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

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

Pure dissipative systems c G c eq

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 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

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

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

Correction of invariance defect C1C1 C3C3 equilibrium invariant manifold quasi-equilibrium manifold corrections (1-P c )J = 0 invariance equation Newton iterations

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

Growing Invariant Flag

Entropic scalar product equilibrium natural parameter entropy

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

One-dimensional dynamics

Separation of times is the eigen value of symmetrised matrix

Two-dimensional dynamics

Visualizing functions: concentration of H “Fast” coordinate

Visualizing functions: concentration of H 2 “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 Physics Reports 396, pp Gorban A, Karlin I, Zinovyev A. Invariant Grids for Reaction Kinetics Physica A, V.333, pp

People Doctor Iliya Karlin ETH, Zurich Professor Alexander Gorban University of Leicester, UK