摘要
随着生命科学与纳米技术的发展,介观化学体系获得了越来越多的关注。介观尺度上的化学反应可以用离散随机过程来描述,内涨落(分子涨落)是其内禀属性。一般说来,内涨落的强度和体系的尺度或者体系内反应分子总数成反比。在介观化学体系中,内涨落可以达到显著的程度,它可能使得体系的动力学以及热力学性质与宏观相比有着显著差异。研究内涨落如何影响介观体系的动力学以及热力学性质,是当前非平衡统计物理研究中的前沿课题,也是本论文的重要研究内容。本论文着重研究了介观化学体系中的如下两类非平衡统计问题:
●介观化学体系内涨落的动力学尺度效应
人们已经知道,在远离平衡条件下,化学体系可以出现丰富的动力学现象,比如合成基因网络中的蛋白质浓度振荡,心肌细胞中的钙螺旋波,纳米粒子表面催化体系的反应速率振荡、螺旋波以及时空混沌等.既然在介观体系中内涨落不可忽略,那么它对于这些动力学现象会有何影响呢?2004年我们组曾在表面和细胞体系中模拟发现内涨落导致的‘最佳尺度效应',结果表明,内涨落在那些靠近霍普夫分岔点的体系中可以诱导出有效振荡,并且振荡的信噪比在合适的体系尺度下可以达到极大。后来我们在许多体系中都发现了这种现象,但是一直不清楚它的形成条件以及机理。在本论文中,作者利用随机过程理论以及非线性动力学的理论方法,建立了霍普夫分岔点附近的普适的随机范式方程,成功解释了內涨落诱导振荡和最佳尺度效应的机理。在两变量布鲁塞尔体系和三变量生理时钟体系中,我们验证了随机范式方程的有效性(这方面的工作发表在ChemPhyschem以及New J.Phys.上);我们还使用随机范式方程研究了色噪声关联性质对噪声诱导振荡的调控应用(此工作发表在Chaos上)。
●介观化学体系的随机热力学
近年来,小体系的非平衡热力学特别是涨落定理和随机热力学的研究引起了人们的广泛关注。我们将小体系热力学的最新研究进展应用到了介观化学体系中,特别是考察了涨落定理在介观振荡体系中的具体形式,以及霍普夫分岔对熵产生等热力学量的影响.在不可逆布鲁塞尔体系中,基于对主方程的动力学蒙特卡洛模拟,我们检验了沿着轨线的总熵变满足的细致涨落定理,且发现平均熵产生对体系尺度的依赖在霍普夫分岔点前后显示出不同的标度率(此工作发表在J.Chem.Phys.上)。进一步,从描述物种浓度演化的朗之万方程出发,我们研究了任意介观化学振荡体系的随机热力学。根据随机过程的路径积分方法以及随机范式理论,得到了霍普夫分岔点附近熵产生的理论表达式,从理论上证实,熵产生对体系尺度的标度关系以及在霍普夫分岔点前后标度指数发生突变是介观化学振荡体系的普适特性。我们的工作表明有可能在体系的动力学分岔与随机热力学性质之间可以建立起某种关联(这方面的工作即将发表在J.Phvs.Chem.B)。我们还对介观化学反应体系单个反应通道的熵变满足的涨落定理进行了初步的研究。
In the recent years,mesoscopic chemical systems had gained growing attention with the rapid progresses in nanotechnology and life science.Chemical reactions in mesocopic scale are discrete stochastic processes described by master equation,so internal noise or the molecular fluctuation is intrinsic property of chemical systems. A mesoscopic chemical system contains only a small amout of molecular,so the internal noise will be significant,and will affect the dynamics and thermodynamics. In this thesis,we have studied the following two kinds of nonequilibrium statistic problems which are relavent to the internal noise in mesoscopic chemical systems:
●Optimal System Size Effect of Internal Noise on Dynamics
When driven away from equilibrium,chemical systems exhibit aboundant dynamic behaviors,such as the oscillation in gene networks,Calcium spiral wave in the myocardial,reaction rate oscillations,spiral wave and spatio-temporal chaos in nano particle surfaces.Since internal noise is significant in mesoscopic systems,how will it affect such dynamic behaviors? In 2004,our group found a novel phenomenon in the mesoscopic chemical systems that are near Hopf bifurcation,which is called 'Optimal systems size effect',i.e.,internal noise could induce oscillation,and the signal-to-noise-ratio of these noise induced oscillations undergoes a clear maximum under a suitable systems size.After that,we apply ourselves to find its underly mechanism.Herein,we contribute to this issue by theoretic analysis.Starting from chemical Langevin equations,using the normal form theorem and stochastic average method,a unified 'stochastic averaged normal form' is obtained.We had succedully explained the internal noise induced oscillation and optimal system size effect.We had also checked the validity of our method in the Brussel model and a Circadian clock model.We had also used our method to study the effect of noise correlation in controlling the noise induced oscillations.
