Matrix hydrodynamics

and 2-D Euler equations

Klas Modin

Chalmers, Sweden

A continuous medium deprived of its physical properties (elasticity, thermal and electrical conductivity, and so on) still retains a definite position in space and still interacts through the mutual pressure of its parts due to Aristotle’s principle that two bodies cannot occupy the same space. Amazingly, it is these elementary interactions that cause the most intricate behavior, including turbulence.

initial

mixing

long-time

Why 2-D Euler equations?

\dot v +\nabla_v v = -\nabla p

\operatorname{div}v = 0

Geometry of the Euler equations

Euler's equations describe Riemannian geodesics on

\operatorname{Diff}_{\mu}(M) = \{ \varphi\in\operatorname{Diff}(M) \mid |D\varphi| = 1 \}

But how?

\dot v +\nabla_v v = -\nabla p

\operatorname{div}v = 0

Riemannian metrics on Lie groups

V \in T_g G

\langle V,V\rangle_g =

Riemannian metrics on Lie groups

V\cdot g^{-1} \in \mathfrak{g}

\langle V,V\rangle_g =

\langle V\cdot g^{-1},V\cdot g^{-1}\rangle_e

Right-invariant Riemannian metric determined by inner product on \(\mathfrak{g}\)

Governing equations

\mathfrak{g}^*

\displaystyle\frac{d}{dt}\frac{\partial L}{\partial \dot g} = \frac{\partial L}{\partial g}

\(G\)

\(T_eG\simeq\mathfrak g\)

L(g,\dot g) = \langle\dot g,\dot g\rangle_g

\displaystyle \dot m + \mathrm{ad}^*_v m = 0

\displaystyle \dot g = v\cdot g

\displaystyle \mathrm{Ad}^*_g m_0 = \langle v,\cdot \rangle_e

\displaystyle m = \langle v,\cdot \rangle_e

Euler-Arnold

(Lie-Poisson)

Euler-Lagrange

Transport

Arnold's example:

the Euler equations

G = \operatorname{Diff}_\mu(M), \quad \mathfrak{g} = \mathfrak{X}_\mu(M) = \{ v\mid \operatorname{div}v=0 \}

Inner product:

\displaystyle\langle v,v\rangle_{L^2} = \int_M \lvert v \rvert^2 \mu

Arnold's theorem: \(\gamma(t)\in \operatorname{Diff}_\mu(M)\) geodesic curve \(\Rightarrow\) vector field \(v(t) = \dot\gamma(t)\circ\gamma(t)^{-1}\) fulfills Euler's equations

Arnold's example:

the 2-D Euler equations

G = \operatorname{Diff}_\mu(S^2), \quad \mathfrak{g} = \{ \nabla^\bot\psi\mid \psi\in C^\infty(S^2) \}

Inner product:

\displaystyle\langle \nabla^\bot\psi,\nabla^\bot\psi\rangle_{L^2} = \int_M \psi (-\Delta \psi)\mu

\displaystyle \dot m + \mathrm{ad}^*_v m = 0\quad\iff

\displaystyle \dot \omega + \{\omega,\psi\} = 0

\quad -\Delta\psi = \omega

Lie-Poisson formulation:

\displaystyle \dot \omega + \{\omega,\psi\} = 0

\quad -\Delta\psi = \omega

What is matrix hydrodynamics?

\displaystyle \dot W + \frac{1}{\hbar}[W,P] = 0

\quad -\Delta_N P = W

Hoppe-Yau Laplacian

\Delta_N P = \frac{1}{\hbar^2}\sum_{\alpha=1}^3 [[P, X_\alpha], X_\alpha]

Numerical experiment:
What is mixing?

Point vortex solutions

\dot\omega + \{\psi,\omega \}= 0

Lie-Poisson system for \(\omega \in\mathfrak{g}^*\)

(strong form)

\langle\dot\omega,\xi\rangle -\langle\omega, \{\psi,\xi \}\rangle= 0

(weak form)

\omega = \sum_i \Gamma_i \delta_{x_i}

(weak co-adjoint orbit)

Statistical mechanics

theories for 2D Euler

Idea by Onsager (1949):

Approximate \(\omega\) by PV for large \(N\)
Apply statistical mechanics and presume ergodicity

Hamiltonian function:

\displaystyle H(x_1,\ldots,x_N) = \sum_{kl} \Gamma_k\Gamma_l G(x_k, x_l)

Statistical mechanics

theories for 2D Euler

Idea by Onsager (1949):

Approximate \(\omega\) by PV for large \(N\)
Apply statistical mechanics and presume ergodicity

Hamiltonian function:

\displaystyle H(x_1,\ldots,x_N) = -\sum_{kl} \Gamma_k\Gamma_l \log |x_k-x_l|

Onsager's observation:

Pos. and neg. strengths \(\Rightarrow\) energy takes values \(-\infty\) to \(\infty\)

Statistical mechanics

theories for 2D Euler

Idea by Onsager (1949):

Approximate \(\omega\) by PV for large \(N\)
Apply statistical mechanics and presume ergodicity

Hamiltonian function:

\displaystyle H(x_1,\ldots,x_N) = -\sum_{kl} \Gamma_k\Gamma_l \log |x_k-x_l|

Onsager's observation:

Pos. and neg. strengths \(\Rightarrow\) energy takes values \(-\infty\) to \(\infty\)

\(\Rightarrow\) phase volume function \(v(E)\) has inflection point

Statistical mechanics

theories for 2D Euler

Idea by Onsager (1949):

Approximate \(\omega\) by PV for large \(N\)
Apply statistical mechanics and presume ergodicity

Hamiltonian function:

\displaystyle H(x_1,\ldots,x_N) = -\sum \Gamma_k\Gamma_l \log |x_k-x_l|

Onsager's prediction

Problems with Onsager's theory

PV solutions far from smooth (\(H^{-1}\) but never \(L^p\))
\(\Rightarrow\) No Casimir functions
But experiments and numerical simulations indicate that Casimirs affect long-time behavior

Numerical experiment:
Long-term behavior

Vanishing total

angular momentum

Non-vanishing total

angular momentum

Numerical experiment:
Long-term behavior

Vanishing total

angular momentum

Non-vanishing total

angular momentum

Numerical experiment:
Long-term behavior

Vanishing total

angular momentum

Non-vanishing total

angular momentum

Low dim, near invariant manifolds?

Small condensates (blobs) merge
Separated blobs \(\Rightarrow\) almost blob-vortex dynamics (BVD)
Dynamics not integrable \(\Rightarrow\) blobs continue to merge
\(k\)-BVD integrable \(\Rightarrow\) quasi-periodicity prevents further mixing

Known about integrability

Point vortex dynamics on \(\mathbb{S}^2\)

3-PVD is integrable (Sakajo, 1999)
4-PVD is integrable for vanishing momentum (Sakajo, 2007)
4-PVD non-integrable in general (Bagrets & Bagrets, 1997)

Symplectic reduction theory:

only \(SO(3)\) symmetry needed in proof

\Rightarrow

Blob vortex dynamics on \(\mathbb{S}^2\)

3-BVD is integrable
4-BVD is integrable for vanishing angular momentum

Alignment with

point-vortex dynamics