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Extended irreversible thermodynamics is a theory that extends the classical framework of nonequilibrium thermodynamics by going beyond the local-equilibrium assumption. A notable example of this is the Maxwell-Cattaneo heat flux model, which introduces a time lag in the heat flux response to temperature gradients. In this paper, we develop a variational formulation of the equations of extended irreversible thermodynamics by introducing an action principle for a nonequilibrium Lagrangian that treats thermodynamic fluxes as independent variables. A key feature of this approach is that it naturally extends both Hamilton's principle of reversible continuum mechanics and the earlier variational formulation of classical irreversible thermodynamics. The variational principle is initially formulated in the material (Lagrangian) description, from which the Eulerian form is derived using material covariance (or relabeling symmetries). The tensorial structure of the thermodynamic fluxes dictates the choice of objective rate in the Eulerian description, leading to the introduction of nonequilibrium stresses arising from both viscous and thermal effects to ensure thermodynamic consistency. This framework naturally results in the Cattaneo-Christov model for heat flux. We also investigate the extension of the approach to accommodate higher-order fluxes and the general form of entropy fluxes. The variational framework presented in this paper has promising applications in the development of structure-preserving and thermodynamically consistent numerical methods. It is particularly relevant for modeling systems where entropy production is a delicate issue that requires careful treatment to ensure consistency with the laws of thermodynamics.

The computational challenges posed by many-particle quantum systems are often overcome by mixed quantum-classical (MQC) models in which certain degrees of freedom are treated as classical while others are retained as quantum. One of the fundamental questions raised by this hybrid picture involves the characterization of the information associated to MQC systems. Based on the theory of dynamical invariants in Hamiltonian systems, here we propose a family of hybrid entropy functionals that consistently specialize to the usual R\'enyi and Shannon entropies. Upon considering the MQC Ehrenfest model for the dynamics of quantum and classical probabilities, we apply the hybrid Shannon entropy to characterize equilibrium configurations for simple Hamiltonians. The present construction also applies beyond Ehrenfest dynamics., Comment: First version. For submission to Lecture Notes in Comput. Sci

A new geometric approach to systems with boundary energy flow is developed using infinite-dimensional Dirac structures within the Lagrangian formalism. This framework satisfies a list of consistency criteria with the geometric setting of finite-dimensional mechanics. In particular, the infinite-dimensional Dirac structure can be constructed from the canonical symplectic form on the system's phase space; the system's evolution equations can be derived equivalently from either a variational perspective or a Dirac structure perspective; the variational principle employed is a direct extension of Hamilton's principle in classical mechanics; and the approach allows for a process of system interconnection within its formulation. This is achieved by developing an appropriate infinite dimensional version of the previously developed Lagrange-Dirac systems. A key step in this construction is the careful choice of a suitable dual space to the configuration space, specifically, a subspace of the topological dual that captures the system's behavior in both the interior and the boundary, while allowing for a natural extension of the canonical geometric structures of mechanics. This paper focuses on systems where the configuration space consists of differential forms on a smooth manifold with a boundary. To illustrate our theory, several examples, including nonlinear wave equations, the telegraph equation, and the Maxwell equations are presented.

We derive variational integrators for stochastic Hamiltonian systems on Lie groups using a discrete version of the stochastic Hamiltonian phase space principle. The structure-preserving properties of the resulting scheme, such as symplecticity, preservation of the Lie-Poisson structure, preservation of the coadjoint orbits, and conservation of Casimir functions, are discussed, along with a discrete Noether theorem for subgroup symmetries. We also consider in detail the case of stochastic Hamiltonian systems with advected quantities, studying the associated structure-preserving properties in relation to semidirect product Lie groups. A full convergence proof for the scheme is provided for the case of the Lie group of rotations. Several numerical examples are presented, including simulations of the free rigid body and the heavy top.

We develop a Lie group geometric framework for the motion of fluids with permeable boundaries that extends Arnold's geometric description of fluid in closed domains. Our setting is based on the classical Hamilton principle applied to fluid trajectories, appropriately amended to incorporate bulk and boundary forces via a Lagrange-d'Alembert approach, and to take into account only the fluid particles present in the fluid domain at a given time. By applying a reduction by relabelling symmetries, we deduce a variational formulation in the Eulerian description that extends the Euler-Poincar\'e framework to open fluids. On the Hamiltonian side, our approach yields a bracket formulation that consistently extends the Lie-Poisson bracket of fluid dynamics and contains as a particular case a bulk+boundary bracket formulation proposed earlier. We illustrate the geometric framework with several examples, and also consider the extension of this setting to include multicomponent fluids, the consideration of general advected quantities, the analysis of higher-order fluids, and the incorporation of boundary stresses. Open fluids are found in a wide range of physical systems, including geophysical fluid dynamics, porous media, incompressible flow and magnetohydrodynamics. This new formulation permits a description based on geometric mechanics that can be applied to a broad class of models.

