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

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

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

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  Mesoscopic Level: Stochastic, Discrete Master Equation Kinetic Monte Carlo Simulation (KMC) Gillespie ’ s algorithm Exactly Approximately Deterministic kinetic equation Internal Noise

9. New: Noise Induced Oscillation FFT  A model system: The Brusselator DeterministicStochastic Noisy Oscillation

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

11. Seems to be common …  Internal Noise Stochastic Resonance in a Circadian Clock System J.Chem.Phys. 119, 11508(2003)  Optimal Particle Size for Rate Oscillation in CO Oxidation on Nanometer-Sized Palladium(Pd) Particles J.Phys.Chem.B 108, 17796(2004)  Internal Noise Stochastic Resonance of synthetic gene network Chem.Phys.Lett. 401,307(2005)  Effects of Internal Noise for rate oscillations during CO oxidation on platinum(Pt) surfaces J.Chem.Phys. 122, 134708(2005)  System size bi-resonance for intracellular calcium signaling ChemPhysChem 5, 1041(2004)  Double-System-Size resonance for spiking activity of coupled HH neurons ChemPhysChem 5, 1602(2004) ? Common mechanism Analytical Study

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 U. Seifert, PRL 2005 A Random Trajectory Trajectory Entropy Total Entropy Change

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

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

24. Brusselator  Detailed FT and the 2 nd law

25. Scaling of Entropy Production  System Size Dependence Simulation SNF Theory 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