Nonlinear Dynamics and Non-equilibrium Thermodynamics in Mesoscopic Chemical Systems

1. Nonlinear Dynamics and Non-equilibrium Thermodynamics in Mesoscopic Chemical Systems Zhonghuai Hou (侯中怀) Shanghai，TACC2008 Email: hzhlj@ustc.edu.cn Department of Chemical Physics Hefei National Lab for Physical Science at Microscale University of Science & Technology of Ch`ina (USTC)

2. Our Research Interests • Nonlinear Dynamics in Mesoscopic Chemical Systems • Dynamics of/on Complex Networks • Nonequilibrium Thermodynamics of Small Systems (Fluctuation Theorem) • Mesoscopic Modeling of Complex Systems Nonequilibrium +Nonlinear+ Complexity

3. Outline • Introduction • Noise effect on Nonlinear Dynamics - Noise Induced Oscillation - Optimal Size Effect - Stochastic Normal Form Theory • Non-equilibrium Thermodynamics - Entropy Production - Fluctuation Theorem • Conclusions

4. Nonlinear Chemical Dynamics Stationary spatial structures in reaction-diffusion systems Two or more stable states under same external constraints Travelling/Target/Spiral/Soliton … waves Temporally Periodic Variations of Concentrations Aperiodic/Initial condition sensitivity/strange attractor… Strange Attractor The Lorenz System Chemical turbulence CO+O2 on Pt Surface Science 2001 Turing Pattern BZ Reaction System PNAS 2003 Synthetic transcriptional oscillator (Repressilator) Nature 2002 Calcium Spiral Wave in Cardiac Tissues Nature 1998 Reactive/Inactive bistabe CO+O2 on Pt filed tip PRL1999 Genetic Toggle Switch In E. Coli Nature 2000 Cellular Pattern CO Oxidation on Pt PRL 2001 Rate Oscillation CO+O2 Nano-particle Catal.Today 2003 PEEM Image CO Oxidation on Pt PRL 1995 • far-from equilibrium, self-organized, complex, spatio-temporal structures • Oscillation • Multistability • Patterns • Waves • Chaos Collective behavior involving many molecular units Nonequilibrium Statistical Mechanics

5. Mesoscopic Reaction System Molecular Fluctuation N, V (Small) ? Chemical Oscillation Regularity Stochasticity Nonlinear Chemical Dynamics • Heterogeneous catalysis - field emitter tips - nanostructured composite surface - small metal particles • Sub-cellular reactions - gene expression - ion-channel gating - calcium signaling ……

6. We already know ... • Noise Induced Pattern Transition • Disorder sustained spiral waves • Taming Chaos by Topological Disorder • Ordering Bursting Chaos in Neuron Networks • M. Wang, Z.Hou, H.Xin. ChemPhysChem 7，579( 2006); Z.Hou, et al., PRL 81, 2854 (1998) Z.Hou, et al., PRL 89, 280601 (2002) F. Qi, Z.Hou, H. Xin, PRL 91, 064102 (2003) • Noise and disorder play constructive roles in nonlinear systems

7. Modeling of Chemical Oscillations • Macro- Kinetics: Deterministic, Cont. N Species, M reaction channels, well-stirred in V Reaction j: Rate: Hopf bifurcation leads to oscillation

8. Modeling of Chemical Oscillations Exactly Kinetic Monte Carlo Simulation (KMC) Gillespie’s algorithm Approximately Internal Noise Deterministic kinetic equation • Mesoscopic Level: Stochastic, Discrete Master Equation

9. New: Noise Induced Oscillation • A model system: The Brusselator Stochastic Deterministic FFT Noisy Oscillation

10. Optimal System Size Best performance Coherence Resonance Optimal System size for mesoscopic chemical oscillation Z. Hou, H. Xin. ChemPhysChem 5, 407(2004)

11. Seems to be common … • Internal Noise Stochastic Resonance in a Circadian Clock SystemJ.Chem.Phys.119, 11508(2003) ? Common mechanism • System size bi-resonance for intracellularcalcium signaling ChemPhysChem 5, 1041(2004) • Double-System-Size resonance for spiking activity of coupledHHneurons ChemPhysChem 5, 1602(2004) • Optimal Particle Size for Rate Oscillation in COOxidationonNanometer-SizedPalladium(Pd) Particles J.Phys.Chem.B 108, 17796(2004) • Effects of Internal Noise for rate oscillations during CO oxidation on platinum(Pt) surfaces J.Chem.Phys.122, 134708(2005) Analytical Study • Internal Noise Stochastic Resonance of syntheticgenenetwork Chem.Phys.Lett. 401,307(2005)

12. Analytical study • Main idea Fact: all happens close to the HB Question: common features near HB? Answer: normal form on center manifold

13. Analytical study • Stochastic Normal Form(SNF)

14. Analytical study • Stochastic Averaging (...) Time scale separation

15. Analytical study(…) • Probability distribution of r Fokker-Planck equation Stationary distribution Most probable radius Noise induced oscillation

16. Analytical study(…) • Auto-correlation function

17. Analytical study(…) • Power spectrum and SNR Optimal system size:

18. Analytical study(…) Universal near HB System Dependent ChemPhysChem 7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007); New J. Phys. 9, 403(Nov. 2007) ;

19. Entropy Production? • Macroscopic Level: Nonequilibrium Statistical Thermodynamics I. Prigogine 1970s

20. Entropy Production? • Mesoscopic Level: Stochastic Thermodynamics Luo,Nicolis 1984; P.Gaspard 2004

21. Entropy Production? • Single Trajectory Level: Path thermodynamics A Random Trajectory Trajectory Entropy Total Entropy Change U. Seifert, PRL 2005

22. Fluctuation Theorems ! • Integrate FT • Detailed FT(NESS)

23. Brusselator • Random Path (State Space) ‘Microscopic’ Reversibility

24. Brusselator • Detailed FT and the 2nd law

25. Scaling of Entropy Production • System Size Dependence SNF Theory Simulation Before bifurcation: Constant value After bifurcation: Linear increase Entropy production and fluctuation theorem along a stochastic limit cycle T Xiao, Z. Hou, H. Xin. J. Chem. Phys. 129, 114508(2008)

26. Conclusion • Nonlinear dynamics - Noise induced oscillation is observed - Optimal System Size exists - Stochastic normal form theory works • Nonequilibrium thermodynamics - FT holds far from equilibrium - Scaling of Entropy production characterize noisy oscillation

27. Acknowledgements Supported by: National science foundation (NSF) Thank you