# Dr Cesare Tronci

## About

### Biography

I obtained a Laurea in Nuclear Engineering in May 2004 at the Politecnico di Torino (Italy). Then, after spending two years (06/2003 - 05/2005) at CERN (Switzerland) working on microwave electronics under the direction of Ugo Amaldi (also at TU München and TERA Foundation), I moved to the Theoretical Division (in the former Plasma Theory Group) of the Los Alamos National Lab (LANL, USA), where I visited Giovanni Lapenta (now at KU Leuven, Belgium) for several months.

In 10/2005, I entered a PhD programme in Applied Mathematics at Imperial College, under the direction of Darryl Holm. Between 2006 and 2007, I also spent two summers at LANL, working with Bruce Carlsten in the former International, Space & Response Division (High Power Electrodynamics Group). In 09/2008, I joined the Mathematics Section of EPF Lausanne (Switzerland) as a research assistant in Tudor Ratiu's group (Geometric Analysis). In 01/2012, I was appointed as a Lecturer (Assistant Professor) at Surrey, where I became Reader (Associate Professor) in 08/2018. In 06/2020, I was also appointed Adjunct Professor at Tulane University (USA) within the Department of Physics & Engineering Physics.

Over the last 8 years, I was an invited participant fellow at several research programs at the Fields Institute (Toronto, CA), the Newton Institute (Cambridge, UK), the Bernoulli Center (Lausanne, CH), and the MSRI (Berkeley, USA). In 2019, I was awarded an Alexander von Humboldt fellowship and spent a year working at the Max Planck Institute for Plasma Physics in Garching bei München (Germany).

## Research

### Research interests

Over the years, I have worked with engineers, physicists and mathematicians and shared with them the excitement of doing research in each of the corresponding fields. I have developed an understanding of the needs and the challenges in these disciplines and this experience is for me a continuous source of inspiration. This diversified path has led me to the discovery of mathematical research and its interconnections with other pure and applied sciences. Then, applied mathematics emerges not only as a discipline that applies mathematical concepts to other fields, but also as an area in which other fields of science serve as a continuous inspiration to develop new exciting directions in mathematical research - this two-way vision is precisely where my research stands. Over the last ten years, I have been working on geometric approaches to the formulation of nonlinear multi-physics models systems with multiple scales. Applications include complex fluids, plasma physics and quantum molecular dynamics.

*Specific interests include:* Momentum maps and reduction by symmetry; applications of symplectic and Poisson geometry; Hamiltonian and Lagrangian techniques in nonlinear dynamics; multi-physics modelling of nonlinear multiscale dynamics; nonlinear modelling of magnetized plasmas; liquid crystals and fluids with internal structure; nonlocal dynamics of aggregation and self-assembly; phase-space methods in classical and quantum mechanics; mixed classical-quantum dynamics and applications in chemistry.

### Research collaborations

*International collaboration network:*Within a 4-year Leverhulme Research Project, I am leading an international collaboration composed of Emanuele Tassi at the Centre of Theoretical Physics in Marseille (France), Giovanni Lapenta at the Centre for Plasma Astrophysics of KU Leuven (Belgium), and Philip J. Morrison at the Institute of Fusion Studies at UT Austin (USA). Within the same project, I also collaborate with Alain J. Brizard (St. Michael College of Vermont, USA), Enrico Camporeale (CWI Amsterdam, The Netherlands), and Maria Elena Innocenti (KU Leuven, Belgium).

*National collaboration network:*Within a London Mathematical Society Scheme 3 award, I am the coordinator of a national Network in Applied Geometric Mechanics involving three groups working on core topics in geometric mechanics: geometric quantum dynamics (Brunel University), geometric imaging science (Imperial College), and geometric fluid dynamics (University of Surrey).

