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