# Dr Cesare Tronci

Reader at University of Surrey, UK
Adjunct Professor at Tulane University, USA
+44 (0)1483 683356
16 AA 04

Department of Mathematics, Quantum Foundations Centre.

### 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

### Publications

Innocenti ME, Tronci C, Markidis S, Lapenta G (2016) Grid coupling mechanism in the semi-implicit adaptive Multi-Level Multi-Domain method, Journal of Physics: Conference Series 719 IOP Publishing
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.
Holm DD, Tronci C (2012) Communications in Mathematical Sciences 10 (1) pp. 191-222
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.
Gay-Balmaz F, Ratiu TS, Tronci C (2012) ACTA APPLICANDAE MATHEMATICAE 120 (1) pp. 127-151 SPRINGER
Nematodynamics is the orientation dynamics of flowless liquid-crystals. We show how Euler-Poincaré reduction produces a unifying framework for various theories, including Ericksen-Leslie, Luhiller-Rey, and Eringen's micropolar theory. In particular, we show that these theories are all compatible with each other and some of them allow for more general configurations involving a non vanishing disclination density. All results are also extended to flowing liquid crystals. © 2012 Springer Science+Business Media B.V.
Gibbons J, Holm DD, Tronci C (2006)
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.
Morrison PJ, Tassi E, Tronci C (2013) Energy stability analysis for a hybrid fluid-kinetic plasma model, In: Nonlinear Physical Systems
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.
Burby J, Tronci Cesare (2017) Plasma Physics and Controlled Fusion 59 (4) 045013 IOP Publishing
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.
Tronci C, Tassi E, Morrison PJ (2015) Journal of Physics A-Mathematical and Theoretical 48 (18) ARTN 18550 IOP PUBLISHING LTD
Different variants of hybrid kinetic-fluid models are considered for
describing the interaction of a bulk fluid plasma obeying MHD and an energetic
component obeying a kinetic theory. Upon using the Vlasov kinetic theory for
energetic particles, two planar Vlasov-MHD models are compared in terms of
their stability properties. This is made possible by the Hamiltonian structures
underlying the considered hybrid systems, whose infinite number of invariants
makes the energy-Casimir method effective for determining stability.
Equilibrium equations for the models are obtained from a variational principle
and in particular a generalized hybrid Grad-Shafranov equation follows for one
of the considered models. The stability conditions are then derived and
discussed with particular emphasis on kinetic particle effects on classical MHD
stability.
Tronci C (2010) Journal of Physics A: Mathematical and Theoretical 43 (37)
Tronci C, Tassi E, Camporeale E, Morrison PJ (2014) PLASMA PHYSICS AND CONTROLLED FUSION 56 (9) ARTN 09500 IOP PUBLISHING LTD
This paper investigates hybrid kinetic-MHD models, where a hot plasma
(governed by a kinetic theory) interacts with a fluid bulk (governed by MHD).
Different nonlinear coupling schemes are reviewed, including the
pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter
scheme suffers from being non-Hamiltonian and to not exactly conserve total
energy. Upon adopting the Vlasov description for the hot component, the
non-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis is
given to the linear stability of Alfv\'en waves, for which it is shown that a
spurious instability appears at high frequency in the non-Hamiltonian version.
This instability is removed in the Hamiltonian version.
Holm DD, Putkaradze V, Tronci C (2007) Comptes Rendus Mathematique 345 (5) pp. 297-302 Elsevier
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.
Holm DD, Putkaradze V, Tronci C (2008) JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 41 (34) ARTN 34401 IOP PUBLISHING LTD
We suggest kinetic models of dissipation for an ensemble of interacting
oriented particles, for example, moving magnetized particles. This is achieved
by introducing a double bracket dissipation in kinetic equations using an
oriented Poisson bracket, and employing the moment method to derive continuum
equations for magnetization and density evolution. We show how our continuum
equations generalize the Debye-Hueckel equations for attracting round
particles, and Landau-Lifshitz-Gilbert equations for spin waves in magnetized
media. We also show formation of singular solutions that are clumps of aligned
particles (orientons) starting from random initial conditions. Finally, we
extend our theory to the dissipative motion of self-interacting curves.
