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Nonlinear Dynamics in Mesoscopic Chemical Systems: Patterns, Waves, and Oscillations

This research delves into the intricate world of nonlinear dynamics in mesoscopic chemical systems, exploring stationary spatial structures and multistability. Key phenomena such as traveling waves, chaotic behavior, and the role of noise in chemical reactions are examined. Significant findings include noise-induced oscillations and optimal system sizes for reaction processes. The paper highlights the importance of stochastic dynamics and molecular fluctuations in achieving sustainable oscillatory behavior, with applications in various fields including heterogeneous catalysis and biological signaling.

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Nonlinear Dynamics in Mesoscopic Chemical Systems: Patterns, Waves, and Oscillations

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  1. Nonlinear Dynamics in Mesoscopic Chemical Systems Zhonghuai Hou (侯中怀) Department of Chemical Physics Hefei National Lab of Physical Science at Microscale University of Science & Technology of China

  2. 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

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

  4. Noise/Disorder • Noise Induced Pattern Transition • Disorder sustained spiral waves • Taming Chaos by Topological Disorder 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 dynamical systems

  5. Stochastic Chemical Kinetics stochastic state variable probability distribution • chemical reactions are essentially stochastic, discrete processes Discrete Brownian Motion of X : Prob. Evolution: Master equation Sample Trajectory: Langevin equation

  6. Chemical Langevin equation (CLE) • Molecular fluctuation (Internal noise) N Species, M reaction channels, well-stirred in V Reaction j: Rate: • Deterministic kinetics for • Each channel contributes independently to internal noise: • Fast numerical simulation

  7. The Brusselator • Deterministic bifurcation Fixed Point: Hopf bifurcation:

  8. Noise Induced Oscillation • Stochastic dynamics FFT

  9. Optimal System Size Optimal System size for mesoscopic chemical oscillation Z. Hou, H. Xin. ChemPhysChem 5, 407(2004)

  10. Seems to be common … • Internal Noise Stochastic Resonance in a Circadian Clock SystemJ.Chem.Phys.119, 11508(2003) • 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) • Internal Noise Stochastic Resonance of syntheticgenenetwork Chem.Phys.Lett. 401,307(2005)

  11. Analytical study • Stochastic Normal Form

  12. Analytical study • Stochastic Averaging

  13. Analytical study • Probability distribution of r Fokker-Planck equation Stationary distribution Most probable radius Noise induced oscillation

  14. Analytical study • Auto-correlation function

  15. Analytical study • Power spectrum and SNR Optimal system size:

  16. Analytical study Universal near HB System Dependent Internal Noise Coherent Resonance for Mesoscopic Chemical oscillations: a Fundamental Study. Z. Hou, … ChemPhysChem 7, 1520(2006)

  17. Summary • In mesoscopic chemical systems, molecular fluctuations can induce oscillation even outside the deterministic oscillatory region • Optimal system size exists, where the noise-induced oscillation shows the best performance, characterized by a maximal SNR, a trade off between strength and regularity • Based on stochastic normal form, analytical studies show rather good agreements with the simulation results, uncovering the mechanism of NIO and OSS

  18. Further questions

  19. Acknowledgements Supported by: National science foundation (NSF) Fok Yin Dong education foundation Thank you

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