To accurately compute data-based prediction of Hamiltonian systems, especially the long-term evolution of such systems, it is essential to utilize methods that preserve the structure of the equations over time. We consider a case that is particularly challenging for data-based methods: systems with interacting parts that do not reduce to pure momentum evolution. Such systems are essential in scientific computations. For example, any discretization of a continuum elastic rod can be viewed as interacting elements that can move and rotate in space, with each discrete element moving on the group of rotations and translations $SE(3)$. We develop a novel method of data-based computation and complete phase space learning of such systems. We follow the original framework of \emph{SympNets} (Jin et al, 2020) building the neural network from canonical phase space mappings, and transformations that preserve the Lie-Poisson structure (\emph{LPNets}) as in (Eldred et al, 2024). We derive a novel system of mappings that are built into neural networks for coupled systems. We call such networks Coupled Lie-Poisson Neural Networks, or \emph{CLPNets}. We consider increasingly complex examples for the applications of CLPNets: rotation of two rigid bodies about a common axis, the free rotation of two rigid bodies, and finally the evolution of two connected and interacting $SE(3)$ components. Our method preserves all Casimir invariants of each system to machine precision, irrespective of the quality of the training data, and preserves energy to high accuracy. Our method also shows good resistance to the curse of dimensionality, requiring only a few thousand data points for all cases studied, with the effective dimension varying from three to eighteen. Additionally, the method is highly economical in memory requirements, requiring only about 200 parameters for the most complex case considered., Comment: 52 pages, 9 figures

The discretization of Cartan's exterior calculus of differential forms has been fruitful in a variety of theoretical and practical endeavors: from computational electromagnetics to the development of Finite-Element Exterior Calculus, the development of structure-preserving numerical tools satisfying exact discrete equivalents to Stokes' theorem or the de Rham complex for the exterior derivative have found numerous applications in computational physics. However, there has been a dearth of effort in establishing a more general discrete calculus, this time for differential forms with values in vector bundles over a combinatorial manifold equipped with a connection. In this work, we propose a discretization of the exterior covariant derivative of bundle-valued differential forms. We demonstrate that our discrete operator mimics its continuous counterpart, satisfies the Bianchi identities on simplicial cells, and contrary to previous attempts at its discretization, ensures numerical convergence to its exact evaluation with mesh refinement under mild assumptions., Comment: 58 pages, 20 figures, Fix erroneous line break

The restricted Siegel disc is a homogeneous space related to the connected component $T_0(1)$ of the Universal Teichm\"uller space via the period mapping. In this paper we show that it is a coadjoint orbit of the universal central extension of the restricted symplectic group or, equivalently, an affine coadjoint orbit of the restricted symplectic group with cocycle given by the Schwinger term., Comment: 16 pages

In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg's principle, we overcame these difficulties by blending Koopman's classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz which introduces the trajectories of computational particles - the koopmons - sampling the Lagrangian classical paths in phase space. In the case of Tully's nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results., Comment: Second version. 40 pages, 15 figures

An accurate data-based prediction of the long-term evolution of Hamiltonian systems requires a network that preserves the appropriate structure under each time step. Every Hamiltonian system contains two essential ingredients: the Poisson bracket and the Hamiltonian. Hamiltonian systems with symmetries, whose paradigm examples are the Lie-Poisson systems, have been shown to describe a broad category of physical phenomena, from satellite motion to underwater vehicles, fluids, geophysical applications, complex fluids, and plasma physics. The Poisson bracket in these systems comes from the symmetries, while the Hamiltonian comes from the underlying physics. We view the symmetry of the system as primary, hence the Lie-Poisson bracket is known exactly, whereas the Hamiltonian is regarded as coming from physics and is considered not known, or known approximately. Using this approach, we develop a network based on transformations that exactly preserve the Poisson bracket and the special functions of the Lie-Poisson systems (Casimirs) to machine precision. We present two flavors of such systems: one, where the parameters of transformations are computed from data using a dense neural network (LPNets), and another, where the composition of transformations is used as building blocks (G-LPNets). We also show how to adapt these methods to a larger class of Poisson brackets. We apply the resulting methods to several examples, such as rigid body (satellite) motion, underwater vehicles, a particle in a magnetic field, and others. The methods developed in this paper are important for the construction of accurate data-based methods for simulating the long-term dynamics of physical systems., Comment: 57 pages, 13 figures

