1. INVARIANT MANIFOLDS for reaction kinetics Andrei Zinovyev Institut des Hautes Études Scientifiques

2. Stoichiometric equations  s1 A 1 + … +  sn A n   s1 A 1 + … +  sn A n n – number of species, s – number of reactions c1 c2 cncn

3. Pure dissipative systems c G c eq

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 EQUILIBRIUM fast motion slow motion Why Invariant? once the point on the manifold, the trajectory will stay on it until the equilibrium 

6. Projector P c on (some) manifold induces new (reduced) dynamics J induced dynamics P c J tangent space  = (1-P c )J - invariance defect   xx

7. Thermodynamic projector J P c J The induced dynamics is dissipative only if

8. Quasi-equilibrium manifold usually is not invariant entropy S  max macroscopic (reduced) variables

9. Correction of invariance defect 0.2 0.4 0.6 0.8 1.0 0.05 0.100.15 0.20 C1C1 C3C3 equilibrium invariant manifold quasi-equilibrium manifold corrections (1-P c )J = 0 invariance equation Newton iterations

10. Invariant grid EQUILIBRIUM tangent space J invariance defect is corrected for every node independently

11. Growing Invariant Flag

12. Entropic scalar product 0 1 2 -2 equilibrium natural parameter entropy

13. Hydrogen burning model reaction 1: H 2  2H 2: O 2  2O 3: H 2 O  H + OH 4: H 2 + O  H + OH 5: O 2 + H  O + OH 6: H 2 + O  H 2 O Conservation laws: 2c H2 +2c H2O +c H +c OH = b H 2c O2 +c H2O +c O +c OH = b O

14. One-dimensional dynamics

15. Separation of times is the eigen value of symmetrised matrix

16. Two-dimensional dynamics

17. Visualizing functions: concentration of H “Fast” coordinate

18. Visualizing functions: concentration of H 2 “Slow” coordinate

19. Visualizing functions: concentration of OH

20. Visualizing functions: Entropy and entropy production Entropy Entropy production

21. Visualizing functions: Separation 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 Doctor Iliya Karlin ETH, Zurich Professor Alexander Gorban University of Leicester, UK