A geometric theory of selective decay with applications in MHD

Gay–Balmaz, François; Holm, Darryl D.

doi:10.1088/0951-7715/27/8/1747

Cited by 16 publications

(25 citation statements)

References 59 publications

(92 reference statements)

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“…We then exploit this setting to show that the entropy in the Souriau model is a Casimir function of the Lie-Poisson bracket with Lie algebra cocycle associated with the nonequivariance cocycle of the momentum map, i.e., it Poisson commutes with every functions. Based on this we formulate a dynamical geometric model for dissipation/production of this Casimir, following the Lie algebraic setting proposed in [37,38]. This allows us to clarify the link between the geometry underlying Souriau symplectic models and that underlying models proposed in [73] in the framework of quantum physics by information geometry for some Lie algebras, see also [5].…”

Section: Souriau Symplectic Model Of Statistical Mechanicsmentioning

confidence: 98%

“…We follow the general Lie algebraic approach developed in [37,38] for Casimir dissipation, slightly extended here to take into account of a cocycle, and to a wider class of dissipation.…”

Section: Dynamics With Casimir Dissipation/productionmentioning

confidence: 99%

“…We denote by : g → g * the flat operator associated to γ. That is, the linear form ξ ∈ g * is given by ξ (η) = γ(ξ, η), for all ξ, η ∈ g. For Θ = 0 and h = k, this is the model proposed in [37,38] and applied there in the infinite dimensional setting, with applications to geophysical fluids and magnetohydrodynamics. because of (2.39) and since {h, h} Θ = 0.…”

Section: Dynamics With Casimir Dissipation/productionmentioning

confidence: 99%

“…They are the critical conditions for the variational principle 38) which is just a special instance of (4.36) applied to central extensions. The existence of a field (g, α) :…”

Section: Multisymplectic Lie Group Variational Integratorsmentioning

confidence: 99%

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Lie Group Cohomology and (Multi)Symplectic Integrators : New Geometric Tools for Lie Group Machine Learning based on Souriau Geometric Statistical Mechanics

Barbaresco¹,

Gay–Balmaz

2020

Preprint

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In this paper we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated to Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

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Section: Souriau Symplectic Model Of Statistical Mechanicsmentioning

confidence: 98%

“…We follow the general Lie algebraic approach developed in [37,38] for Casimir dissipation, slightly extended here to take into account of a cocycle, and to a wider class of dissipation.…”

Section: Dynamics With Casimir Dissipation/productionmentioning

confidence: 99%

Section: Dynamics With Casimir Dissipation/productionmentioning

confidence: 99%

“…They are the critical conditions for the variational principle 38) which is just a special instance of (4.36) applied to central extensions. The existence of a field (g, α) :…”

Section: Multisymplectic Lie Group Variational Integratorsmentioning

confidence: 99%

See 2 more Smart Citations

Lie Group Cohomology and (Multi)Symplectic Integrators : New Geometric Tools for Lie Group Machine Learning based on Souriau Geometric Statistical Mechanics

Barbaresco¹,

Gay–Balmaz

2020

Preprint

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“…This equation shows that the system is forced towards a position where the right-hand side vanishes, which is a condition compatible with an equilibrium solution of the original deterministic system. We refer to [37] for an extensive discussion on this condition.…”

Section: The Double Bracket Dissipationmentioning

confidence: 99%

Structure preserving noise and dissipation in the Toda lattice

Arnaudon¹

2018

J. Phys. A: Math. Theor.

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In this paper, we use Flaschka's change of variables of the open Toda lattice and its interpretation in term of the group structure of the LU factorisation as a coadjoint motion on a certain dual of Lie algebra to implement a structure preserving noise and dissipation. Both preserve the structure of coadjoint orbit, that is the space of symmetric tri-diagonal matrices and arise as a new type of multiplicative noise and nonlinear dissipation of the Toda lattice. We investigate some of the properties of these deformations and in particular the continuum limit as a stochastic Burger equation with a nonlinear viscosity. This work is meant to be exploratory, and open more questions that we can answer with simple mathematical tools and without numerical simulations.

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From Variational to Bracket Formulations in Nonequilibrium Thermodynamics of Simple Systems

Gay–Balmaz

Yoshimura

2019

Lecture Notes in Computer Science

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A variational formulation for nonequilibrium thermodynamics was recently proposed in Gay-Balmaz and Yoshimura [2017a,b] for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systematic way to derive from a unified perspective several bracket formulations for nonequilibrium thermodynamics proposed earlier in the literature, such as the single generator bracket and the double generator bracket. In the case of a linear relation between the thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC bracket is recovered. We also show how the processes of reduction by symmetry can be applied to these brackets. In the reduced setting, we also consider the case in which the coadjoint orbits are preserved and explain the link with double bracket dissipation. A similar development has been presented for continuum systems in Eldred and Gay-Balmaz [2019] and applied to multicomponent fluids.

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A geometric theory of selective decay with applications in MHD

Cited by 16 publications

References 59 publications

Lie Group Cohomology and (Multi)Symplectic Integrators : New Geometric Tools for Lie Group Machine Learning based on Souriau Geometric Statistical Mechanics

Lie Group Cohomology and (Multi)Symplectic Integrators : New Geometric Tools for Lie Group Machine Learning based on Souriau Geometric Statistical Mechanics

Structure preserving noise and dissipation in the Toda lattice

From Variational to Bracket Formulations in Nonequilibrium Thermodynamics of Simple Systems

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