Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Newton's equations on diffeomorphisms and densities
Klas Modin
Joint work with
Boris Khesin
University of Toronto
Gerard Misiolek
University of Notre Dame
Arnold's 1966 discovery
\dot u + \nabla_u u = -\nabla p
Euler's equations for inviscid incompressible fluid on \(M\)
Arnold (1966): diffeomorphism \(\varphi(t)\) generated by \(u(t)\) is geodesic curve on \(\mathrm{Diff}_\mu(M)\) w.r.t. \[ G_\varphi(\dot\varphi,\dot\varphi) = \int_M |\dot\varphi|^2 \mu \]
Led to geometric and topological hydrodynamics
\(\Rightarrow\) stability results (Arnold and Khesin)
\(\Rightarrow\) well-posedness results (Ebin and Marsden)
Newton's equations on \(\mathrm{Diff}(M)\)
\displaystyle
\dot \varphi + \nabla_{\dot\varphi} \dot\varphi +\nabla \frac{\delta \bar U}{\delta\varrho}(\varphi_*\mu)\circ\varphi = 0
Aim: extend Arnold's framework
Ingredients:
- Riemannian metric \(G\) on \(\mathrm{Diff}(M)\)
- Potential function \(U:\mathrm{Diff}(M)\to \mathbb{R}\)
\(\mathrm{Diff}_\mu(M)\) symmetry:
- \(G_{\varphi\circ\eta}(\dot\varphi\circ\eta,\dot\varphi\circ\eta) = G_\varphi(\dot\varphi,\dot\varphi)\qquad\forall \,\eta\in \mathrm{Diff}_\mu(M)\)
- \(V(\varphi\circ\eta) = V(\varphi)\)
Riemannian submersion
\mathrm{Diff}(M)
\mathrm{Dens}(M)
\mathrm{Id}
\mu
\varrho
\pi(\varphi)=\varphi_*\mu
Moser 1965:
Principal bundle
\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)
\(L^2\) metric on \(\mathrm{Diff}(M)\)
G_\varphi(\dot\varphi,\dot\varphi) = \int_{M}\left\vert \dot\varphi \right\vert^2 \mu
Induces Otto metric
{\overline{G}}_\varrho(\dot\varrho,\dot\varrho) \Rightarrow d_W^2(\mu,\varrho)
\mathrm{Hor}
\displaystyle
U(\varphi) = \bar U(\varphi_*\mu)\quad \bar U\colon\mathrm{Dens}(M)\to\mathbb{R}
\mathrm{Dens}(M)=\{ \varrho\in\Omega^n(M)\mid \varrho>0, \int_M \varrho = 1\}
Smooth probability densities
Induced potential function
Symplectic reduction viewpoint
T^*\mathrm{Diff}(M)
T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu}(M)\simeq \mathrm{Dens}(M) \times \mathfrak{X}^*(M)
T^*\mathrm{Diff}(M)//\mathrm{Diff}_{\mu}(M)\simeq T^*\mathrm{Dens}(M)
Poisson reduction
Symplectic reduction
\dot \rho + \mathrm{div}(\rho\nabla \theta) = 0,\; \dot \theta + \frac{1}{2}|\nabla \theta|^2 + \frac{\delta \bar U}{\delta\varrho}(\varrho) = 0
\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot u + \nabla_u u + \nabla\frac{\delta \bar U}{\delta\varrho}(\varrho) = 0
Example: barotropic fluid
\displaystyle\bar U(\varrho) = \int_M e(\rho)\varrho
Gives compressible (barotropic) Euler equations
\displaystyle\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot u + \nabla_u u + \frac{1}{\rho}\nabla P(\rho) = 0
\(\displaystyle P(\rho) = e'(\rho)\rho^2\) is the pressure function
"Potential solutions" \(u = \nabla\theta\) \(\Rightarrow\) horizontal solutions
Example: Schrödinger equations
\displaystyle\bar U(\varrho) = \int_M F(\rho)\mu + \frac{1}{2}\int_M \frac{|\nabla \rho|^2}{\rho}\mu
Consider horizontal solutions \(\Rightarrow\) system on \(T^*\mathrm{Dens}(M)\)
\displaystyle\dot \rho + \mathrm{div}(\rho u) = 0,\; \dot\theta + \frac{1}{2}|\nabla \theta|^2 + f(\rho) - \frac{2}{\sqrt{\rho}}\nabla\sqrt{\rho} = 0
Fisher functional
From \((\varrho,\theta)\in T^*\mathrm{Dens}(M)\) construct wave function
\displaystyle \psi = \sqrt{\rho\mathrm{e}^{i\theta}}
Theorem (Madelung 1927, von Renesse 2011):
Wave function fulfills (nonlinear) Schrödinger equation \[ i\dot\psi + \Delta\psi - f(|\psi|^2)\psi=0\]
Madelung transform
Madelung transform as symplectomorphism
Theorem: Madelung transform induces symplectomorphism \[ \Phi\colon T^*\mathrm{Dens}(M)\to PC^\infty(M,\mathbb{C}\backslash\{0\})\] (Fréchet topology of smooth functions)
Geometric quantum mechanics (Kibble 1979):
wave function \(\Rightarrow\) element of complex projective space
PC^\infty(M,\mathbb{C})
Kähler manifold
Madelung transform as
Kähler morphism
Theorem: Madelung transform induces Kähler morphism \[ \Phi\colon T^*\mathrm{Dens}(M)\to PC^\infty(M,\mathbb{C}\backslash\{0\})\] (Fréchet topology of smooth functions)
Is Madelung transform isometry?
Canonical metric on \(PC^\infty(M,\mathbb{C})\): Fubini-Study metric
Fisher-Rao metric on \(\mathrm{Dens}(M)\):
\displaystyle G_\varrho(\dot\varrho,\dot\varrho) = \int_M \frac{\dot\rho^2}{\rho}\mu
Sasaki-Fisher-Rao metric
on \(T^*\mathrm{Dens}(M)\):
\displaystyle G_{(\varrho,\theta)}(\dot\varrho,\dot\theta,\dot\varrho,\dot\theta) = \int_M (\frac{\dot\rho^2}{\rho} + \dot\theta^2\rho)\mu
Some consequences
- Link between classical information geometry, quantum information geometry, and hydrodynamics
- New, geometric understanding of several PDE: 2-component Camassa-Holm, vortex filament flow, ...
- Geometric and hydrodynamic interpretation of relativistic quantum mechanics:
Klein-Gordon equation as stationary solution of infinite-dimensional Neumann problem on Minkowski space
THANKS!
Geometric hydrodynamics via Madelung transform,
PNAS, 2018
Geometry of the Madelung transform,
arXiv preprint, 2018
Slides available at: slides.com/kmodin
Newton's equations on diffeomorphisms and densities
By Klas Modin
Newton's equations on diffeomorphisms and densities
Presentation given 2017-11-08 at the GSI'17 conference in Paris.
