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Nonlinear Dynamics and Non-equilibrium Thermodynamics in
Mesoscopic Chemical SystemsZhonghuai Hou ()ShanghaiTACC2008
Email: [email protected] of Chemical PhysicsHefei
National Lab for Physical Science at MicroscaleUniversity of
Science & Technology of Ch`ina (USTC)
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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
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OutlineIntroduction Noise effect on Nonlinear Dynamics - Noise
Induced Oscillation - Optimal Size Effect - Stochastic Normal Form
TheoryNon-equilibrium Thermodynamics - Entropy Production -
Fluctuation Theorem Conclusions
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Nonlinear Chemical Dynamicsfar-from equilibrium, self-organized,
complex, spatio-temporal structuresOscillation
MultistabilityPatternsWavesChaosCollective behavior involving many
molecular unitsNonequilibrium Statistical Mechanics
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Mesoscopic Reaction SystemSub-cellular reactions- gene
expression- ion-channel gating- calcium signaling Heterogeneous
catalysis- field emitter tips- nanostructured composite surface-
small metal particles
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We already know ... Noise and disorder play constructive roles
in nonlinear systems
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Modeling of Chemical OscillationsMacro- Kinetics: Deterministic,
Cont.N Species, M reaction channels, well-stirred in VReaction j:
Rate: Hopf bifurcation leads to oscillation
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Modeling of Chemical OscillationsMesoscopic Level: Stochastic,
Discrete Master Equation
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New: Noise Induced Oscillation A model system: The
BrusselatorFFTDeterministicStochasticNoisy Oscillation
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Optimal System SizeOptimal System size for mesoscopic chemical
oscillation Z. Hou, H. Xin. ChemPhysChem 5, 407(2004)Best
performanceCoherence Resonance
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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
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Analytical study Main idea Fact: all happens close to the
HBQuestion: common features near HB?Answer: normal form on center
manifold
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Analytical study Stochastic Normal Form(SNF)
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Analytical study Stochastic Averaging (...)Time scale
separation
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Analytical study() Probability distribution of rFokker-Planck
equationStationary distributionMost probable radius Noise induced
oscillation
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Analytical study() Auto-correlation function
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Analytical study() Power spectrum and SNROptimal system
size:
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Analytical study() Universalnear HBSystem DependentChemPhysChem
7, 1520(July 2006) ; J. Phys.Chem.A 111, 11500(Nov. 2007); New J.
Phys. 9, 403(Nov. 2007) ;
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Entropy Production?Macroscopic Level: Nonequilibrium Statistical
ThermodynamicsI. Prigogine 1970s
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Entropy Production?Mesoscopic Level: Stochastic
ThermodynamicsLuo,Nicolis 1984; P.Gaspard 2004
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Entropy Production?Single Trajectory Level: Path
thermodynamicsU. Seifert, PRL 2005 A Random TrajectoryTrajectory
EntropyTotal Entropy Change
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Fluctuation Theorems !Integrate FTDetailed FT(NESS)
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BrusselatorRandom Path (State Space)Microscopic
Reversibility
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BrusselatorDetailed FT and the 2nd law
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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)
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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
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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. )