arXiv subjects

François Gay-Balmaz

Publications and source records attributed to François Gay-Balmaz.

At least 55 records · Page 3Linked to original sources

arXiv2021

From quantum hydrodynamics to Koopman wavefunctions I

Based on Koopman's theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classical wavefunctions. Upon applying the Madelung transform from quantum hydrodynamics in the new context, we show how the Koopman wavefunction picture is insufficient to reproduce arbitrary classical distributions. However, this problem is entirely overcome by resorting to von Neumann operators. Indeed, we show that the latter also allow for singular $δ-$like profiles of the Liouville density, thereby reproducing point particles in phase space.

math-ph↗

arXiv2021

From quantum hydrodynamics to Koopman wavefunctions II

Based on the Koopman-van Hove (KvH) formulation of classical mechanics introduced in Part I, we formulate a Hamiltonian model for hybrid quantum-classical systems. This is obtained by writing the KvH wave equation for two classical particles and applying canonical quantization to one of them. We illustrate several geometric properties of the model regarding the associated quantum, classical, and hybrid densities. After presenting the quantum-classical Madelung transform, the joint quantum-classical distribution is shown to arise as a momentum map for a unitary action naturally induced from the van Hove representation on the hybrid Hilbert space. While the quantum density matrix is positive by construction, no such result is currently available for the classical density. However, here we present a class of hybrid Hamiltonians whose flow preserves the sign of the classical density. Finally, we provide a simple closure model based on momentum map structures.

math-ph↗

arXiv2021

Multisymplectic variational integrators for barotropic and incompressible fluid models with constraints

We present a structure preserving discretization of the fundamental spacetime geometric structures of fluid mechanics in the Lagrangian description in 2D and 3D. Based on this, multisymplectic variational integrators are developed for barotropic and incompressible fluid models, which satisfy a discrete version of Noether theorem. We show how the geometric integrator can handle regular fluid motion in vacuum with free boundaries and constraints such as the impact against an obstacle of a fluid flowing on a surface. Our approach is applicable to a wide range of models including the Boussinesq and shallow water models, by appropriate choice of the Lagrangian.

math.NA↗

arXiv2021

Thermodynamically consistent semi-compressible fluids: a variational perspective

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing \textit{low-Mach} dynamics, since they are \textit{soundproof}. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density $s$, specific entropy $η$, buoyancy $b$, temperature $T$, potential temperature $θ$ and a generic entropic variable $χ$; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating the elliptic equation for the pseudoincompressible equations with irreversible dynamics.

physics.flu-dyn↗

arXiv2020

A Finite Element Method for MHD that Preserves Energy, Cross-Helicity, Magnetic Helicity, Incompressibility, and $\operatorname{div} B = 0$

We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint $\operatorname{div} B = 0$ to machine precision, both at the spatially and temporally discrete levels.

math.NA↗

arXiv2020

Geometric variational approach to the dynamics of porous media filled with incompressible fluid

We derive the equations of motion for the dynamics of a porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an illustration of the method, the equations of motion for both the elastic matrix and the fluid are derived in the spatial (Eulerian) frame. Such an approach is of relevance e.g. for biological problems, such as sponges in water, where the elastic porous media is highly flexible and the motion of the fluid has a 'primary' role in the motion of the whole system. We then analyze the linearized equations of motion describing the propagation of waves through the media. In particular, we derive the propagation of S-waves and P-waves in an isotropic media. We also analyze the stability criteria for the wave equations and show that they are equivalent to the physicality conditions of the elastic matrix. Finally, we show that the celebrated Biot's equations for waves in porous media are obtained for certain values of parameters in our models.

physics.flu-dyn↗

arXiv2020

Vortex sheets in ideal 3D fluids, coadjoint orbits, and characters

We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of $\mathbb{R}^3$ associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces enclosing a given volume and endowed with a closed 1-form describing the vorticity density. If the vorticity density has discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle. We determine the Hamilton equations for such vortex sheets relative to the Hamiltonian function suggested in Khesin (2012) and prove that there are no stationary solutions having rotational symmetries. These coadjoint orbits are shown to be prequantizable if the period group of the 1-form and the volume enclosed by the surface satisfy an Onsager-Feynman relation, as argued in Goldin et al. (1991) for the case of open vortex sheets (tubes/ribbons). We find a character for the prequantizable coadjoint orbits, as well as a polarization group on which the character extends, which is a first step beyond prequantization.

math.SG↗

arXiv2020

Madelung transform and probability densities in hybrid classical-quantum dynamics

This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint classical-quantum density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

math-ph↗

arXiv2019

A Conservative Finite Element Method for the Incompressible Euler Equations with Variable Density

We construct a finite element discretization and time-stepping scheme for the incompressible Euler equations with variable density that exactly preserves total mass, total squared density, total energy, and pointwise incompressibility. The method uses Raviart-Thomas or Brezzi-Douglas-Marini finite elements to approximate the velocity and discontinuous polynomials to approximate the density and pressure. To achieve exact preservation of the aforementioned conserved quantities, we exploit a seldom-used weak formulation of the momentum equation and a second-order time-stepping scheme that is similar, but not identical, to the midpoint rule. We also describe and prove stability of an upwinded version of the method. We present numerical examples that demonstrate the order of convergence of the method.

math.NA↗

arXiv2019

A Variational Finite Element Discretization of Compressible Flow

We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart-Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation that doesn't seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations.

math.NA↗

arXiv2019

Dirac structures in nonequilibrium thermodynamics for simple open systems

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.

math-ph↗

arXiv2019

Koopman wavefunctions and classical-quantum correlation dynamics

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect -- the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive-definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical-quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

math-ph↗

arXiv2019

Dirac structures and port-Lagrangian systems in thermodynamics

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-d'Alembert 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-d'Alembert-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.

math-ph↗

arXiv2019

From variational to bracket formulations in nonequilibrium thermodynamics of simple systems

A variational formulation for nonequilibrium thermodynamics was recently proposed in \cite{GBYo2017a,GBYo2017b} 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 \cite{ElGB2019} and applied to multicomponent fluids.

math-ph↗

arXiv2019

From Lagrangian mechanics to nonequilibrium thermodynamics: a variational perspective

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite dimensional case of discrete systems as well as for the infinite dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton's principle, we show, with the help of thermodynamic systems with gradually increasing level complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat and mass transfer, both in the adiabatically closed and in the open cases. On the continuum side, we illustrate our theory with the example of multicomponent Navier-Stokes-Fourier systems.

math-ph↗

arXiv2019

Variational integrator for the rotating shallow-water equations on the sphere

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincaré reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincaré equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and the excellent conservational properties of the discrete variational integrator.

math.NA↗

arXiv2019

Variational integrators for anelastic and pseudo-incompressible flows

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincaré variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighed velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases.

math.NA↗

arXiv2018

Towards a geometric variational discretization of compressible fluids: the rotating shallow water equations

This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

math.NA↗