*Other collaborations:*Over the last 10 years, I have worked with several (mainly international) collaborators on different mathematical problems, with special emphasis on applications. Recent collaborators include: Joshua W. Burby (Courant Institute - NYU, USA), Bin Cheng (U. Surrey, UK), François Gay-Balmaz (ENS-CNRS Paris, France), Darryl D. Holm (Imperial College, UK), Tomoki Ohsawa (UT Dallas, USA), Tudor S. Ratiu (EPF Lausanne, Switzerland), Vakhtang Putkaradze (U. Alberta, Canada), Endre Süli (Oxford, UK), Cornelia Vizman (U. West Timisoara, Romania).

### Research grants

Leverhulme Research Project Grant: In 2014, I was awarded the Leverhulme Research Grant RPG2014-112 “From geometry to kinetic-fluid systems (and back)”. Total budget: £252K. Duration: 4 years (started 7/1/15). This grant involves two 2-years research assistants and three international collaborations in Europe and USA.

Small awards: Over the last five years, I have been awarded a series of small grants (<£3K) by the London Mathematical Society, the Institute of Mathematics and Applications and the internal Faculty Research Support Fund at the University of Surrey.

### Selected invited talks

*Multi-physics models for hybrid kinetic-fluid and classical-quantum systems*, Hamiltonian Systems, from Topology to Applications through Analysis I, 8-12 October 2018, Mathematical Sciences Research Institute, Berkeley CA, USA

*Symmetry methods for quantum variational principles and expectation value dynamics*, PACM Colloquium, Princeton University, November 6 2017, Princeton NJ, USA

*Variational formulations of low frequency kinetic-MHD in the current-coupling scheme*, Mini-conference on "New Developments in Algorithms and Verification of Gyro Kinetic Simulations", 58th Annual Meeting of the APS Division of Plasma Physics, October 31 - November 4, 2016, San Jose CA, USA. (Solicited talk)

*Modeling efforts in hybrid kinetic-MHD and fully kinetic theories*, Princeton Plasma Physics Laboratory, October 27 2016, Princeton NJ, USA

*Multiphysics models for hybrid kinetic-fluid systems*, Courant Institute of Mathematical Sciences, New York University, April 7 2016, New York NY, USA

*Classical-quantum variational principles*, Classic and Stochastic Geometric Mechanics Workshop, June 8-11, 2015, École Polytechnique Fédérale de Lausanne, Switzerland

*Relabeling symmetry in fluid dynamics*, Geometry and Fluids, April 7-11 2014, Clay Mathematics Institute, Oxford, UK

*Hydrodynamic vorticity and helicity of conservative liquid crystal flows*, Isaac Newton Institute for Mathematical Sciences, June 11 2103, Cambridge, UK

*Geometry and symmetry in multi-physics models for magnetized plasmas*, Fields Institute for Research in Mathematical Sciences, July 9 2012, Toronto, Canada

## Teaching

Teaching portfolio: *Linear Algebra & Vector Calculus (MAT1037), Geometric Mechanics* (MATM032), *Geometric Mechanics* (MAGIC080)

Summary: Since I joined the University of Surrey, I introduced a new 4th year module (thought 6 times) in Geometric Mechanics. The module introduces symmetry methods in mechanical problems for different applications, from rigid body dynamics to quantum mechanics. I introduced a new partial assessment method that requires students to give presentations to the rest of the class. Also, I have thought (5 times) an online mini-course (10 hours) in Geometric Mechanics (with more advanced content) to PhD students across the UK, within the Mathematics Access Grid group of universities. Since 2016, I have also been teaching a first-year module in Linear Algebra & Vector Calculus (including a small part on MATLAB).

## Publications

The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether's conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called `collective'. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ℏ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ℏ≠0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called 'Bohmions', which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles. Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.

We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian and symplecticgeometric point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics.

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems. Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.

In plasma physics, a hybrid fluid-kinetic model is composed of a magnetohydrodynamics (MHD) part that describes a bulk fluid component and a Vlasov kinetic theory part that describes an energetic plasma component. While most hybrid models in the plasma literature are non-Hamiltonian, this paper investigates a recent Hamiltonian variant in its two-dimensional configuration. The corresponding Hamiltonian structure is described along with its Casimir invariants. Then, the energy-Casimir method is used to derive explicit sufficient stability conditions, which imply a stable spectrum and suggest nonlinear stability.

