# The reversibility paradox in matrix hydrodynamics

## Euler's equations

\dot v + v\cdot \nabla v = -\nabla p
\operatorname{div}v = 0

## Euler's equations

\operatorname{curl}(\dot v + v\cdot \nabla v) = -\operatorname{curl}\nabla p
\operatorname{div}v = 0

## Euler's equations

\frac{\partial}{\partial t}\operatorname{curl} v + \operatorname{curl}(v\cdot \nabla v) = 0
\operatorname{div}v = 0

(2-D)

\underbrace{\phantom{curl a}}_{\omega}
\underbrace{\phantom{curl a curl bb}}_{v\cdot\nabla\omega}

## Euler's equations

\dot\omega + v\cdot \nabla\omega = 0
\operatorname{div}v = 0

(2-D)

\omega = \operatorname{curl} v

## Euler's equations

\dot\omega + v\cdot \nabla\omega = 0
\operatorname{div}v = 0

(2-D)

\omega = \operatorname{curl} \nabla^\bot \psi
\underbrace{\phantom{v}}_{\nabla^\bot\psi}

## Euler's equations

\dot\omega + \nabla^\bot\psi\cdot \nabla\omega = 0

(2-D)

\omega = \Delta\psi

## Euler's equations

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

(2-D)

\omega = \Delta\psi

Poisson bracket

## Euler-Zeitlin equations

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

(2-D)

W = \Delta_N P

Commutator

$$N\times N$$ matrices

## Matrix hydrodynamics

### approach for spatial discretization

Poisson algebra $$(M,\{\cdot,\cdot\})$$ $$\iff$$ matrix Lie algebra $$(\mathfrak{g},\frac{1}{\hbar}[\cdot,\cdot])$$

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

Thm [M. & Viviani]

Solutions $$\omega(t)\in H^6(M)$$ and $$W(t)\in \mathfrak{u}(N)$$ for $$t\in [0,T]$$

Then

$$||T_N^*W(t)-\omega(t)||_{L^2} = \mathcal{O}(N^{-1})$$

## Low dim, near integrable (quasi) invariant manifolds?

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

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

# But...

Euler-Zeitlin dynamics reversible/Hamiltonian $$\Rightarrow$$ Poincaré recurrence

Blob "convergence" due to numerical dissipation?

Insightful question:

Recurrence theorem: volume-preserving flow on compact phase space eventually returns arbitrarily close to initial state

H-theorem: in closed system of interacting, idealized gas particles, entropy increases with time

## Numerical time-discretization errors

1. Local truncation error $$|| \Phi_h - \Psi_h ||$$

2. Root-finding error
(tolerance of Newton/fixed-point iterations)

3. Round-off errors

"Convergence" due to discretization errors?

YES

NO

?

By Klas Modin

# The reversibility paradox in matrix hydrodynamics

CAM-seminar, January 2024.

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