\(W\)-flows on Weyl manifolds and Gaussian thermostats
type: Item
instance of: scholarly article
title: \(W\)-flows on Weyl manifolds and Gaussian thermostats
zbMATH Open document ID: 0969.37017
DOI: 10.1016/S0021-7824(00)00176-8
author: Maciej P. Wojtkowski
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2001-09-25T00:00:00Z
review text: This is a very clearly written paper on Weyl structures, including examples. A Weyl structure on a compact Riemannian \(n\)-dimensional manifold \(M\) consists of a family of torsion-free connections with the parallel transport preserving conformal class of metrics. The author fixes the Riemannian metric, which plays the role of a physical space, and takes a Weyl flow on \(UM\), whose orbits are geodesics of the Weyl connection. The Weyl structure coincides with the Levi-Civita connection on a rescaled metric. He calls \(W\)-flows those defined by this new metric. He points out several physical settings where this situation holds: on the cotangent bundle, in the Hamiltonian formulation of Gaussian thermostats, which are models in nonequilibrium statistical mechanics. Also, in conformally symplectic formulations of isokinetic dynamics. The author then uses the Jacobi equation for the geodesic flow of a Weyl connection and straightforwardly obtains the linear approximation of the flow in terms of the curvature of the Weyl connection. The symmetric component is a multiple of the identity but the antisymmetric part corresponds to sectional curvatures in plane directions. By analysing their sign he obtains that sectional curvatures for the Weyl structures defined by a constant vector field on a flat two dimensional torus are zero, but this is not the case in higher dimensions. He then turns to study extensively the hyperbolic properties of \(W\)-flows. Finally, taking a Riemannian manifold \(M\) with boundary, he analyses a \(W\)-flow in two dimensions with collisions, as defined in [the author and \textit{P. Maciej}, Commun. Math. Phys. 126, 507-533 (1990; Zbl 0695.70007)], and observes that the splitting obtained for this case, does not occur in dimensions 3 and higher. For other articles on Weyl manifolds, see \textit{A. Yoshioka} [Rep. Math. Phys. 43, No. 1-2, 357-366 (1999; Zbl 0957.53049) and Lobachevskii J. Math. 4, 177-206 (1999; Zbl 0963.53055)] for those on non…
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 82C10
Mathematics Subject Classification ID: 82C40
zbMATH DE Number: 1557019
zbMATH Keywords: Weyl manifold
zbMATH Keywords: geodesic flows
zbMATH Keywords: isokinetic dynamics
zbMATH Keywords: Gaussian thermostats
zbMATH Keywords: hyperbolic flows
zbMATH Keywords: Sinai billiards
zbMATH Keywords: Lorentz gas
reviewed by: L. Del Riego
MaRDI profile type: Publication
Wikidata QID: Q125966230
full work available at URL: s0021-7824(00)00176-8
OpenAlex ID: W2034585515
Recommended article: Q4549222
Recommended article: Weingarten flows in Riemannian manifolds
Recommended article: Weyl-flow and the conformally symplectic structure of thermostatted billiards: the problem of the hyperbolicity
Recommended article: Gaussian thermostats as geodesic flows of nonsymmetric linear connections
Recommended article: Magnetic flows and Gaussian thermostats on manifolds of negative curvature
Recommended article: On invariant MASAs for endomorphisms of the Cuntz algebras
Recommended article: On the thermodynamic formalism for the Gauss map
Recommended article: The heat equation on manifolds as a gradient flow in the Wasserstein space
Recommended article: The dynamics of the Weil-Petersson flow
Recommended article: On thermodynamics of rational maps on the Riemann sphere
description: scientific article; zbMATH DE number 1557019