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Invariant grids: method of complexity reduction in reaction networksAndrei ZinovyevInstitut Curie, ParisInstitut des Hautes tudes Scientifiques
European conference on complex systems, Paris, November 2005
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Stoichiometric equationsas1A1 + + asnAn bs1A1 + + bsnAnn number of species, s number of reactionsc1c2cn
European conference on complex systems, Paris, November 2005
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What is Model Reduction?1 Shorten list of species eliminate some create integrated components2 Shorten list of reactions eliminate some freeze fast reactions3 Decompose motion into fast and slow
European conference on complex systems, Paris, November 2005
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Approaching steady state
European conference on complex systems, Paris, November 2005
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Positively Invariant ManifoldSteady statefast motionslow motionWhy Invariant? once the point on the manifold, the trajectory will stay on it until the equilibriumW
European conference on complex systems, Paris, November 2005
- Why for do we need invariant manifold?Model reduction:Macroscopic system descriptionxRN detailed description yRm macroscopic description. m
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Why for do we need invariant manifold?Dynamics visualization
European conference on complex systems, Paris, November 2005
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Other useful non-invariant manifoldsQuasy steady-stateFast variables are steady
Quasi-equilibriumManifolds maximizing entropy
Intrinsic low-dimensional manifoldDecomposition of Jacobian fields
European conference on complex systems, Paris, November 2005
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Projector Pc on (some) manifoldinduces new (reduced) dynamicsJinduced dynamicsPc Jtangent space = (1-Pc)J - invariance defectWTxW
European conference on complex systems, Paris, November 2005
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Quasi-equilibrium manifoldis not necessarily invariantentropy Smaxmacroscopic (reduced) variables
European conference on complex systems, Paris, November 2005
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Class of dissipative systemscGceqLyapunov function
European conference on complex systems, Paris, November 2005
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Thermodynamic projectorJPc JThe induced dynamics is dissipative only if
European conference on complex systems, Paris, November 2005
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Correction of invariance defect0.20.40.60.81.00.050.100.150.20C1C3equilibriuminvariant manifoldinitial approximationcorrections(1-Pc)J = 0invariance equationNewton iterations
European conference on complex systems, Paris, November 2005
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Invariant gridEQUILIBRIUMtangent spacetangent spaceJinvariance defect is corrected for every node independently
European conference on complex systems, Paris, November 2005
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Invariant grid
European conference on complex systems, Paris, November 2005
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Growing Invariant FlagPhase space
European conference on complex systems, Paris, November 2005
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Entropic scalar product012-1-2equilibriumnatural parameterentropy
European conference on complex systems, Paris, November 2005
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Hydrogen burning model reaction1: H2 2H 2: O2 2O 3: H2O H + OH 4: H2 + O H + OH5: O2 + H O + OH6: H2 + O H2O Conservation laws: 2cH2 +2cH2O+cH+cOH = bH2cO2+cH2O+cO+cOH = bO
European conference on complex systems, Paris, November 2005
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One-dimensional dynamicsequilibrium
European conference on complex systems, Paris, November 2005
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Separation of timesl is the eigen valueof symmetrised
matrix
European conference on complex systems, Paris, November 2005
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Two-dimensional dynamics
European conference on complex systems, Paris, November 2005
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Visualizing functions:concentration of HFast coordinate
European conference on complex systems, Paris, November 2005
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Visualizing functions:concentration of H2Slow coordinate
European conference on complex systems, Paris, November 2005
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Visualizing functions:concentration of OH
European conference on complex systems, Paris, November 2005
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Visualizing functions:Entropy and entropy productionEntropyEntropy production
European conference on complex systems, Paris, November 2005
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Visualizing functions:Separation of relaxation times2/13/2
European conference on complex systems, Paris, November 2005
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Open system as closed system in a flowflow Entropy does not increase everywhere
Non-uniqueness of stationary states, auto-oscillations, etc.
inertial manifold often exists
European conference on complex systems, Paris, November 2005
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Zero-order approximationConstruct the invariant manifold W for W
European conference on complex systems, Paris, November 2005
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First-order approximationNew invariance equation:Fast and slow flow:W
European conference on complex systems, Paris, November 2005
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ConclusionsInvariant grids: constructive method for chemical kineticsclass of dissipative systemsextension to open systems
Use of thermodynamics: metrics in the phase spaceunique thermodynamic projector
Possibility to visualize and explore system dynamics globally
European conference on complex systems, Paris, November 2005
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PapersGorban 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
European conference on complex systems, Paris, November 2005
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PeopleDoctor Iliya Karlin ETH, Zurich
Professor Alexander GorbanUniversity of Leicester, UK
European conference on complex systems, Paris, November 2005