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

Abstract

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) .

On New Ideal (Noninteracting) Gases in Supercritical Thermodynamics

“…Как мы уже говорили в [12], в общем случае, главный член асимптотики решения можно определить каноническим оператором, который учитывает также и точки поворота, приводящие к появлению в асимптотике функций Эйри. В работе [13] рассмотрен общий случай сложных каустик.…”

“…Понятие лагранжева многообразия в фазовом пространстве, введенное автором, как и следующие из него понятия контактного и лежандрова многообразия, а также понятие канонического оператора на них, позволяет рассматривать с единой точки зрения самые разные физические явления и разные математические концепции. В [12] уже говорилось, что с точки зрения топологических свойств лагранжева многообразия и канонического оператора на нем, правило площадей Максвелла при фазовом переходе и правило площадей Римана для получения скачка на ударной волне для волны Римана -это в математическом смысле одно явление. Но главный момент заключается в том, что точка, в которой площади равны нулю (критическая точка и точка возникновения ударной волны) -это явления одного и того же типа, описываемые туннельным каноническим оператором на идентичном лагранжевом многообразии.…”