We present explicit expressions of the helicity conservation in nematic liquid-crystal flows, for both the Ericksen–Leslie and Landau–de Gennes theories. This is done by using a minimal coupling argument that leads to an Euler-like equation for a modified vorticity involving both velocity and structure fields (e.g. director and alignment tensor). This equation for the modified vorticity shares many relevant properties with ideal fluid dynamics, and it allows for vortex-filament configurations, as well as point vortices, in two dimensions. We extend all these results to particles of arbitrary shape by considering systems with fully broken rotational symmetry.

After summarizing the variational approach to splitting mean flow and fluctuation kinetics in the standard Vlasov theory, the same method is applied to the drift-kinetic equation from Littlejohn’s theory of guiding-center motion. This process sheds a new light on driftordered fluid (drift-fluid) models, whose anisotropic pressure tensor is then considered in detail. In addition, current drift-fluid models are completed by the insertion of magnetization terms ensuring momentum conservation. Magnetization currents are also shown to lead to challenging aspects when drift-fluid models are coupled to Maxwell’s equations for the evolution of the electromagnetic field. In order to overcome these difficulties, a simplified guiding-center theory is proposed along with its possible applications to hybrid kinetic-fluid models.

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest’s theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the Ehrenfest group. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated to another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models previously appeared in the chemical physics literature.

While the treatment of conical intersections in molecular dynamics generally requires nonadiabatic approaches, the Born-Oppenheimer adiabatic approximation is still adopted as a valid alternative in certain circumstances. In the context of Mead-Truhlar minimal coupling, this paper presents a new closure of the nuclear Born-Oppenheimer equation, thereby leading to a molecular dynamics scheme capturing geometric phase effects. Specifically, a semiclassical closure of the nuclear Ehrenfest dynamics is obtained through a convenient prescription for the nuclear Bohmian trajectories. The conical intersections are suitably regularized in the resulting nuclear particle motion and the associated Lorentz force involves a smoothened Berry curvature identifying a loop-dependent geometric phase. In turn, this geometric phase rapidly reaches the usual topological index as the loop expands away from the original singularity. This feature reproduces the phenomenology appearing in recent exact nonadiabatic studies, as shown explicitly in the Jahn-Teller problem for linear vibronic coupling. Likewise, a newly proposed regularization of the diagonal correction term is also shown to reproduce quite faithfully the energy surface presented in recent nonadiabatic studies.

The Lagrange, Euler, and Euler-Poincar____'{e} variational principles for the guiding-center Vlasov-Maxwell equations are presented. Each variational principle presents a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived by Noether method, where the guiding-center stress tensor is now shown to be explicitly symmetric.

A new symplectic variational approach is developed for modeling dissipation in kinetic equations. This approach yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations. The Vlasov example admits measure-valued single-particle solutions. Such solutions are reversible; and the total entropy is a Casimir, and thus is preserved.

The Vlasov equation for the collisionless evolution of the single-particle probability distribution function (PDF) is a well-known example of coadjoint motion. Remarkably, the property of coadjoint motion survives the process of taking moments. That is, the evolution of the moments of the Vlasov PDF is also coadjoint motion. We find that {____it geodesic} coadjoint motion of the Vlasov moments with respect to powers of the single-particle momentum admits singular (weak) solutions concentrated on embedded subspaces of physical space. The motion and interactions of these embedded subspaces are governed by canonical Hamiltonian equations for their geodesic evolution.

We formulate Euler–Poincaré and Lagrange–Poincaré equations for systems with broken symmetry. We specialize the general theory to present explicit equations of motion for nematic systems, ranging from single nematic molecules to biaxial liquid crystals. The geometric construction applies to order parameter spaces consisting of either unsigned unit vectors (directors) or symmetric matrices (alignment tensors). On the Hamiltonian side, we provide the corresponding Poisson brackets in both Lie–Poisson and Hamilton–Poincaré formulations. The explicit form of the helicity invariant for uniaxial nematics is also presented, together with a whole class of invariant quantities (Casimirs) for two-dimensional incompressible flows.