The paper investigates a systematic approach to modeling in nonequilibrium thermodynamics by focusing upon the notion of interconnections, where we propose a novel Lagrangian variational formulation of such interconnected systems by extending the variational principle of Hamilton in mechanics. In particular, we show how a nonequilibrium thermodynamic system can be regarded as an interconnected system of primitive physical elements or subsystems throughout an interconnection. While this approach is new in nonequilibrium thermodynamics, this idea has been known as a useful tool for the modeling of complicated systems in networks as well as in mechanics. Hence, the setting developed in this paper yields a promising direction for building a unifying description in various areas of modern science via thermodynamic principles, while being at the same time related to the early developments of variational mechanics.

If a porous media is being damaged by excessive stress, the elastic matrix at every infinitesimal volume separates into a 'solid' and a 'broken' component. The 'solid' part is the one that is capable of transferring stress, whereas the 'broken' part is advecting passively and is not able to transfer the stress. In previous works, damage mechanics was addressed by introducing the damage parameter affecting the elastic properties of the material. In this work, we take a more microscopic point of view, by considering the transition from the 'solid' part, which can transfer mechanical stress, to the 'broken' part, which consists of microscopic solid particles and does not transfer mechanical stress. Based on this approach, we develop a thermodynamically consistent dynamical theory for porous media including the transfer between the 'broken' and 'solid' components, by using a variational principle recently proposed in thermodynamics. This setting allows us to derive an explicit formula for the breaking rate, i.e., the transition from the 'solid' to the 'broken' phase, dependent on the Gibbs' free energy of each phase. Using that expression, we derive a reduced variational model for material breaking under one-dimensional deformations. We show that the material is destroyed in finite time, and that the number of 'solid' strands vanishing at the singularity follows a power law. We also discuss connections with existing experiments on material breaking and extensions to multi-phase porous media.

Mixed quantum-classical models have been proposed in several contexts to overcome the computational challenges of fully quantum approaches. However, current models typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg's uncertainty principle. Here, we present a fully Hamiltonian theory of quantum-classical dynamics that appears to be the first to ensure a series of consistency properties, beyond positivity of quantum and classical densities. Based on Lagrangian phase-space paths, the model possesses a quantum-classical Poincar\'e integral invariant as well as infinite classes of Casimir functionals. We also exploit Lagrangian trajectories to formulate a finite-dimensional closure scheme for numerical implementations., Comment: 10 pages, no figure. To appear in Lecture Notes in Comput. Sci

Respecting the laws of thermodynamics is crucial for ensuring that numerical simulations of dynamical systems deliver physically relevant results. In this paper, we construct a structure-preserving and thermodynamically consistent finite element method and time-stepping scheme for heat conducting viscous fluids, with general state equations. The method is deduced by discretizing a variational formulation for nonequilibrium thermodynamics that extends Hamilton's principle for fluids to systems with irreversible processes. The resulting scheme preserves the balance of energy and mass to machine precision, as well as the second law of thermodynamics, both at the spatially and temporally discrete levels. The method is shown to apply both with insulated and prescribed heat flux boundary conditions, as well as with prescribed temperature boundary conditions. We illustrate the properties of the scheme with the Rayleigh-B\'enard thermal convection. While the focus is on heat conducting viscous fluids, the proposed discrete variational framework paves the way to a systematic construction of thermodynamically consistent discretizations of continuum systems., Comment: 37 pages, 5 figures, 2 tables

Mixed quantum-classical spin systems have been proposed in spin chain theory and, more recently, in magnon spintronics. However, current models of quantum-classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg's uncertainty principle. Here, we present a fully Hamiltonian theory of quantum-classical spin dynamics that appears to be the first to ensure an entire series of consistency properties, including positivity of both the classical and the quantum density, so that Heisenberg's principle is satisfied at all times. We show how this theory may connect to recent energy-balance considerations in measurement theory and we present its Poisson bracket structure explicitly. After focusing on the simpler case of a classical Bloch vector interacting with a quantum spin observable, we illustrate the extension of the model to systems with several spins, and restore the presence of orbital degrees of freedom., Comment: Submitted as a contribution to the Special Collection "Koopman methods in classical and quantum-classical mechanics" in the Journal of Physics A. Revised version

It has been a long standing question how to extend the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang-Mills theory.

