Optimal Thermodynamic Processes For Gases

Kushner, Alexei G.; Lychagin, Valentin; Roop, Mikhail

doi:10.3390/e22040448

Cited by 15 publications

(25 citation statements)

References 16 publications

(23 reference statements)

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“…Note that this curve is of the similar form as that separating domains where

and

. This means that we get a more accurate applicability condition for van der Waals model, and it may be a considerable contribution to the theory of phase transitions, since, as we know, it is the sign changing of

that forces the first order phase transitions in van der Waals model [ 4 , 5 , 7 ].…”

Section: Discussionmentioning

confidence: 99%

“…In this section, we briefly recall how contact geometry naturally appears in the context of measurement, as well as how symmetric k -forms represent k th central moments. For details, we refer to Reference [ 4 , 7 , 9 ].…”

Section: Geometry Measurement Thermodynamicsmentioning

confidence: 99%

“…It is natural to require that all the even order symmetric forms must be positive on Legendrian manifolds. In case of

, this condition leads to the notion of phases [ 4 , 7 ]. Further, we will elaborate symmetric forms of orders

and

in more detail for ideal and van der Waals models of gases.…”

Section: Geometry Measurement Thermodynamicsmentioning

confidence: 99%

“…The geometrical interpretation of thermodynamic systems in equilibrium goes back already to the 19th century [ 1 ] and is reflected recently in Reference [ 2 ]. In modern terms, it is clear that thermodynamic states are Legendrian submanifolds, i.e., maximal integral manifolds of the structure contact form (for more details, see Reference [ 3 , 4 , 5 ]) in the contact space where the mentioned structure form is the first law of thermodynamics. Additional structures, such as Riemannian structures on these Legendrian manifolds, were studied in, for example, Reference [ 6 ].…”

Section: Introductionmentioning

confidence: 99%

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On Higher Order Structures in Thermodynamics

Lychagin

Roop

2020

Entropy

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We present the development of the approach to thermodynamics based on measurement. First of all, we recall that considering classical thermodynamics as a theory of measurement of extensive variables one gets the description of thermodynamic states as Legendrian or Lagrangian manifolds representing the average of measurable quantities and extremal measures. Secondly, the variance of random vectors induces the Riemannian structures on the corresponding manifolds. Computing higher order central moments, one drives to the corresponding higher order structures, namely the cubic and the fourth order forms. The cubic form is responsible for the skewness of the extremal distribution. The condition for it to be zero gives us so-called symmetric processes. The positivity of the fourth order structure gives us an additional requirement to thermodynamic state.

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“…Note that this curve is of the similar form as that separating domains where

and

that forces the first order phase transitions in van der Waals model [ 4 , 5 , 7 ].…”

Section: Discussionmentioning

confidence: 99%

Section: Geometry Measurement Thermodynamicsmentioning

confidence: 99%

“…It is natural to require that all the even order symmetric forms must be positive on Legendrian manifolds. In case of

, this condition leads to the notion of phases [ 4 , 7 ]. Further, we will elaborate symmetric forms of orders

and

in more detail for ideal and van der Waals models of gases.…”

Section: Geometry Measurement Thermodynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Higher Order Structures in Thermodynamics

Lychagin

Roop

2020

Entropy

Self Cite

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“…for all x t ∈ M and t ∈ R. In particular, from (17) and (18) the following global internal energy functional…”

Section: Proposition 1 the Functional Manifold Gmentioning

confidence: 99%

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Balinsky

Blackmore

Kycia

et al. 2020

Entropy

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We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

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Nonequilibrium thermodynamic process with hysteresis and metastable states—A contact Hamiltonian with unstable and stable segments of a Legendre submanifold

Goto

2022

Journal of Mathematical Physics

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In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state. Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves. Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory. Then, a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold and that another pruned segment is a unstable fixed point set. This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one. To show the statements above explicitly, an Ising type spin model with long-range interactions, called the Husimi–Temperley model, is focused, where this model exhibits a phase transition.

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Optimal Thermodynamic Processes For Gases

Cited by 15 publications

References 16 publications

On Higher Order Structures in Thermodynamics

On Higher Order Structures in Thermodynamics

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Nonequilibrium thermodynamic process with hysteresis and metastable states—A contact Hamiltonian with unstable and stable segments of a Legendre submanifold

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