Metriplectic Systems Research Articles

Energetically consistent model reduction for metriplectic systems

The metriplectic formalism is useful for describing complete dynamical systems which conserve energy and produce entropy. This creates challenges for model reduction, as the elimination of high-frequency information will generally not preserve the metriplectic structure which governs long-term stability of the system. Based on proper orthogonal decomposition, a provably convergent metriplectic reduced-order model is formulated which is guaranteed to maintain the algebraic structure necessary for energy conservation and entropy formation. Numerical results on benchmark problems show that the proposed method is remarkably stable, leading to improved accuracy over long time scales at a moderate increase in cost over naive methods.

Metriplectic Structure of a Radiation-Matter-Interaction Toy Model.

A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines a Leibniz bracket; and a free energy, formed by a “Hamiltonian” and an entropy, playing the role of dynamics generator. Generally, these tensors are a Poisson bracket tensor, describing the Hamiltonian part of the dynamics, and a symmetric metric tensor, that models purely dissipative dynamics. In this paper, the metriplectic system describing a simplified two-photon absorption by a two-level atom is disclosed. The Hamiltonian component is sufficient to describe the free electromagnetic radiation. The metric component encodes the radiation–matter coupling, driving the system to an asymptotically stable state in which the excited level of the atom is populated due to absorption, and the radiation has disappeared. First, a description of the system is used, based on the real–imaginary decomposition of the electromagnetic field phasor; then, the whole metriplectic system is re-written in terms of the phase–amplitude pair, named Madelung variables. This work is intended as a first result to pave the way for applying the metriplectic formalism to many other irreversible processes in nonlinear optics.

Metriplectic integrators for the Landau collision operator

We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations, in general, independently of the mesh configuration.

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

This lecture is a short review on the role entropy plays in those classical dissipative systems whose equations of motion may be expressed via a Leibniz Bracket Algebra (LBA). This means that the time derivative of any physical observable f of the system is calculated by putting this f in a “bracket” together with a “special observable” F, referred to as a Leibniz generator of the dynamics. While conservative dynamics is given an LBA formulation in the Hamiltonian framework, so that F is the Hamiltonian H of the system that generates the motion via classical Poisson brackets or quantum commutation brackets, an LBA formulation can be given to classical dissipative dynamics through the Metriplectic Bracket Algebra (MBA): the conservative component of the dynamics is still generated via Poisson algebra by the total energy H, while S, the entropy of the degrees of freedom statistically encoded in friction, generates dissipation via a metric bracket. The motivation of expressing through a bracket algebra and a motion-generating function F is to endow the theory of the system at hand with all the powerful machinery of Hamiltonian systems in terms of symmetries that become evident and readable. Here a (necessarily partial) overview of the types of systems subject to MBA formulation is presented, and the physical meaning of the quantity S involved in each is discussed. Here the aim is to review the different MBAs for isolated systems in a synoptic way. At the end of this collection of examples, the fact that dissipative dynamics may be constructed also in the absence of friction with microscopic degrees of freedom is stressed. This reasoning is a hint to introduce dissipation at a more fundamental level.

Representation of irreversible systems in a metric thermodynamic phase space

This paper studies geometric properties of a class of irreversible dynamical systems, referred to in the literature as metriplectic systems. This class of systems, related to generalized (or dissipative) Hamiltonian systems, are generated by a conserved component and a dissipative component and appear, for example, in non-equilibrium thermodynamics. In non-equilibrium thermodynamics, the two potentials generating the dynamics are interpreted as generalized energy and generalized entropy, respectively. Stability and stabilization results for metriplectic systems have been presented in the literature, however, some aspects are still poorly understood, in particular the existence of dynamical invariants such as periodic orbits. In this note, we study the properties of metriplectic systems by considering a lift from the n-dimensional state space to a (2n+1)-dimensional contact space, following an approach introduced in recent years to study irreversible control systems. This lift leads to a deeper geometric characterization of metriplectic systems in the extended space. An example is provided to illustrate the approach proposed in this paper.

The Synchronization of Two Metriplectic Systems Arisen from the Lü System

The concept of the metriplectic system was introduced in 1980s by Kaufman (see [). These have important applications in a lot of different mathematical fields, in fluid mechanics or information security. Beginning with the Hamilton-Poisson realization of the Lü system (see [ for details) we construct the associated metriplectic system by adding a dissipation term of a special form. Then, using Pecora and Carroll method, we discuss the synchronization of the two coupled metriplectic systems. In the last part we present an example where the synchronization is used.

Feedback Stabilization of Metriplectic Systems

This paper considers the problem of stabilizing control affine systems where the drift dynamics is generated by a metriplectic structure. These systems can be viewed as an extension of generalized (or dissipative) Hamiltonian systems, where two potentials, interpreted as generalized energy and entropy, are generating the dynamics. The proposed approach consists in generating, by homotopy centered at an equilibrium of the system, a mixed potential for the metriplectic system, and in using the obtained potential to construct a damping state feedback controller. Stability of the closed-loop system is then considered.

Numerical Integration and Synchronization for the 3‐Dimensional Metriplectic Volterra System

The main purpose of this paper is to study the metriplectic system associated to 3‐dimensional Volterra model. For this system we investigate the stability problem and numerical integration via Kahan′s integrator. Finally, the synchronization problem for two coupled metriplectic Volterra systems is discussed.

Asymptotic stability for a class of metriplectic systems

Using the framework of metriplectic systems on Rn we will describe a constructive geometric method to add a dissipation term to a Hamilton-Poisson system such that any solution starting in a neighborhood of a nonlinear stable equilibrium converges toward a certain invariant set. The dissipation term depends only on the Hamiltonian function and the Casimir functions.

A paradigm for joined Hamiltonian and dissipative systems

A paradigm for describing dynamical systems that have both Hamiltonian and dissipative parts is presented. Features of generalized Hamiltonian systems and metric systems are combined to produce what are called metriplectic systems. The phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric Poisson bracket part and a symmetric dissipative part. Flows are obtained by means of this bracket together with a quantity called the generalized free energy, which is composed of an energy and a generalized entropy. The generalized entropy is some function of the Casimir invariants of the Poisson bracket. Two examples are considered: (1) a relaxing free rigid body and (2) a plasma collision operator that can be tailored so that the equilibrium state is an arbitrary monotonic function of the energy.