While electron kinetic effects are well known to be of fundamental importance in several situations, the electron mean-flow inertia is often neglected when lengthscales below the electron skin depth become irrelevant. This has led to the formulation of different reduced models, where electron inertia terms are discarded while retaining some or all kinetic effects. Upon considering general full-orbit particle trajectories, this paper compares the dispersion relations emerging from such models in the case of the Weibel instability. As a result, the question of how lengthscales below the electron skin depth can be neglected in a kinetic treatment emerges as an unsolved problem, since all current theories suffer from drawbacks of different nature. Alternatively, we discuss fully kinetic theories that remove all these drawbacks by restricting to frequencies well below the plasma frequency of both ions and electrons. By giving up on the lengthscale restrictions appearing in previous works, these models are obtained by assuming quasi-neutrality in the full Maxwell-Vlasov system.

We formulate Euler-Poincar____'e equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing ____delta-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group of volume-preserving automorphisms of a principal bundle. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.

We propose an innovative linear accelerating structure, particularly suited for hadrontherapy applications. Its two main features are compactness and good power efficiency at low beam velocities: the first is achieved through a high working frequency and a consequent high accelerating gradient, the second is obtained by coupling several H-mode cavities together. The structure is called CLUSTER, which stands for "Coupled-cavity Linac USing Transverse Electric Radial field". In order to compare the performance of this structure with other hadrontherapy linac designs involving high frequencies, a conceptual study has been performed for an operating frequency of 3 GHz. Moreover a proof of principle has been obtained through RF measurements on a prototype operating at 1 GHz. An accelerator complex using a CLUSTER linac is also considered for protontherapy purposes. The whole complex is called cyclinac and is composed of a commercial cyclotron injecting the beam in a high-frequency linac.

We investigate the emergence of singular solutions in a nonlocal model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of nonlocality on the system’s stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.

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.

The variational formulations of guiding-center Vlasov-Maxwell theory based on Lagrange, Euler, and Euler-Poincaré variational principles are presented. Each variational principle yields a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived by Noether method, where the guiding-center stress tensor is now shown to be explicitly symmetric.

Upon combining Northrop’s picture of charged particle motion with modern liquid crystal theories, this paper provides a new description of guiding center dynamics (to lowest order). This new perspective is based on a rotation gauge field (gyrogauge) that encodes rotations around the magnetic field. In liquid crystal theory, an analogue rotation field is used to encode the rotational state of rod-like molecules. Instead of resorting to sophisticated tools (e.g. Hamiltonian perturbation theory and Lie series expansions) that still remain essential in higher-order gyrokinetics, the present approach combines the WKB method with a simple kinematical ansatz, which is then replaced into the charged particle Lagrangian. The latter is eventually averaged over the gyrophase to produce Littlejohn’s guiding-center equations. A crucial role is played by the vector potential for the gyrogauge field. A similar vector potential is related to liquid crystal defects and is known as wryness tensor in Eringen’s micropolar theory.

The IDRA project (fig.l) - The Institute for Diagnostic and RAdiotherapy - consists of a 30 MeV commercial cyclotron for the production of isotopes for PET, SPECT and other radionuclides for diagnostic and therapy combined with a proton linac for the treatment of deep seated tumours. A 3 GHz Side Coupled Linac named LIBO (Linac BOoster) post-accelerates the 30 MeV protons from the cyclotron up to 210 MeV in a length of 16.6 metres. The combination of a cyclotron and a linac has been dubbed "cyclinac".

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- Poincaré 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.

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the `Ehrenfest group'. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated to another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models previously appeared in the chemical physics literature.

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered in arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy’s Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.

