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INVARIANT MANIFOLDS for reaction kinetics

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What is Description Reduction? 1 Shorten list of species. eliminate some ... Doctor Iliya Karlin. ETH, Zurich. Professor Alexander Gorban. University of Leicester, UK ... – PowerPoint PPT presentation

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Title: INVARIANT MANIFOLDS for reaction kinetics


1
INVARIANT MANIFOLDSfor reaction kinetics
  • Andrei Zinovyev
  • Institut des Hautes Études Scientifiques

2
Stoichiometric equations
as1A1 asnAn ? bs1A1 bsnAn
n number of species, s number of reactions
cn
c1
c2
3
Pure dissipative systems
G
c
ceq
4
What is Description Reduction?
  • 1 Shorten list of species
  • eliminate some
  • create integrated components
  • 2 Shorten list of reactions
  • eliminate some
  • freeze fast reactions
  • 3 Decompose motion into fast and slow

5
Positively Invariant Manifold
fast motion
slow motion
EQUILIBRIUM
W
Why Invariant? once the point on the manifold,
the trajectory will stay on it until the
equilibrium
6
Projector Pc on (some) manifoldinduces new
(reduced) dynamics
induced dynamics
?
tangent space
J
Pc J
W
TxW
? (1-Pc)J - invariance defect
7
Thermodynamic projector
J
Pc J
The induced dynamics is dissipative only if
8
Quasi-equilibrium manifoldusually is not
invariant
entropy S?max
macroscopic (reduced) variables
9
Correction of invariance defect
invariant manifold
corrections
C1
invariance equation
1.0
(1-Pc)J 0
0.8
0.6
Newton iterations
equilibrium
0.4
0.2
0.20
0.05
0.10
0.15
C3
quasi-equilibrium manifold
10
Invariant grid
tangent space
EQUILIBRIUM
J
tangent space
invariance defect is corrected for every node
independently
11
Growing Invariant Flag
12
Entropic scalar product
2
1
0
equilibrium
-1
natural parameter entropy
-2
13
Hydrogen burning model reaction
1 H2 ? 2H 2 O2 ? 2O 3 H2O ? H OH 4
H2 O ? H OH 5 O2 H ? O OH 6 H2 O ?
H2O
Conservation laws
2cH2 2cH2OcHcOH bH 2cO2cH2OcOcOH bO
14
One-dimensional dynamics
15
Separation of times
l is the eigen value of symmetrised matrix
16
Two-dimensional dynamics
17
Visualizing functionsconcentration of H
Fast coordinate
18
Visualizing functionsconcentration of H2
Slow coordinate
19
Visualizing functionsconcentration of OH
20
Visualizing functionsEntropy and entropy
production
Entropy production
Entropy
21
Visualizing functionsSeparation of relaxation
times
?2/?1
?3/?2
22
Papers
Gorban A, Karlin I, Zinovyev A. Constructive
Methods of Invariant Manifolds for Kinetic
Problems 2004. Physics Reports 396, pp.197-403.
Gorban A, Karlin I, Zinovyev A. Invariant Grids
for Reaction Kinetics 2004. Physica A, V.333,
pp.106-154
23
People
Professor Alexander Gorban University of
Leicester, UK
Doctor Iliya Karlin ETH, Zurich
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