defined field theory of matter. It is well known that this is accomplished in a most natural way when both theories are formulated as relativistic, Lagrangian field theories, as is the case with Einstein-Maxwell theory. Special matter models of this type have ben available; here a more general thermodynamical model that allows for vortex flows is presented.In a wider context, the problem of subjecting hydrodynamics and thermodynamics to an action principle is one that has been pursued for at least 150 years. A solution to this problem has been known for some time, but only under the strong restriction to potential flows.A variational principle for general flows has become available. It represents a development of the Navier-Stokes-Fourier approach to fluid dynamics. The principal innovation is the recognition that two kinds of flow velocity fields are needed, one the gradient of a scalar field and the other the time derivative of a vector field that is closely associated with vorticity. In the relativistic theory that is presented here the latter is the Hodge dual of an exact 3-form, well known as the notoph field of Ogievetskij and Palubarinov, the B-field of Kalb and Ramond and the vorticity field of Lund and Regge. The total number of degrees of freedom of a unary system, including the density and the two velocity fields is 4, as expected.The present paper deals with the relativistic context, Special Relativity and General Relativity. The energy momentum tensor has a structure that is more general than that of Tolman, and different from proposed generalizations; it appears to be well suited to represent rotational flows in General Relativity. The current of mass flow is conserved: the theory incorporates the hydrodynamical equation of continuity.

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“…More recently, Kapranov has also given a thermodynamical interpretation of the moment map for toric varieties [97] and Pavlov, thermodynamics from the differential geometry standpoint [98].…”

Section: Revisited Souriau Symplectic Model Of Statistical Physicsmentioning

confidence: 99%

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

Barbaresco

2016

Entropy

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Abstract:We introduce the symplectic structure of information geometry based on Souriau's Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.Keywords: Lie group thermodynamics; moment map; Gibbs density; Gibbs equilibrium; maximum entropy; information geometry; symplectic geometry; Cartan-Poincaré integral invariant; geometric mechanics; Euler-Poincaré equation; Fisher metric; gauge theory; affine group Lorsque le fait qu'on rencontre est en opposition avec une théorie régnante, il faut accepter le fait et abandonner la théorie, alors même que celle-ci, soutenue par de grands noms, est généralement adoptée -Claude Bernard in "Introduction à l'Étude de la Médecine Expérimentale" [1] Au départ, la théorie de la stabilité structurelle m'avait paru d'une telle ampleur et d'une telle généralité, qu'avec elle je pouvais espérer en quelque sorte remplacer la thermodynamique par la géométrie, géométriser en un certain sens la thermodynamique, éliminer des considérations thermodynamiques tous les aspects à caractère mesurable et stochastiques pour ne conserver que la caractérisation géométrique correspondante des attracteurs.-René Thom in "Logos et théorie des Catastrophes" [2] Entropy

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Thermodynamics from the differential geometry standpoint

Cited by 10 publications

References 4 publications

Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics

Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics

Relativistic thermodynamics, a Lagrangian field theory for general flows including rotation

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

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