The Multi-Level Multi-Domain (MLMD) method is a semi-implicit adaptive method for Particle-In-Cell plasma simulations. It has been demonstrated in the past in simulations of Maxwellian plasmas, electrostatic and electromagnetic instabilities, plasma expansion in vacuum, magnetic reconnection [1, 2, 3]. In multiple occasions, it has been commented on the coupling between the coarse and the refined grid solutions. The coupling mechanism itself, however, has never been explored in depth. Here, we investigate the theoretical bases of grid coupling in the MLMD system. We obtain an evolution law for the electric field solution in the overlap area of the MLMD system which highlights a dependance on the densities and currents from both the coarse and the refined grid, rather than from the coarse grid alone: grid coupling is obtained via densities and currents.

Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov-Maxwell kinetic theory. The celebrated Koopman-von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is noncanonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant of the Koopman-von Neumann construction that carries a different phase dynamics. Going back to van Hove's prequantum theory, the corresponding Koopman-van Hove equation provides an alternative Clebsch representation which is then coupled to the electromagnetic fields. Finally, the role of gauge transformations in the new context is discussed in detail.

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.

We derive equations of motion for the dynamics of anisotropic particles directly from the dissipative Vlasov kinetic equations, with the dissipation given by the double bracket approach (Double Bracket Vlasov, or DBV). The moments of the DBV equation lead to a nonlocal form of Darcy's law for the mass density. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the DBV form. The moment dynamics for these double bracket kinetic equations is expressed as Lie-Darcy continuum equations for densities of mass and orientation. We also show how to obtain a Smoluchowski model from a cold plasma-like moment closure of DBV. Thus, the double bracket kinetic framework serves as a unifying method for deriving different types of dynamics, from density--orientation to Smoluchowski equations. Extensions for more general physical systems are also discussed.

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs associated to left and right group actions. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to different realizations of the density operator, which are of interest in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmann’s density matrix and Koopman wavefunctions are shown to be special cases of this construction.

Three different hybrid Vlasov-fluid systems are derived by applying reduction by symmetry to Hamilton's variational principle. In particular, the discussion focuses on the Euler- Poincaré formulation of three major hybridMHD models, which are compared in the same framework. These are the current-coupling scheme and two different variants of the pressure-coupling scheme. The Kelvin-Noether theorem is presented explicitly for each scheme, together with the Poincaré invariants for its hot particle trajectories. Extensions of Ertel's relation for the potential vorticity and for its gradient are also found in each case, as well as new expressions of cross helicity invariants.

This paper exploits the theory of geometric gradient ﬂows to introduce an alternative regularization of the thin-ﬁlm equation valid in the case of large-scale droplet spreading—the geometric diffuse-interface method. The method possesses some advantages when compared with the existing models of droplet spreading, namely the slip model, the precursor ﬁlm method and the diffuse-interface model. These advantages are discussed and a case is made for using the geometric diffuse-interface method for the purpose of numerical simulations. The mathematical solutions of the geometric diffuse interface method are explored via such numerical simulations for the simple and well-studied case of large-scale droplet spreading for a perfectly wetting ﬂuid— we demonstrate that the new method reproduces Tanner’s Law of droplet spreading via a simple and robust computational method, at a low computational cost. We discuss potential avenues for extending the method beyond the simple case of perfectly wetting ﬂuids.

Multipolar order in complex fluids is described by statistical correlations. This paper presents a novel dynamical approach, which accounts for microscopic effects on the order parameter space. Indeed, the order parameter field is replaced by a statistical distribution function that is carried by the fluid flow. Inspired by Doi's model of colloidal suspensions, the present theory is derived from a hybrid moment closure for Yang-Mills Vlasov plasmas. This hybrid formulation is constructed under the assumption that inertial effects dominate over dissipative phenomena, so that the total energy is conserved. After presenting the basic geometric properties of the theory, the effect of Yang-Mills fields is considered and a direct application is presented to magnetized fluids with quadrupolar order (spin nematic phases). Hybrid models are also formulated for complex fluids with symmetry breaking. For the special case of liquid crystals, the moment method can be applied to the hybrid formulation to study to the dynamics of cubatic phases.