Motivated by compartmental analysis in engineering and biophysical systems, we present a variational framework for the nonequilibrium thermodynamics of systems involving both distributed and discrete (finite dimensional) subsystems by specifically using the ideas of interconnected systems. We focus on the process of non-isothermal diffusion and show how the resulting form of the entropy equation naturally yields phenomenological expressions for the diffusion and entropy fluxes between two compartments, which results in generalized forms of Robin type interface conditions.

In this paper, we develop a Hamiltonian variational formulation for the nonequilibrium thermodynamics of simple adiabatically closed systems that is an extension of Hamilton's phase space principle in mechanics. We introduce the Hamilton-d'Alembert principle for thermodynamic systems by considering nonlinear nonholonomic constraints of thermodynamic type. In particular, for the case in which the given Lagrangian is degenerate, we construct the Hamiltonian by incorporating the primary constraints via Dirac's theory of constraints. We illustrate our Hamiltonian variational formulation with some examples of systems with friction, with internal matter transfer as well as with chemical reactions.

We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of the Hamilton principle for the relativistic particle, we deduce two classes of reduced variational principles that are associated to either spacetime covariance, which is an axiom of the continuum theory, or material covariance, which is related to particular properties of the system such as isotropy. The covariance hypotheses and the Lagrangian reduction process are efficiently formulated by making explicit the dependence of the theory on given material and spacetime tensor fields that are transported by the world-tube of the continuum via the push-forward and pull-back operations. It is shown that the variational formulation, when augmented with the Gibbons-Hawking-York (GHY) boundary terms, also yields the Israel-Darmois junction conditions between the solution at the interior of the relativistic continua and the solution describing the gravity field produced outside from it. The expression of the first variation of the GHY term with respect to the hypersurface involves some extensions of previous results that we also derive in the paper. We consider in details the application of the variational framework to relativistic fluids and relativistic elasticity. For the latter case, our setting also allows to clarify the relation between formulations of relativistic elasticity based on the relativistic right Cauchy-Green tensor or on the relativistic Cauchy deformation tensor. The setting developed here will be further exploited for modelling purpose in subsequent parts of the paper., Comment: 56 pages, 2 diagrams, 2 tables

This paper presents the variational discretization of the compressible Navier-Stokes-Fourier system, in which the viscosity and the heat conduction terms are handled within the variational approach to nonequilibrium thermodynamics as developed by one of the authors. In a first part, we review the variational framework for the Navier-Stokes-Fourier (NSF) system in the smooth setting. In a second part, we review a discrete exterior calculus based on discrete diffeomorphisms then proceed to establish the spatially discretized variational principle for the NSF system through the use of this discrete exterior calculus, which yields a semi-discrete nonholonomic variational principle, as well as semi-discrete evolution equations. In order to avoid important technical difficulties, further treatment of the phenomenological constraint is needed. In a third part we discretize in time the spatial variational principle underlying the NSF system by extending previous work of the authors, which at last yields a nonholonomic variational integrator for the NSF system, as well as fully discrete evolution equations.

A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincar\'e integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics., Comment: New version. 35 pages in total

We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy-Green deformation tensors, which allows for the development of a general discrete geometric setting for frame indifferent isotropic hyperelastic models. The resulting discrete framework is in perfect adequacy with the multisymplectic discretization of fluids proposed earlier by the authors. Thanks to the unified discrete setting, a geometric variational discretization can be developed for the coupled dynamics of a fluid impacting and flowing on the surface of an hyperelastic body. The variational treatment allows for a natural inclusion of incompressibility and impenetrability constraints via appropriate penalty terms. We test the resulting integrators in 2D and 3D with the case of a barotropic fluid flowing on incompressible rubber-like nonlinear models., Comment: 37 pages, 11 figures

We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum-classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times, thereby addressing a series of stringent consistency requirements. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems., Comment: Second version. Largely revised. Comments welcome!

Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of non-moving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper we derive the equations of motion for the dynamics of a deformable porous media which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics. This theory extends the recently developed variational derivation of the mechanics of deformable porous media to include thermodynamic processes and can easily include incompressibility constraints. The model for the combined fluid-matrix system, written in the spatial frame, is developed by introducing mechanical and additional variables describing the thermal energy part of the system, writing the action principle for the system, and using a nonlinear, nonholonomic constraint on the system deduced from the second law of thermodynamics. The resulting equations give us the general version of possible friction forces incorporating thermodynamics, Darcy-like forces and friction forces similar to those used in the Navier-Stokes equations. The equations of motion are valid for arbitrary dependence of the kinetic and potential energies on the state variables. The results of our work are relevant for geophysical applications, industrial applications involving high pressures and temperatures, food processing industry, and other situations when both thermodynamics and mechanical considerations are important., Comment: arXiv admin note: text overlap with arXiv:2007.02605, arXiv:2107.03661

