Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics

Abstract

Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed.

38 pages

References in corpus (5)

• Contact Symmetries and Hamiltonian Thermodynamics
• Liouville's Theorem and the canonical measure for nonconservative systems from contact geometry
• Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics
• Para-Sasakian geometry in thermodynamic fluctuation theory
• Contact Geometry of the Pontryagin Maximum Principle

Cited by in corpus (22)

• Contact Hamiltonian Mechanics
• New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries
• Contact variational integrators
• A $k$-contact Lagrangian formulation for nonconservative field theories
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
• Higher-order contact mechanics
• Nonautonomous k-contact field theories
• Contact Lie systems
• Application of Herglotz's Variational Principle to Electromagnetic Systems with Dissipation
• The Herglotz variational principle for dissipative field theories
• Exact Baker-Campbell-Hausdorff formula for the contact Heisenberg algebra
• Inverse problem and equivalent contact systems
• A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian
• Nonequilibrium thermodynamic process with hysteresis and metastable states -- A contact Hamiltonian with unstable and stable segments of a Legendre submanifold
• Expectation variables on a para-contact metric manifold exactly derived from master equations
• Diffusion equations from master equations -- A discrete geometric approach --
• Skinner-Rusk formalism for k-contact systems
• Constrained Lagrangian dissipative contact dynamics
• Hessian-information geometric formulation of Hamiltonian systems and generalized Toda's dual transform
• From the Fokker-Planck equation to a contact Hamiltonian system
• Affine geometric description of thermodynamics