Amaldi U, Citterio A, Crescenti M, Giuliacci A, Tronci C, Zennaro R (2007) NUCLEAR PHYSICS B-PROCEEDINGS SUPPLEMENTS 172 pp. 277-279
A novel linear accelerating structure is proposed particularly suited for hadrontherapy applications. The main features are compactness, due to the high frequency and consequent large gradients, and good power efficiency, especially at low beam velocities, obtained by using coupled H-mode cavities. The structure is called CLUSTER which stands for "Coupled cavity Linac USing Transverse Electric Radial field". In order to compare the performances of this structure with other hadrontherapy linac designs operating at high frequencies, a conceptual study has been performed by choosing the frequency of 3 GHz. Moreover a proof of principle of the radiofrequency behaviour of the cavity has been obtained by RF measurements on a prototype running at 1 GHz. © 2007 Elsevier B.V. All rights reserved.
Gay-Balmaz F, Ratiu TS, Tronci C (2011) Equivalent theories of liquid crystal dynamics, arXiv
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.
Holm DD, O Náraigh L, Tronci C (2009) Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 79 (1) 016601 American Physical Society
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.
Holm DD, Tronci C (2009) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465 (2102) pp. 457-476
Holm DD, Náraigh L, Tronci C (2008) Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 77 (3) 036211 American Physical Society
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.
Gibbons J, Holm DD, Tronci C (2007) Physics Letters A 372 (7) pp. 1024-1033 Elsevier
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.
Holm DD, Putkaradze V, Tronci C (2008) Kinetic models of oriented self-assembly, Journal of Physics A: Mathematical and Theoretical 41 (34) 344010 IOP PUBLISHING
New kinetic models of dissipation are proposed for the dynamics of an ensemble of interacting oriented particles, for example, moving magnetized nano-particles. This is achieved by introducing double-bracket dissipation into kinetic equations by using an oriented Poisson bracket and employing the moment method to derive continuum equations for the evolution of magnetization and mass density. These continuum equations generalize the Debye?Hückel equations for attracting round particles, and Landau?Lifshitz?Gilbert equations for spin waves in magnetized media. The dynamics of self-assembly is investigated as the emergent concentration into singular clumps of aligned particles (orientons) starting from random initial conditions. Finally, the theory is extended to describe the dissipative motion of self-interacting curved filaments.
Gay-Balmaz F, Tronci C (2011) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467 (2128) pp. 1197-1213
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.
Gay-Balmaz F, Ratiu TS, Tronci C (2013) Equivalent Theories of Liquid Crystal Dynamics, Archive for Rational Mechanics and Analysis 210 (3) pp. 773-811
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 includes 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 may include 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. © 2013 Springer-Verlag Berlin Heidelberg.
Bonet Luz Esther, Tronci C (2015)
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.
Amaldi U, Citterio A, Crescenti M, Giuliacci A, Tronci C, Zennaro R (2007) Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 579 (3) pp. 924-936
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.
Tronci C (2008)
The Vlasov equation of kinetic theory is introduced and the Hamiltonian structure of its moments is presented. Then we focus on the geodesic evolution of the Vlasov moments. As a first step, these moment equations generalize the Camassa-Holm equation to its multi-component version. Subsequently, adding electrostatic forces to the geodesic moment equations relates them to the Benney equations and to the equations for beam dynamics in particle accelerators. Next, we develop a kinetic theory for self assembly in nano-particles. Darcy's law is introduced as a general principle for aggregation dynamics in friction dominated systems (at different scales). Then, a kinetic equation is introduced for the dissipative motion of isotropic nano-particles. The zeroth-moment dynamics of this equation recovers the classical Darcy's law at the macroscopic level. A kinetic-theory description for oriented nano-particles is also presented. At the macroscopic level, the zeroth moments of this kinetic equation recover the magnetization dynamics of the Landau-Lifshitz-Gilbert equation. The moment equations exhibit the spontaneous emergence of singular solutions (clumpons) that finally merge in one singularity. This behaviour represents aggregation and alignment of oriented nano-particles. Finally, the Smoluchowski description is derived from the dissipative Vlasov equation for anisotropic interactions. Various levels of approximate Smoluchowski descriptions are proposed as special cases of the general treatment. As a result, the macroscopic momentum emerges as an additional dynamical variable that in general cannot be neglected.