Energetic particle effects in magnetic confinement fusion dev ices are commonly studied by hybrid kinetic-fluid simulation codes whose unde rlying continuum evolu- tion equations often lack the correct energy balance. While two different kinetic-fluid coupling options are available (current-coupling and pres sure-coupling), this paper applies the Euler-Poincar ́e variational approach to formu late a new conservative hybrid model in the pressure-coupling scheme. In our case th e kinetics of the en- ergetic particles are described by guiding center theory. T he interplay between the Lagrangian fluid paths with phase space particle trajectori es reflects an intricate variational structure which can be approached by letting th e 4-dimensional guiding center trajectories evolve in the full 6-dimensional phase space. Then, the redundant perpendicular velocity is integrated out to recover a four- dimensional description. A second equivalent variational approach is also reported, which involves the use of phase space Lagrangians. Not only do these variational st ructures confer on the new model a correct energy balance, but also they produce a cr oss-helicity invariant which is lost in the other pressure-coupling schemes report ed in the literature.

A new fully kinetic system is proposed for modeling collisionless magnetic reconnection. The formulation relies on fundamental principles in Lagrangian dynamics, in which the inertia of the electron mean flow is neglected in the expression of the Lagrangian, rather then enforcing a zero electron mass in the equations of motion. This is done upon splitting the electron velocity into its mean and fluctuating parts, so that the latter naturally produce the corresponding pressure tensor. The model exhibits a new Coriolis force term, which emerges from a change of frame in the electron dynamics. Then, if the electron heat flux is neglected, the strong electron magnetization limit yields a hybrid model, in which the electron pressure tensor is frozen into the electron mean velocity.

Hybrid kinetic-MHD models describe the interaction of an MHD bulk uid with an ensemble of hot particles, which obeys a kinetic equation. In this work we apply Hamilton's variational principle to formulate new current-coupling kinetic-MHD models in the low-frequency approximation (i.e. large Larmor frequency limit). More particularly, we formulate current-coupling schemes, in which energetic particle dynamics are expressed in either guiding center or gyrocenter coordinates. When guiding center theory is used to model the hot particles, we show how energy conservation requires corrections to the standard magnetization term. On the other hand, charge and momentum conservation in gyrokinetic-MHD lead to extra terms in the usual de nition of the hot current density as well modi cations to conventional gyrocenter dynamics. All these new features arise naturally from the underlying variational structure of the proposed models.

We prove the global-in-time existence of large-data nite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rare ed particles of one species; the incompressible Navier{Stokes system for the bulk uid; and a parabolic evolution equation, involving magnetic di usivity, for the magnetic eld. The physical derivation of our model is given. It is also shown that the weak solution, whose existence is established, has nonincreasing total energy, and that it satis es a number of physically relevant properties, including conservation of the total momentum, conservation of the total mass, and nonnegativity of the probability density function for the energetic particles. The proof is based on a one-level approximation scheme, which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions, in conjunction with a weak compactness argument for the sequence of approximating solutions. The key technical challenges in the analysis of the mathematical model are the nondissipative nature of the Vlasov-type particle equation and passage to the weak limits in the multilinear coupling terms.

There are two competing descriptions of nematic liquid crystal dynamics: the Ericksen-Leslie director theory and the Eringen micropolar approach. Up to this day, these two descriptions have remained distinct in spite of several attempts to show that the micropolar theory comprises the director theory. In this paper we show that this is the case by using symmetry reduction techniques and introducing a new system that is equivalent to the Ericksen-Leslie equations and includes disclination dynamics. The resulting equations of motion are verified to be completely equivalent, although one of the two different reductions offers the possibility of accounting for orientational defects. After applying these two approaches to the ordered micropolar theory of Lhuiller and Rey, all the results are eventually extended to flowing complex fluids, such as nematic liquid crystals.

The Vlasov equation for the collisionless evolution of the single-particle probability distribution function (PDF) is a well-known Lie-Poisson Hamiltonian system. Remarkably, the operation of taking the moments of the Vlasov PDF preserves the Lie-Poisson structure. The individual particle motions correspond to singular solutions of the Vlasov equation. The paper focuses on singular solutions of the problem of geodesic motion of the Vlasov moments. These singular solutions recover geodesic motion of the individual particles.