Many parts of biological organisms are comprised of deformable porous media. The biological media is both pliable enough to deform in response to an outside force and can deform by itself using the work of an embedded muscle. For example, the recent work (Ludeman et al., 2014) has demonstrated interesting 'sneezing' dynamics of a freshwater sponge, when the sponge contracts and expands to clear itself from surrounding polluted water. We derive the equations of motion for the dynamics of such an active porous media (i.e., a deformable porous media that is capable of applying a force to itself with internal muscles), filled with an incompressible fluid. These equations of motion extend the earlier derived equation for a passive 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. We then proceed to extend this theory by computing the case when both the active porous media and the fluid are incompressible, with the porous media still being deformable, which is often the case for biological applications. For the particular case of a uniform initial state, we rewrite the equations of motion in terms of two coupled telegraph-like equations for the material (Lagrangian) particles expressed in the Eulerian frame of reference, particularly suitable for numerical simulations, formulated for both the compressible media/incompressible fluid case and the doubly incompressible case. We derive interesting conservation laws for the motion, perform numerical simulations in both cases and show the possibility of self-propulsion of a biological organism due to particular running wave-like application of the muscle stress., Comment: 51 pages, 7 figures. arXiv admin note: text overlap with arXiv:2007.02605

Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics (MHD) both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total energy and magnetic helicity, and preserves mass and the constraint $ \operatorname{div}B = 0$ to machine precision, both at the spatially and temporally discrete levels. In particular, conservation of energy and magnetic helicity hold at the discrete levels in the ideal case. It is observed that cross helicity is well conserved in our simulation in the ideal case., Comment: 6 figures

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 $\delta-$like profiles of the Liouville density, thereby reproducing point particles in phase space., Comment: 8 pages, no figures. To appear in Lecture Notes in Comput. Sci

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., Comment: 8 pages, 1 figure. To appear in Lecture Notes in Comput. Sci

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.

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 $\eta$, buoyancy $b$, temperature $T$, potential temperature $\theta$ and a generic entropic variable $\chi$; 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., Comment: 40 pages, 1 figure

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.

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., Comment: 34 pages, 6 figures, to appear in Acta Mechanica

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.

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., Comment: 32 pages, 2 figures, 2 tables

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., Comment: 25 pages, 1 figure

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.

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 Poincar\'e 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., Comment: Fully revised. To appear in Nonlinearity

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.

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.

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., Comment: 7 figures

The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known variational formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian formulation that involves a Poisson bracket structure. These variational and bracket structures underlie many of the most basic principles that we know about geophysical fluid flows, such as conservation laws. However, real geophysical flows also include irreversible processes, such as viscous dissipation, heat conduction, diffusion and phase changes. Recent work has demonstrated that the variational formulation can be extended to include irreversible processes and non-equilibrium thermodynamics, through the new concept of thermodynamic displacement. By design, and in accordance with fundamental physical principles, the resulting equations automatically satisfy the first and second law of thermodynamics. Irreversible processes can also be incorporated into the bracket structure through the addition of a dissipation bracket. This gives what are known as the single and double generator bracket formulations, which are the natural generalizations of the Hamiltonian formulation to include irreversible dynamics. Here the variational formulation for irreversible processes is shown to underlie these bracket formulations for fully compressible, multicomponent, multiphase geophysical fluids with a single temperature and velocity. Many previous results in the literature are demonstrated to be special cases of this approach. Finally, some limitations of the current approach (especially with regards to precipitation and nonlocal processes such as convection) are discussed, and future directions of research to overcome them are outlined.

We propose a variational formulation for the nonequilibrium thermodynamics of discrete open systems, i.e., discrete systems which can exchange mass and heat with the exterior. Our approach is based on a general variational formulation for systems with time-dependent nonlinear nonholonomic constraints and time-dependent Lagrangian. For discrete open systems, the~time-dependent nonlinear constraint is associated with the rate of internal entropy production of the system. We show that this constraint on the solution curve systematically yields a constraint on the variations to be used in the action functional. The proposed variational formulation is intrinsic and provides the same structure for a wide class of discrete open systems. We illustrate our theory by presenting examples of open systems experiencing mechanical interactions, as well as internal diffusion, internal heat transfer, and their cross-effects. Our approach yields a systematic way to derive the complete evolution equations for the open systems, including the expression of the internal entropy production of the system, independently on its complexity. It might be especially useful for the study of the nonequilibrium thermodynamics of biophysical systems.

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler-Poincar\'e variations, the derivation of free surface variations, and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically., Comment: 19 pages, 3 figures