The reversibility paradox in matrix hydrodynamics

Klas Modin

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

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

Sectional curvature convergence

Smooth initial conditions

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

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

Smooth initial conditions (vanishing momentum)

Alignment with

point-vortex dynamics

But...

Euler-Zeitlin dynamics reversible/Hamiltonian \(\Rightarrow\) Poincaré recurrence

Blob "convergence" due to numerical dissipation?

Insightful question:

Brief: reversibility paradox

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

?

Same initial conditions forward and backward in time

Same initial conditions forward and backward in time

Forward - pause - backward

Relative error \(||W_{0\leftarrow n}-W_0||/||W_0|| \)

The reversibility paradox in matrix hydrodynamics

By Klas Modin

The reversibility paradox in matrix hydrodynamics

CAM-seminar, January 2024.

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