1. Novel size effect in mesoscopic chemical oscillation 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. 理论化学的三个层面 夸克 美洲豹 电子结构理论 量子力学 结果：反应个体的物理化学性质；基本参数 反应动力学 量子+统计
结果：反应势能面，反应途径，速率常数
统计行为
统计+唯象
结果：宏观量时空演化行为 －实际体系功能 夸克
美洲豹

3. Nonlinear Chemical Dynamics
far-from equilibrium, self-organized, complex, spatio-temporal structures
Oscillation
Multistability
Patterns
Waves
Chaos
Aperiodic/Initial condition sensitivity/strange attractor…
Strange Attractor
The Lorenz System
Chemical turbulence
CO+O2 on Pt Surface
Science 2001
Travelling/Target/Spiral/Soliton … waves
PEEM Image
CO Oxidation on Pt
PRL 1995
Calcium Spiral Wave in Cardiac Tissues
Nature 1998
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
Stationary spatial structures in reaction-diffusion systems
Cellular Pattern
CO Oxidation on Pt
PRL 2001
Turing Pattern
BZ Reaction System
PNAS 2003
Temporally Periodic Variations of Concentrations
Rate Oscillation
CO+O2 Nano-particle Catal.Today 2003
Synthetic transcriptional oscillator (Repressilator)
Nature 2002
Collective behavior involving many molecular units
Nonlinear chemical dynamics is a field to study … far from equilibrium. Typical behaviors include …
Oscillation
Mul..
Patterns…
Different from patterns, waves as non-stationary. There are a few types of waves,
Chaos has now been a quite popular word to us. It lacks a clear definition, nevertheless, they share some features like …
One notes that all these behaviors concerns the spatio-temporal evolution of some ‘macroscopic’ state variable, such as the concentration or total number of molecules of some reaction species, rather than the ‘microscopic’ state, given by the position and momentum of all the molecules.
Nonequilibrium Statistical Mechanics

4. Mesoscopic Reaction System
Heterogeneous catalysis
- field emitter tips
- nanostructured
composite surface
- small metal particles
Sub-cellular reactions
- gene expression
- ion-channel gating
- calcium signaling
… …
N, V
(Small)
Molecular Fluctuation
Recently years, growing attention has been paid to the properties of mesoscopic systems, due to the fast development of nano-science technology and life science. For a mesoscopic reaction system, the total number of molecules N or the system size V is small. An important feature of mesoscopic system is that molecular fluctuation is large, i.e., large deviation from the average exists. Generally, the standard deviation of some macroscopic state variable is proportional to 1 divided by square root of the system size. Examples of such meso-l reaction systems include … and … . As just mentioned, a lot of nonlinear dynamic behavior are observed in these systems. Therefore, our basic question is: how the large molecular fluctuations would influence the nonlinear dynamics in these meso- reaction systems.
Nonlinear Chemical Dynamics ?
Chemical Oscillation
Regularity Stochasticity

5. Why Noise/Disorder ?
Noise and disorder play constructive roles in nonequilibrium systems
Taming Chaos by Topological Disorder
F. Qi, Z.Hou, H. Xin, PRL 91, (2003)
Disorder sustained spiral waves
Z.Hou, et al., PRL 89, (2002)
Noise Induced Pattern Transition
Z.Hou, et al., PRL 81, 2854 (1998)
Another main reason for us to study m-f is: it is now well-known that noise and disorder can often play rather constructive roles in nonlinear dynamic systems. There have been a large amount of literature regarding this issue, and here I will only show a few of our previous works. For example, …
These results give a hint that molecular fluctuations, or internal noise may also have rather interesting, constructive effects.

6. Modeling of Chemical Oscillations
N Species, M reaction channels, well-stirred in V
Reaction j:
Rate:
Macroscopic level: Deterministic, Cont.
Hopf bifurcation leads to oscillation

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

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

9. Optimal System Size Best performance
As in literature, we can use the effective SNR, defined as the H in the PS divided by the half-H width, to measure the performance of the SO. Consequently, we find that the SNR shows a clear maximum at an optimal system size, also indicating an optimal noise level, as shown in the figure. Therefore, we demonstrate an interesting effect of internal noise in mesoscopic chemical oscillation systems.
Optimal System size for mesoscopic chemical oscillation
Z. Hou, H. Xin. ChemPhysChem 5, 407(2004)

