invariant manifolds for reaction kinetics andrei zinovyev institut des hautes Études scientifiques
TRANSCRIPT
INVARIANT MANIFOLDSfor reaction kinetics
Andrei Zinovyev
Institut des Hautes Études Scientifiques
Stoichiometric equations
Jc
s1A1 + … + snAn s1A1 + … + snAn
n – number of species, s – number of reactions
si
n
jjs
n
jjsss
jj ckckJ..1 11
)(
c1
c2
cn
Pure dissipative systems
cG
ceq
n
ieqi
ii c
ccG
1
1ln
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
Positively Invariant Manifold
EQUILIBRIUM
fast motion
slow motion
Why Invariant? once the point on the manifold, the trajectory will stay on it until the equilibrium
Projector Pc on (some) manifoldinduces new (reduced) dynamics
J
induced dynamics
Pc J
tangent space
= (1-Pc)J - invariance defect
x
Thermodynamic projector
GJ
Pc J
The induced dynamics is dissipative only if
0))(),(( cJPcG C
Quasi-equilibrium manifoldusually is not invariant
entropy Smax
macroscopic (reduced) variables
Correction of invariance defect
0.2
0.4
0.6
0.8
1.0
0.05 0.10 0.15 0.20
C1
C3
equilibrium
invariant manifold
quasi-equilibrium manifold
corrections
(1-Pc)J = 0
invariance equation
Newton iterations
0
0))()(1(
xP
xx
JxJP
C
C
Invariant grid
EQUILIBRIUM
tangent space
tangent space
J
invariance defect is corrected for every node independently
Growing Invariant Flag
Entropic scalar product
),(
),(,
2
2
Hxxx
Hyxyxc
GH
0 1
2
-1-2
equilibrium
natural parameterentropy
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
21 k12 k13 k
34 10k
35 10k
26 10k
Conservation laws:
2cH2 +2cH2O+cH+cOH = bH
2cO2+cH2O+cO+cOH = bO
One-dimensional dynamics
Separation of times
is the eigen valueof symmetrised
matrixc
J
Two-dimensional dynamics
Visualizing functions:concentration of H
“Fast” coordinate
Visualizing functions:concentration of H2
“Slow” coordinate
Visualizing functions:concentration of OH
Visualizing functions:Entropy and entropy production
Entropy Entropy production
Visualizing functions:Separation of relaxation times
2/1 3/2
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 Kinetics2004. Physica A, V.333, pp.106-154
People
Doctor Iliya Karlin ETH, Zurich
Professor Alexander GorbanUniversity of Leicester, UK