Citterio A, Amaldi U, Allegretti S, Bcrra P, Bourhaleb F, Braccini S, Crescenti M, Giuliacci A, Fagnola G, Pearce P, Toncelli S, Tronci C, Weiss M, Zennaro R, Rosso E, Szeless B (2006) Frascati Physics Series 40 pp. 355-357
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".
Tronci C (2012) A Lagrangian kinetic model for collisionless magnetic reconnection, Plasma Physics and Controlled Fusion
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.
Holm DD, Putkaradze V, Tronci C (2007) Proceedings of the Royal Society A 466 pp. 2991-3012
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.
Tronci C, Camporeale E (2015) PHYSICS OF PLASMAS 22 (2) ARTN 02070 AMER INST PHYSICS
The low-frequency limit of Maxwell equations is considered in the
Maxwell-Vlasov system. This limit produces a neutral Vlasov system that
captures essential features of plasma dynamics, while neglecting radiation
effects. Euler-Poincar\'e reduction theory is used to show that the neutral
Vlasov kinetic theory possesses a variational formulation in both Lagrangian
and Eulerian coordinates. By construction, the model recovers all collisionless
neutral models employed in plasma simulations. Then, comparisons between the
neutral Vlasov system and hybrid kinetic-fluid models are presented in the
linear regime.
Noguchi K, Tronci C, Zuccaro G, Lapenta G (2007) PHYSICS OF PLASMAS 14 (4) ARTN 04230 AMER INST PHYSICS
A new formulation is presented for the implicit moment method applied to the time-dependent relativistic Vlasov-Maxwell system. The new approach is based on a specific formulation of the implicit moment method that allows us to retain the same formalism that is valid in the classical case despite the formidable complication introduced by the nonlinear nature of the relativistic equations of motion. To demonstrate the validity of the new formulation, an implicit finite difference algorithm is developed to solve the Maxwell's equations and equations of motion. A number of benchmark problems are run: two stream instability, ion acoustic wave damping, Weibel instability, and Poynting flux acceleration. The numerical results are all in agreement with analytical solutions. © 2007 American Institute of Physics.
Close ARD, Tronci C (2015) PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 471 (2183) ARTN 20150 ROYAL SOC
Within the framework of liquid crystal flows, the Qian & Sheng (QS) model for
Q-tensor dynamics is compared to the Volovik & Kats (VK) theory of biaxial
nematics by using Hamilton's variational principle. Under the assumption of
rotational dynamics for the Q-tensor, the variational principles underling the
two theories are equivalent and the conservative VK theory emerges as a
specialization of the QS model. Also, after presenting a micropolar variant of
the VK model, Rayleigh dissipation is included in the treatment. Finally, the
treatment is extended to account for nontrivial eigenvalue dynamics in the VK
model and this is done by considering the effect of scaling factors in the
evolution of the Q-tensor.
Gay-Balmaz F, Tronci C, Vizman C (2010) arXiv
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.
Gibbons J, Holm DD, Tronci C (2008) Physics Letters, Section A: General, Atomic and Solid State Physics 372 (23) pp. 4184-4196
Holm DD, Tronci C (2009) Journal of Geometric Mechanics 1 (2) pp. 181-208
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.
Gay-Balmaz F, Ratiu TS, Tronci C (2012) Euler-Poincaré approaches to nematodynamics, Acta Applicandae Mathematicae 120 (1) pp. 127-151
Nematodynamics is the orientation dynamics of flowless liquid-crystals. We show how Euler-Poincaré reduction produces a unifying framework for various theories, including Ericksen-Leslie, Luhiller-Rey, and Eringen's micropolar theory. In particular, we show that these theories are all compatible with each other and some of them allow for more general configurations involving a non vanishing disclination density. All results are also extended to flowing liquid crystals. © 2012 Springer Science+Business Media B.V.
Gay-Balmaz F, Tronci C (2010) Physica D: Nonlinear Phenomena 239 (20-22) pp. 1929-1947 Elsevier
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.
Bonet Luz Esther, Tronci C (2015) JOURNAL OF MATHEMATICAL PHYSICS 56 (8) ARTN 08210 AMER INST PHYSICS
This paper presents the geometric setting of quantum variational principles
and extends it to comprise the interaction between classical and quantum
degrees of freedom. Euler-Poincar\'e reduction theory is applied to the
Schr\"odinger, Heisenberg and Wigner-Moyal dynamics of pure states. This
construction leads to new variational principles for the description of mixed
quantum states. The corresponding momentum map properties are presented as they
arise from the underlying unitary symmetries. Finally, certain
semidirect-product group structures are shown to produce new variational
principles for Dirac's interaction picture and the equations of hybrid
classical-quantum dynamics.