10. Seems to be common … ? Common mechanism Analytical Study
Internal Noise Stochastic Resonance in a Circadian Clock System J.Chem.Phys. 119, 11508(2003)
? Common mechanism
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)
Optimal Particle Size for Rate Oscillation in CO Oxidation on Nanometer-Sized Palladium(Pd) Particles
J.Phys.Chem.B 108, 17796(2004)
Such phenomenon seems to be common in mesoscopic oscillation systems. For instance, we have found very similar behaviors in clock, cal..., CO.., gene.... These brings the question to us whether they share some common mechanisms. Very recently, we have been able to perform an analytical study on such behaviors, and the analytical results show rather good agreements with the numerical results.
Effects of Internal Noise for rate oscillations during CO oxidation on platinum(Pt) surfaces
J.Chem.Phys. 122, (2005)
Analytical Study
Internal Noise Stochastic Resonance of synthetic gene network Chem.Phys.Lett. 401,307(2005)

11. Analytical study Fact: all happens close to the HB
Main idea
Fact: all happens close to the HB
Question: common features near HB?
In the following part, I will briefly outline the basic steps of the analytical study. I would not go into the details, but focus on the main idea. Since NIO and optimal size effect are observed near the HB, we believe that they must be related to some common features of the HB. From the dynamical system theory, we know the dynamics near the HB can be described by a normal form on the center manifold, which involves the evolution of the oscillation amplitude and phase angle. Therefore, we believe that ‘normal form’ is the key for analytical study.
Answer: normal form on center manifold

12. Analytical study
Stochastic Normal Form

13. Analytical study
Stochastic Averaging

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

15. Analytical study(…)
Auto-correlation function

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

17. Analytical study(…) Universal near HB System Dependent
Here is a few remarks of the analytical study. Since the analysis is based on the normal form, we can conclude that NIO and Optimal size effect is a universal behavior near the HB. When the system size goes small, i.e., the internal noise goes large, the oscillation amplitude increases, while the correlation time decreases, both monotonically. The SNR shows a maximum at intermediate values of r and tau_c, therefore the best performance is a trade off between oscillation strength and regularity. Though the behavior is common near the HB, the location of the optimal size is system dependent, through the parameter Cr and eps2, which is related to the details of all the elementary reactions. (This analytical work is published very recently. )
System
Dependent
ChemPhysChem 7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007);
New J. Phys. 9, 403(Nov. 2007) ;

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

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

20. Entropy Production? Single Trajectory Level: Dynamic Irreversibity
A Random Trajectory
Trajectory Entropy
Total Entropy Change
U. Seifert, PRL 2005

21. Fluctuation Theorems !
Integrate FT
Detailed FT(NESS)

22. Application to Brusselator
Detailed FT holds

23. Application to Brusselator
System Size Dependence
Simulation
SNF Theory

24. Summary Noise Induced Oscillation  Stochastic Modeling is important
Optimal system size
 Noise + Nonlinearity
Success Analytical Study
 Universality + Underlying mechanism
Fluctuation Theorem
 Single Trajectory Thermodynamics
+ Dynamic Irreversibility

25. Statistical physics in mesoscopic chemical systems
Future directions
How does fluctuation influence the properties of small chemical systems
Thermodynamics
Size dependence
Jarzynski equality
Fluctuation theorems…
Dynamics
Nucleation and growth
Transport/relaxation
Nonlinear dynamics …
Here I would like to take sometime to outlook some related questions in our future studies. The basic question of us is that how …, including thermodynamics and dynamics. For instance, for small systems, ‘size’ is a new parameter, and thermodynamic functions should be size-dependent, hence new thermodynamic relations may be required. This is the main topic of the book written by Terill Hill, “Thermodynamics of small systems’. Also in recent years, the Jarzynsky equation, which relates the free-energy difference of two equilibrium states to the exponential average of irreversible work, has gained much attention, which is also relevant with some so-called FTs. It is interesting to apply these new ideas to small chemical reaction systems, where the system is modeled as stochastic Markov process. We all also interested in how dynamic processes, such as nucleation and growth, transport and relaxation processes, are affected by large molecular fluctuations. What I talked today actually mainly concern the nonlinear dynamics, that is only one small part of the whole question: statistical mechanics of mesoscopic chemical systems. I really hope that we can move forward toward these interesting and important directions in our future works.
Statistical physics in mesoscopic chemical systems

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

27. Details