Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
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 \( (C^\infty(M),\{\cdot,\cdot\})\) \(\iff\) Lie algebra \( (\mathfrak{u}(N),\frac{1}{\hbar}[\cdot,\cdot])\)
\dot\omega + \{\psi,\omega \}= 0
\dot W + \frac{1}{\hbar}[P,W]= 0
Convergence of order 1/2:
\iff
||T_N^*W(t)-\omega(t)||_{L^2} = \mathcal{O}(N^{-1})
Sectional curvature convergence
Smooth initial conditions
Smooth initial conditions
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
- 4-PVD is integrable for vanishing momentum
- 4-PVD non-integrable in general
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)
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
- Local truncation error \( || \Phi_h - \Psi_h || \)
- Root-finding error
(tolerance of Newton/fixed-point iterations)
- 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
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.
