Image: D.D. Holm, T. Schmah & C. Stoica. Geometric Mechanics and Symmetry:

From Finite to Infinite Dimensions, Oxford University Press (2009).

### Second AGM Meeting

Geometric Hydrodynamics: Quantum and Classical

The second meeting will be held on Friday 22 July 2022 at the University of Surrey. If you would like to attend the meeting, please register by sending an email to c.tronci@surrey.ac.uk or h.alemi-ardakani@exeter.ac.uk

The focus on hydrodynamic models finds its justification in the high importance that continuum fluid models possess within many areas of physics and technology, including turbulence, meteorology, plasmas, and quantum systems (among many others). In particular, an innovative direction will be pursued in terms of hybrid quantum-classical models. The latter represents a long-sought method in molecular dynamics: as the curse of dimensions in many-particle quantum systems poses severe computational challenges, some degrees of freedom are treated as classical while other are left quantum. This leads to the unanswered question of quantum-classical coupling. Other directions will also involve nonlinear waves, oceanography, and modulation theory. The apparently diverse set of topics in this endeavour all share different concepts in geometry and symmetry that serve to unify the derivations of their fundamental equations and the interpretations of their solutions. For example, hydrodynamic models and methods of data assimilation for weather prediction are also used in fundamental applications of shape analysis for the comparison of biomedical images. Likewise, the study of geodesic flows in geometric mechanics (from rigid body to fluid dynamics) is of central importance also in the analysis of the Fubini-Study geodesics in quantum dynamics.

Topics expected to be discussed include:

(1) Geometry of fluid models: Geometric aspects of fluid systems will be discussed. These involve highly mathematical concepts in Poisson geometry, such as dual pairs, as well as geometric flows such as geodesics on diffeomorphism groups and vortex filament dynamics. Part of the meeting will discuss applications of these concepts, particularly in multiscale systems and shallow water waves. Stability issues will also be considered along with the theory of Relative Equilibria. Euler-Poincaré variational principles for the generalised Green-Naghdi and Whitham equations will be discussed.

(2) Hybrid quantum fluid models: These models couple fluid models describing a classical subsystem with quantum equations governing the degrees of freedom of a quantum component such as electrons or spins. Various aspects of these models will be discussed including their geometric properties and formal stability considerations.

Friday 22 July 2022, University of Surrey.

Registration and coffee breaks will take place in the Thomas Telford Building, Room 39AA04 (Maths Common Room), and the Scientific Talks will take place in the Lewis Carroll Building, Room 05AC03.

The Thomas Telford Building is labelled Academic Building 1 (in square 4D) and the Lewis Carroll Building is labelled Academic Building 3 (in square 4E) on this map:

https://www.surrey.ac.uk/sites/default/files/2022-02/Campus-map-update-v4.pdf

Visit the following website for directions to the University of Surrey:

https://www.surrey.ac.uk/visit-university/how-get-here

Timetable

09:45–10:30 Coffee (Room 39AA04, Maths Department)

10:30–11:15 Mauro Spera (Università Cattolica del Sacro Cuore, Brescia)

11:15–11:45 Coffee break (Room 39AA04, Maths Department)

11:45–12.30 Jacques Vanneste (University of Edinburgh)

12:30–14:00 Lunch

14:00–14:30 Rosa Kowalewski (University of Bath)

14:30–15:00 Oliver Street (Imperial College London)

15:00–15:30 Coffee break (Room 39AA04, Maths Department)

15:30–16:00 Cesare Tronci (University of Surrey)

Titles and Abstracts

Mauro Spera (Università Cattolica del Sacro Cuore, Brescia)

Some Geometric Aspects of Generalized Schroedinger Equations

In this talk we review several geometric aspects concerning the Schroedinger and the Pauli equations, mostly based on [Spe16],[Spe18]. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schroedinger equation. The probability current provides an equivariant moment map for the group G = sDiff(R^3) of volume-preserving diffeomorphisms of R^3 (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture extends, mutatis mutandis, to the Pauli equation and to generalized Schroedinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is outlined via consideration of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Connections with geometric quantum mechanics, coherent state and Fisher information theoretic aspects are highlighted throughout, and contact with recent developments [KMM19, FT20] is established.

References

[FT20] M.S. Foskett, C. Tronci, arXiv:2003.08664v1 [math-ph]

[KMM19] B. Khesin, G. MisioŁek & K. Modin, Geometry of the Madelung Transform, Arch. Rational Mech. Anal. 234 (2019) 549–573, https://doi.org/10.1007/s00205-019-01397-2

[Spe16] M. Spera, Moment map and gauge geometric aspects of the Schroedinger and Pauli equations International Journal of Geometric Methods in Modern Physics Vol. 13, No. 4 (2016) 1630004 (36 pages) DOI: 10.1142/S021988781630004X

[Spe18] M. Spera, On Some Geometric Aspects of Coherent States, in Coherent States and Their Applications, J.-P. Antoine et al. (eds.), Springer Proceedings in Physics 205, Springer Nature 2018, https://doi.org/10.1007/978-3-319-76732-1_8, Chapter 8, pp 157-172.

Jacques Vanneste (University of Edinburgh)

A Geometric Look at Momentum Flux and Stress in Fluid Dynamics

I will discuss a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form-valued 2-forms, and their divergence as a covariant exterior derivative. I will show how a variational derivation automatically yields a symmetric momentum flux for inviscid fluid models. I will also consider the geometric form of the viscous stress in the Naiver-Stokes equations.

Joint work with Andrew D. Gilbert.

Rosa Kowalewski (University of Bath)

Navier-Stokes Equations on a Riemannian Manifold from a Non-Conservative Action Principle

We derive the equations of motion for a viscous barotropic fluid on a manifold using a non-conservative action principle in a geometric setting. Our work contributes to the ongoing discussion of which Laplace operator is the correct choice when generalising the Navier-Stokes equations from a flat space to a curved Riemannian manifold. Making the assumption that the fluid is isotropic and Newtonian, we derive the compressible Navier-Stokes equations where the Laplace operator Def* Def is defined using the deformation tensor Def (where Def* is the adjoint).

Oliver Street (Imperial College London)

A Structure Preserving Stochastic Perturbation of Classical Water Wave Theory

The classical water wave equations, as well as the Euler equations from which they are derived, have a rich geometric structure. Due to the simplifications required to derive a closed set of equations for the free surface problem, and the richness of the dynamics found in reality, a stochastic perturbation is desirable to account for the many sources of uncertainty. We will assume the bulk flow satisfies a stochastic version of the Euler equations, designed to preserve their geometric properties following Holm (2015 Proc. R. Soc. A). It will be shown that this leads to a stochastic perturbation of the water wave problem which, by design, has a natural extension of the Hamiltonian structure found by Zakharov (1968 J. Appl. Mech. Tech. Phys.).

Cesare Tronci (University of Surrey)

Geometry of Quantum Hydrodynamics in Quantum Chemistry

After an overview of the geometric structures appearing in Madelung's hydrodynamic description of quantum evolution, this talk will exploit this geometric setting to provide new molecular dynamics algorithms for quantum chemistry.

The London Mathematical Society administers a Caring Supplementary Grant Scheme. Further information about this scheme can be found on the LMS website: www.lms.ac.uk/grants/caring-supplementary-grants

For any enquiries, please contact Hamid Alemi Ardakani.