Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDirac structures in nonequilibrium thermodynamics
90
0
0.0
(
0
)
Added by
Fran\\c{c}ois Gay-Balmaz
Publication date
2017
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Dirac structures are geometric objects that generalize both Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems. In this paper, we show that the evolution equa- tions for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures, both on the Lagrangian and the Hamiltonian settings. In absence of irreversible processes these Dirac structures reduce to canonical Dirac structures associated to canonical symplectic forms on phase spaces. Our geometric formulation of nonequilibrium thermodynamic thus consistently extends the geometric formulation of mechanics, to which it reduces in absence of irreversible processes. The Dirac structures are associated to the variational formulation of nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2016a,b] and are induced from a nonlinear nonholonomic constraint given by the expression of the entropy production of the system.
rate research
Read More
Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in structuring a dynamical system through the energy flow between its subsystems and elements. In this paper, we show that the evolution equations for open thermodynamic systems, i.e., systems exchanging heat and matter with the exterior, admit an intrinsic formulation in terms of Dirac structures. We focus on simple systems, in which the thermodynamic state is described by a single entropy variable. A main difficulty compared to the case of closed systems lies in the explicit time dependence of the constraint associated to the entropy production. We overcome this issue by working with the geometric setting of time-dependent nonholonomic mechanics. We define three type of Dirac dynamical systems for the nonequilibrium thermodynamics of open systems, based either on the generalized energy, the Lagrangian, or the Hamiltonian. The variational formulations associated to the Dirac systems formulations are also presented.
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit field equations obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Furthermore, we show that any multi-Dirac structure naturally gives rise to a multi-Poisson bracket. We treat the case of field theories with nonholonomic constraints, showing that the integrability of the constraints is equivalent to the integrability of the underlying multi-Dirac structure. We finish with a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields and the electromagnetic field.
In this paper, we introduce the notion of port-Lagrangian systems in nonequilibrium thermodynamics, which is constructed by generalizing the notion of port-Lagrangian systems for nonholonomic mechanics proposed in Yoshimura and Marsden [2006c], where the notion of interconnections is described in terms of Dirac structures. The notion of port-Lagrangian systems in nonequilibrium thermodynamics is deduced from the variational formulation of nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2017a,2017b]. It is a type of Lagrange-dAlembert principle associated to a specific class of nonlinear nonholonomic constraints, called phenomenological constraints, which are associated to the entropy production equation of the system. To these phenomenological constraints are systematically associated variational constraints, which need to be imposed on the variations considered in the principle. In this paper, by specifically focusing on the cases of simple thermodynamic systems with constraints, we show how the interconnections in thermodynamics can be also described by Dirac structures on the Pontryagin bundle as well as on the cotangent bundle of the thermodynamic configuration space. Each of these Dirac structures is induced from the variational constraint. Furthermore, the variational structure associated to this Dirac formulation is presented in the context of the Lagrange-dAlembert-Pontryagin principle. We illustrate our theory with some examples such as a cylinder-piston with ideal gas as well as an LCR circuit with entropy production due to a resistor.
In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role in several topics of modern Mathematics, in particular in the study of the geometry of moduli spaces of a class of black holes, the geometry of NS-5 brane solutions of type II supergravity theories and BPS solitons in some string theories. We obtain a generalized Lichnerowicz formula, decompositions of the Dirac and spin-Dirac operators and their squares in terms of the standard Dirac and spin-Dirac operators and using the fact that spinor fields (sections of a spin-Clifford bundle) have representatives in the Clifford bundle we present also a noticeable relation involving the spin-Dirac and the Dirac operators.
We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space $R^n$, the hyperbolic space $H^n$ and a Riemannian manifold $mathfrak{S^n}$ ($ngeq 3$) with the Schwarzschild metric to any Riemannian manifold $N$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