●Stochastic Thermodynamics
In recent years,nonequilibrium thermodynamics of small systems has gained extensive attention.Of particular interests are the fluctuation theorem and stochastic thermodynamics.Herein,fluctuation theorem and stochastic thermodynamics in mesoscopic oscillate chemical systems are discussed.By direct simulation in the irreversible Brusselator,we had checked the validity of the detailed fluctuation theorem for the entropy producton along a stochastic trajectory,and found that the entropy producton on system size have different scaling on system size when the control parameter is befor or afer Hopf bifurcation.Further more,basing on the chemical Langevin equation,we had studied the stochastic thermodynamics of mesoscopic chemical reaction networks.Using the path intergral and stochastic normal form theory,we had derive explicit theoretical expressions for the mean entropy production in the stationary state,and demonstrated that the scaling law of entropy production on system size and the scaling exponent change abruptly at Hopf bifurcation point is universal in mesoscopic chemical systems,suggesting a relation between dynamic bifurcation and stochastic thermodynamics behaviors.We had also studied the fluctuation theorem of entropy production of single reaction channel.
引文
[1]C.W.Gardiner,‘Handbook of Stochastic Methods for physics,chemistry,and the natural science’,Springer-Verlag(1983)
[2]N.G.Van Kampen,‘Stochastic Processes in Physics and Chemistry’,North-Holland:Amsterdam(1987)
[3]D.T.Gillespie,‘A general method for numerically simulating the stochastic time evolution of coupled chemical reactions’,J.Comput.Phys.22,403 (1976);D.T.Gillespie,‘Exact stochastic simulation of coupled chemical reactions’,J.Phys.Chem.81,2340(1977)
[4]D.T.Gillespie,‘Chemical Langevin Equation’,J.Chem.Phys.113,297(2000)
[5]D.T.Gillespie,‘Approximate accelerated stochastic simulation of chemically reacting systems’,J.Chem.Phys.115,1716(2001)
[6]H.Risken,‘The Fokker-Planck Equation’,Springer,New Tork(1983)
[7]Y.Suchorski,J.Beben,E.W.James,J.W.Evans and R.Imbihl,‘Fluctuation-induced transitions in a bistable surface reaction:Catalytic CO oxidation on a Pt field emitter tip’,Phys.Rev.Lett.82,1907 (1999)
[8]T.S.Gardner,C.R.Cantor and J.J.Collins,‘Construction of a genetic toggle switch in Escherichia coli’,Nature 403,339(2000)
[9]J.Hasty,D.Mcmillen and J.J.Collins,‘Engineered gene circuits’,Nature 420,224(2002)
[10]M.B.Elowitz and S.Leibler,‘A synthetic oscillatory network of transcriptional regulators'Nature 403,335 (2000)
[11]M.M.Slinko,A.A.Ukharskii,N.V.Peskov and N.I.Jaeger,‘Chaos and synchronisation in heterogeneous catalytic systems:CO oxidation over Pd zeolite catalysts’,Catal.Today 70,341(2001)
[12]M.Falcke,‘Reading the patterns in living cells:the physics of Ca~(2+) signaling’,Adv.Phys.,53(3),255-440(2004)
[13]M.Kim et al.,‘Controlling Chemical Turbulence by Global Delayed Feedback:Pattern Formation in Catalytic CO Oxidation on Pt(llO)’,Science 292,1357(2001)
[14]J.P.Gollub and L.S.Langer,‘Pattern Formation in Nonequilibrium Physics’,Rev.Mod.Phys.71,S396(1999)
[15]J.Lechleiter,S.Girard,E.Peralta,and D.Clapham,‘Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes’,Science 252,123(1991)
[16]R.M.Eiswirth,K.Krischer and G.Ertl,‘Nonlinear dynamics in the CO oxidation on Pt single crystal surface’,Appl.Phys.A,51,79(1990)
[17]M.Eiswirth,M.Bar and H.H.Rotermund,‘ Spatiotemporal self organization on isothermal catalysts’,Physica D 84,40(1995)
[18]M.Eiswirth,‘Chaos in Chemistry and Biochemistry’,World Scientific,Singapore (1993)
[19]W.Gerstner and W.M.Kistler,‘Spiking Neuron Models:single neurons,populations,plasticity’,Cambridge University Press(2002)
[20]H.H.Mcadams and A.Arkin,‘It's a noisy business! Genetic regulation at the nanomolar scale’,Trend Genet.15,65(1999)
