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
v˙+v⋅∇v=−∇p
\dot v + v\cdot \nabla v = -\nabla p
divv=0
\operatorname{div}v = 0
Euler's equations
curl(v˙+v⋅∇v)=−curl∇p
\operatorname{curl}(\dot v + v\cdot \nabla v) = -\operatorname{curl}\nabla p
divv=0
\operatorname{div}v = 0
Euler's equations
∂t∂curlv+curl(v⋅∇v)=0
\frac{\partial}{\partial t}\operatorname{curl} v + \operatorname{curl}(v\cdot \nabla v) = 0
divv=0
\operatorname{div}v = 0
(2-D)
ωcurla
\underbrace{\phantom{curl a}}_{\omega}
v⋅∇ωcurlacurlbb
\underbrace{\phantom{curl a curl bb}}_{v\cdot\nabla\omega}
Euler's equations
ω˙+v⋅∇ω=0
\dot\omega + v\cdot \nabla\omega = 0
divv=0
\operatorname{div}v = 0
(2-D)
ω=curlv
\omega = \operatorname{curl} v
Euler's equations
ω˙+v⋅∇ω=0
\dot\omega + v\cdot \nabla\omega = 0
divv=0
\operatorname{div}v = 0
(2-D)
ω=curl∇⊥ψ
\omega = \operatorname{curl} \nabla^\bot \psi
∇⊥ψv
\underbrace{\phantom{v}}_{\nabla^\bot\psi}
Euler's equations
ω˙+∇⊥ψ⋅∇ω=0
\dot\omega + \nabla^\bot\psi\cdot \nabla\omega = 0
(2-D)
ω=Δψ
\omega = \Delta\psi
Euler's equations
ω˙+{ψ,ω}=0
\dot\omega + \{ \psi , \omega \}= 0
(2-D)
ω=Δψ
\omega = \Delta\psi
Poisson bracket
Euler-Zeitlin equations
W˙+ℏ1[P,W]=0
\dot W + \frac{1}{\hbar}[P , W]= 0
(2-D)
W=ΔNP
W = \Delta_N P
Commutator
N×N matrices
Matrix hydrodynamics
approach for spatial discretization
Poisson algebra (M,{⋅,⋅}) ⟺ matrix Lie algebra (g,ℏ1[⋅,⋅])
ω˙+{ψ,ω}=0
\dot\omega + \{\psi,\omega \}= 0
W˙+ℏ1[P,W]=0
\dot W + \frac{1}{\hbar}[P,W]= 0
Thm [M. & Viviani]
Solutions ω(t)∈H6(M) and W(t)∈u(N) for t∈[0,T]
Then
∣∣TN∗W(t)−ω(t)∣∣L2=O(N−1)
Sectional curvature convergence
Smooth initial conditions
Low dim, near integrable (quasi) invariant manifolds?
- Small condensates (blobs) merge
- Separated blobs ⇒ almost blob-vortex dynamics (BVD)
- Dynamics not integrable ⇒ blobs continue to merge
- k-BVD integrable ⇒ quasi-periodicity prevents further mixing
Known about integrability
Point vortex dynamics on S2
- 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 S2
- 3-BVD is integrable
- 4-BVD is integrable for vanishing angular momentum
Alignment with
point-vortex dynamics
But...
Euler-Zeitlin dynamics reversible/Hamiltonian ⇒ 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 ∣∣Φh−Ψh∣∣
- 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
Relative error ∣∣W0←n−W0∣∣/∣∣W0∣∣
The reversibility paradox in matrix hydrodynamics Klas Modin
The reversibility paradox in matrix hydrodynamics
By Klas Modin
The reversibility paradox in matrix hydrodynamics
CAM-seminar, January 2024.
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