Geometric quantization of thermodynamics, phase transitions, and asymptotes at critical points
type: Item
instance of: scholarly article
title: Geometric quantization of thermodynamics, phase transitions, and asymptotes at critical points
zbMATH Open document ID: 0845.58035
DOI: 10.1007/BF02362416
author: V. P. Maslov
published in: Mathematical Notes
publication date: 1996-09-16T00:00:00Z
review text: If \(\psi\) is a one-form on the manifold \(Q\), we may consider it as a mapping from \(Q\) to \(T^*Q\) which satisfies the equation \(\psi^* \vartheta = \vartheta\), where \(\vartheta \) is the canonical one-form on the cotangent bundle \(T^*Q\). This allows to establish a natural one-to-one correspondence between the Lagrangian submanifolds of \(T^*Q\) and the closed one-forms on \(Q\). If the one-form corresponding to the Lagrangian submanifold is of the form \(dU\), \(U\) is called a generating function for the submanifold. Here the author considers nine-dimensional configurational manifold of intensive thermodynamic parameters, the temperature \(T\), the pressure \(P\), the electric field strength \(\vec E\), the magnetic field strength \(\vec H\) and the electrostatic potential \(\varphi\). Together with the corresponding momenta which are chosen to be the extensive specific quantities -- the entropy \(s\), the volume \(V\), the polarization \(\vec p\), the magnitization \(\vec \mu\) and the charge \(e\), they form an eighteen-dimensional phase space \(\mathbb{R}^{18}\). The one-form \(dU=T ds-P ds+\vec E.d\vec p+\vec H.d\vec\mu+\varphi de\) defines a new thermodynamic quantity and a Lagrangian submanifold \(\Lambda\) in \(\mathbb{R}^{18}\). With \(\Lambda\) one can associate the so-called ``canonical tunnel operator'' \(K_{\Lambda, \hbar}\). Specification of the canonical tunnel operator defines axiomatically the geometric quantization of thermodynamics. This line is not traced out further but after taking the classical limit, an integral equation is derived from which the local equations of state follow.
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 830732
zbMATH Keywords: phase transitions
zbMATH Keywords: asymptotes
zbMATH Keywords: Lagrangian submanifolds
zbMATH Keywords: integral equations
zbMATH Keywords: geometric quantization
reviewed by: Ivaïlo Mladenov
MaRDI profile type: Publication
cites work: Q4040558
description: scientific article; zbMATH DE number 830732