Gay-Balmaz F, Tronci C (2011) arXiv
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.
Tronci C (2012) Journal of Mathematical Physics 53 (11)
The Euler-Poincaré approach to complex fluids is used to derive multiscale equations for computationally modeling Euler flows as a basis for modeling turbulence. The model is based on a kinematic sweeping ansatz (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zero-mean) motion of fluid parcels at the unresolved scales. Even in the simplest two-scale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modeling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods. © 2012 American Institute of Physics.
Brizard AJ, Tronci C (2016) Physics of Plasmas 23 (6)
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.
Brizard AJ, Tronci C (2016)
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.
Tronci C (2010) arXiv
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.
Holm DD, Putkaradze V, Tronci C (2008) Physica D: Nonlinear Phenomena 237 (22) pp. 2952-2965
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.
Cheng B, Suli E, Tronci C (2017) Proceedings of the London Mathematical Society 115 (4) pp. 854-896 London Mathematical Society
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.
Tronci Cesare, Bonet-Luz E (2016) Proceedings of the Royal Society A 472 (2189) The Royal Society
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.
Camporeale E, Tronci C (2017) Journal of Plasma Physics 83 (3) 705830301 Cambridge University Press
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.
Ohsawa Tomoki, Tronci Cesare (2017) Journal of Mathematical Physics 58 (9) pp. 1-19 AIP Publishing
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.
Close Alex, Burby Joshua W, Tronci Cesare (2018) Journal of Plasma Physics 84 (4) 905840401 Cambridge University Press
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.
Ardakani H. Alemi, Bridges T.J., Gay-Balmaz F., Huang Y.H., Tronci C. (2019) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Royal Society
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.
Foskett Michael S., Holm Darryl D., Tronci Cesare (2019) Acta Applicandae Mathematicae 162 (1) pp. 63-103 Springer
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 ? by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ?`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.
Holm DD, Putkaradze V, Tronci C (2011) arXiv 6 (2) pp. 429-458 AMER INST MATHEMATICAL SCIENCES
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.
Tronci Cesare (2019) Journal of Geometric Mechanics American Institute of Mathematical Sciences
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.
Energetic particle effects in magnetic confinement fusion devices are commonly studied by simulation codes utilising the equations of a hybrid kinetic-fluid model. Typically the underlying continuum equations lack the correct energy balance. This thesis studies the two main hybrid models used in fusion plasma studies (current-coupling and pressure-coupling schemes) in the light of geometric techniques such as geometric reduction, variational principles and Hamiltonian methods. New results in the study of Euler-Poincar´e reduction for semidirect product group structures are presented. Further innovations to suit the drift-kinetic approximation are also presented. Outcomes of the study and development of geometric methods include the explanation of the geometric relationship between the two coupling schemes, and variational and Poisson bracket derivations for a newly conservative (energy-conserving) hybrid model in the pressure-coupling scheme, with energetic particles undergoing guiding centre motion. Kelvin circulation theorems for the new model are presented. The bridge between variational and Poisson structures is considered. This results in a construction that yields the variational and Poisson structures of a generalised, non-canonical Maxwell-Vlasov model in both Lagrangian and Eulerian variables. Achievements are summarised and avenues of future research identified.
Bondar Denys I., Gay-Balmaz François, Tronci Cesare (2019) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Royal Society
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.
Holm Darryl D., Náraigh Lennon Ó, Tronci Cesare (2020) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 (2233) The Royal Society
This paper exploits the theory of geometric gradient ?ows to introduce an alternative regularization of the thin-?lm 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 ?lm 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 ?uid? 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 ?uids.
Tronci Cesare (2020) arXiv.org Cornell University
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.
Gay-Balmaz Francois, Tronci Cesare (2020) Nonlinearity IOP Publishing
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.
Rawlinson Jonathan I., Tronci Cesare (2020) Physical Review A American Physical Society
While the treatment of conical intersections in molecular dynamics generally requires