Nonlinear Dynamics and Non-equilibrium Thermodynamics in Mesoscopic Chemical SystemsZhonghuai Hou ()ShanghaiTACC2008

Email: hzhlj@ustc.edu.cnDepartment of Chemical PhysicsHefei National Lab for Physical Science at MicroscaleUniversity of Science & Technology of Ch`ina (USTC)

Our Research InterestsNonlinear Dynamics in Mesoscopic Chemical SystemsDynamics of/on Complex Networks Nonequilibrium Thermodynamics of Small Systems (Fluctuation Theorem)Mesoscopic Modeling of Complex Systems Nonequilibrium +Nonlinear+ Complexity

OutlineIntroduction Noise effect on Nonlinear Dynamics - Noise Induced Oscillation - Optimal Size Effect - Stochastic Normal Form TheoryNon-equilibrium Thermodynamics - Entropy Production - Fluctuation Theorem Conclusions

Nonlinear Chemical Dynamicsfar-from equilibrium, self-organized, complex, spatio-temporal structuresOscillation MultistabilityPatternsWavesChaosCollective behavior involving many molecular unitsNonequilibrium Statistical Mechanics

Mesoscopic Reaction SystemSub-cellular reactions- gene expression- ion-channel gating- calcium signaling Heterogeneous catalysis- field emitter tips- nanostructured composite surface- small metal particles

We already know ... Noise and disorder play constructive roles in nonlinear systems

Modeling of Chemical OscillationsMacro- Kinetics: Deterministic, Cont.N Species, M reaction channels, well-stirred in VReaction j: Rate: Hopf bifurcation leads to oscillation

Modeling of Chemical OscillationsMesoscopic Level: Stochastic, Discrete Master Equation

New: Noise Induced Oscillation A model system: The BrusselatorFFTDeterministicStochasticNoisy Oscillation

Optimal System SizeOptimal System size for mesoscopic chemical oscillation Z. Hou, H. Xin. ChemPhysChem 5, 407(2004)Best performanceCoherence Resonance

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 mechanismAnalytical Study

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

Analytical study Stochastic Normal Form(SNF)

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

Analytical study() Probability distribution of rFokker-Planck equationStationary distributionMost probable radius Noise induced oscillation

Analytical study() Auto-correlation function

Analytical study() Power spectrum and SNROptimal system size:

Analytical study() Universalnear HBSystem DependentChemPhysChem 7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007); New J. Phys. 9, 403(Nov. 2007) ;

Entropy Production?Macroscopic Level: Nonequilibrium Statistical ThermodynamicsI. Prigogine 1970s

Entropy Production?Mesoscopic Level: Stochastic ThermodynamicsLuo,Nicolis 1984; P.Gaspard 2004

Entropy Production?Single Trajectory Level: Path thermodynamicsU. Seifert, PRL 2005 A Random TrajectoryTrajectory EntropyTotal Entropy Change

Fluctuation Theorems !Integrate FTDetailed FT(NESS)

BrusselatorRandom Path (State Space)Microscopic Reversibility

BrusselatorDetailed FT and the 2nd law

Scaling of Entropy ProductionSystem Size DependenceSimulationSNF TheoryBefore bifurcation: Constant valueAfter bifurcation: Linear increaseEntropy production and fluctuation theorem along a stochastic limit cycle T Xiao, Z. Hou, H. Xin. J. Chem. Phys. 129, 114508(2008)

ConclusionNonlinear 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

AcknowledgementsSupported by: National science foundation (NSF)Thank you

Nonlinear chemical dynamics is a field to study far from equilibrium. Typical behaviors include Oscillation Mul..PatternsDifferent 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.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.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.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.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. 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.

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. )