Geometric Perturbation Theory in Physics

Omohundro, Stephen M.

doi:10.1142/0287

Cited by 23 publications

(18 citation statements)

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“…Chandrashekhar [5] gives a rare treatment of this subject in a textbook. An analogy of thermodynamics with mechanics and optics is mentioned occasionally [6,[8][9][10]. The most detailed geometrical formulation is in the review article Ref.…”

Section: Classical Thermodynamicsmentioning

confidence: 99%

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A Hamilton–Jacobi formalism for thermodynamics

Rajeev

2008

Annals of Physics

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We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/ volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is 2n þ 1-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The 'time' variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the CurieWeiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.

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Section: Classical Thermodynamicsmentioning

confidence: 99%

“…Such problems arise not only in statistical mechanics and thermodynamics, but also in other areas of physics and even in economics [10,11].…”

Section: Contact Geometrymentioning

confidence: 99%

A Hamilton–Jacobi formalism for thermodynamics

Rajeev

2008

Annals of Physics

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“…For instance, in the energy representation [14], one currently considers for a fluid the symplectic 2-form dT ∧ dS − dP ∧ dV in the corresponding 4-dimensional space. Likewise, in the entropy representation [15], the fundamental symplectic 2-form is…”

Section: Introduction and Outlinementioning

confidence: 99%

Hamiltonian structure of thermodynamics with gauge

Balian

Valentin²

2001

Eur. Phys. J. B

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Denoting by q i (i = 1, ..., n) the set of extensive variables which characterize the state of a thermodynamic system, we write the associated intensive variables γ i , the partial derivatives of the entropy S = S q 1 , ..., q n ≡ q 0 , in the form γ i = −p i /p 0 where p 0 behaves as a gauge factor. When regarded as independent, the variables q i , p i (i = 0, ..., n) define a space T having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a n + 1-dimensional gauge-invariant Lagrangean submanifold M of T. Any thermodynamic process, even dissipative, taking place on M is represented by a Hamiltonian trajectory in T, governed by a Hamiltonian function which is zero on M. A mapping between the equations of state of different systems is likewise represented by a canonical transformation in T. Moreover a Riemannian metric arises naturally from statistical mechanics for any thermodynamic system, with the differentials dq i as contravariant components of an infinitesimal shift and the dp i 's as covariant ones. Illustrative examples are given.

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“…A synthetic derivation of a Lagrangian submanifold from the MEP with nonlinear constraints is contained in [16].…”

Section: Introductionmentioning

confidence: 99%

Isotropic submanifolds generated by the Maximum Entropy Principle and Onsager reciprocity relations

Favretti

2005

Journal of Functional Analysis

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We show that the Maximum Entropy Principle (MEP) (Phys. Rev. 106 (Part I and II) (1957) 620-630; Phys. Rev. 108 (1957) 171-630), when considered as a constrained extremization problem, defines in a natural way a Morse Family and a related isotropic (Lagrangian in the finite-dimensional case) submanifold of an infinite-dimensional linear symplectic space. This geometric approach becomes useful when dealing with the MEP with nonlinear constraints and it allows to derive Onsager-like reciprocity relations as a consequence of the isotropy.

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Geometric Perturbation Theory in Physics

Cited by 23 publications

References 0 publications

A Hamilton–Jacobi formalism for thermodynamics

A Hamilton–Jacobi formalism for thermodynamics

Hamiltonian structure of thermodynamics with gauge

Isotropic submanifolds generated by the Maximum Entropy Principle and Onsager reciprocity relations

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