New parastatistics leading to classical thermodynamics: Physical interpretation. II

Maslov, V. P.

doi:10.1134/s0001434614090120

Math Notes

2014

DOI: 10.1134/s0001434614090120

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New parastatistics leading to classical thermodynamics: Physical interpretation. II

V. P. Maslov

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Cited by 3 publications

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“…The projection of such a focal point was studied in detail by Arnold and his team. 1 Once the famous mathematician Poisson presented a physical counter example to Fresnel's wave theory. He used Fresnel's theory to obtain a single illuminated point in the deep shadow behind a smooth ball.…”

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Violation of Carathéodory axioms at the critical point of a gas. Frenkel point as the critical point of the transition “liquid-amorphous solid” in the region of negative pressures

Maslov

2014

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We give a geometric interpretation of the thermodynamic potential, free and internal energy, and enthalpy in terms of a Lagrangian manifold in the phase space of pairs (T, −S), (−μ, N ), and (P, V ) of intensive and extensive variables. The Lagrangian manifold is viewed as the dequantization of the tunnel canonical operator. With this approach, the critical point is a point where the equilibrium quasi-static process described by the Carath´eodory axioms is violated. For a hard liquid with negative pressure, we present a model of a multi-modulus medium.1. The papers [1], [2] deal with the Carath´eodory axioms. Let us return once more to these remarkable axioms and to the notion of quasi-static process introduced by Carath´eodory. Equilibrium thermodynamics is a consequence of Carath´eodory's definition of such a process.Equilibrium thermodynamics deals with pairs of dual variables, namely, intensive variables, which include temperature T , pressure P , and the chemical potential μ, and extensive variables, which include the entropy S, the volume V , and the number N of particles. If we consider the phase space in which T , P , and μ (in economics, the chemical potential with the minus sign has the meaning of the nominal interest rate) are coordinates and −S (negated entropy), V , and N are the dual moments, then the equation of state is a three-dimensional Lagrangian manifold in the six-dimensional phase space. One of the variables, say, the number N of particles or the volume V , can usually be assumed to be constant, and then one deals with a four-dimensional phase space, say, of the variables (T, −S, −μ, N ) and the two-dimensional surface corresponding to the equation of state in this space.However, we should keep in mind that we describe an equilibrium system and, according to the Carath´eodory axioms, a quasi-static process. In particular, this means that we should introduce a small viscosity ν and take this parameter into account by considering the time t satisfying t τ , where τ is the relaxation time, which depends on ν.The asymptotics as ν → 0 (in the principal term) is given by the tunnel canonical operator on the above-mentioned Lagrangian manifold. It provides a one-to-one projection of the Lagrangian manifold onto the coordinate plane (the gas-liquid phase transition).Let us explain this in more detail.In the one-dimensional case, the semiclassical approximation of quantum mechanics outside the turning points is well known to deal with sums of exponentials of the form Ψ ∼ 2 k=1 ϕ k (x)e (i/h)S k (x) .

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Violation of Carathéodory axioms at the critical point of a gas. Frenkel point as the critical point of the transition “liquid-amorphous solid” in the region of negative pressures

Maslov

2014

Math Notes

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Negative nominal interest rate as a guarantee of equilibrium for the class of oligarchs and the middle class

Howl-Maslova¹,

Maslov²

2014

Math Notes

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Calculation of the number of collective degrees of freedom and of the admissible cluster size for isotherms in the Van-der-Waals model in supercritical states

Maslov

2014

Russ. J. Math. Phys.

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New parastatistics leading to classical thermodynamics: Physical interpretation. II

Cited by 3 publications

References 10 publications

Violation of Carathéodory axioms at the critical point of a gas. Frenkel point as the critical point of the transition “liquid-amorphous solid” in the region of negative pressures

Violation of Carathéodory axioms at the critical point of a gas. Frenkel point as the critical point of the transition “liquid-amorphous solid” in the region of negative pressures

Negative nominal interest rate as a guarantee of equilibrium for the class of oligarchs and the middle class

Calculation of the number of collective degrees of freedom and of the admissible cluster size for isotherms in the Van-der-Waals model in supercritical states

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