[21]P.S.Swain,M.B.Elowitz and E.D.Siggia,‘Intrinsic and Extrinsic contributions to stochasticity in gene expression’,Proc.Natl.Acad.Sci.USA 99,12795(2002)
[22]M.B.Elowitz,A.J.Levine,E.D.Siggia and P.S.Swain,‘Stochastic gene expression in a single cell’,Science 297,1183(2002)
[23]J.Paulsson,‘Summing up the noise in gene networks’,Nature 427,415(2004)
[24]C.V.Rao,D.M.Wolf and A.Arkin,‘Control,exploitation and tolerance of intracellular noise’,Nature 420,231(2002)
[25]M.Thattai and A.V.Oudenaarden,‘Intrinsic noise in gene regulatory networks’,Proc.Natl.Acad.Sci.USA 98,8614(2001)
[26]HH.McAdams and A.Adam,‘Stochastic mechanisms in gene expression’,Proc.Natl.Acad.Sci.USA 94,814(1997)
[27]J.Hasty and J.J.Collins,‘Translating the noise’,Nature Genetics 31,13(2001)
[28]W.J.Blake,M.Kasrn,C.R.Cantor and J.J.Collins,‘Noise in eukaryotic gene expression’,Nature 422,633(2003)
[29]J.C.Dunlap,‘Molecular bases for circadian clocks’,Cell 96,271(1999)
[30]D.Gonze,J.Halloy and A.Goldbeter,‘Robustness of circadian rhythms with respect to molecular noise’,Proc.Natl.Acad.Sci.USA 99,673(2002)
[31]N.Barkai and S.Leibler,‘Circadian clocks limited by noise’,Nature 403,267(2000)
[32]J.M.G Vilar,N.Y.Kueh,N.Barkai and S.Leibler,‘Mechanisms of noise-resistance in genetic oscillators’,Proc.Natl.Acad.Sci.USA 99,5988(2002)
[33]Y.Suchorski,et.al,‘Fluctuation-Induced Transitions in a Bistable Surface Reaction:Catalytic CO Oxidation on a Pt Field Emitter Tip’,Phys.Rev.Lett.82,1907 (1999)
[34]C.Sachs,et.al,‘Spatiotemporal Self-Organization in a Surface Reaction:From the Atomic to the Mesoscopic Scale’,Science 293,1635(2001)
[35]M.M.Slinko,A.A.Ukharskii,N.V.Peskov and N.I.Jaeger,‘Chaos and synchronisation in heterogeneous catalytic systems:CO oxidation over Pd zeolite catalysts’,Catal.Today 70, 341(2001)
[36]N.V.Peskov,M.M.Slinko andN.I.Jaeger,‘Stochastic model of reaction rate oscillations in the CO oxidation on nm-sized palladium particles’,J.Chem.Phys.116,2098(2002)
[37]Z.H.Hou and H.W.Xin,‘Internal noise stochastic resonance in a circadian clock system’,J.Chem.Phys.119,11508(2003)
[38]Z.H.Hou and H.W.Xin,‘Optimal System Size For Mesoscopic Chemical Oscillation’,ChemPhysChem 5,407(2004)
[39]Y.B.Gong,Z.H.Hou and H.W.Xin,‘Optimal Particle Size for Reaction Rate Oscillation in CO Oxidation on Nanometer-Sized Palladium Particles’,J.Phys.Chem.B 108,17796(2004)
[40]Y.B.Gong,Z.H.Hou and H.W.Xin,‘Internal Noise Stochastic Resonance in NO Reduction by CO on Platinum Surfaces’,J.Phys.Chem.A 109,2741(2005)
[41]M.S.Wang,Z.H.Hou and H.W.Xin,‘Internal noise-enhanced phase synchronization of coupled chemical chaotic oscillators’,J.Phys.A 38,145(2005)
[42]Z.W.Wang,Z.H.Hou and H.W.Xin,‘Internal noise stochastic resonance of synthetic gene network’,Chem.Phys.Lett.401,307(2005)
[43]H.Y.Li,Z.H.Hou and H.W.Xin,‘Internal Noise Enhanced Harmonal Signal Detection’,Chem.Phys.Lett.402,444(2005)
[44]Z.H.Hou,T.Rao and H.W.Xin,‘Effects of internal noise for rate oscillations during CO oxidation on platinum surfaces’,.!.Chem.Phys.122,134708(2005)
[45]J.Q.Zhang,Z.H.Hou and H.W.Xin,‘System size bi-resonance for intracellular calcium signaling’,ChemPhysChem 5,1041(2004)
[46]M.S.Wang,Z.H.Hou and H.W.Xin,‘Double-System-Size Resonance for Spiking Activity of Coupled Hodgkin-Huxley Neurons’,ChemPhysChem 5,1602(2004)
[47]Z.H.Hou,J.Q.Zhang and H.W.Xin,‘Two system-size-resonance behaviors for calcium signaling:For optimal cell size and for optimal network size’,Phys.Rev.E 74,031901(2006)
[48]P.Gaspard,‘The correlation time of mesoscopic chemical clocks’,J.Chem.Phys.117,8905(2002)
[49]Hofer,‘Model of intercellular calcium oscillations in hepatocytes:synchronization of heterogeneous cells'.7.Biophysical J.77,1244(1999)
[50]W.Gerstner,W.M.Kistler,‘Spiking Neuron Models:single neurons,populations,plasticity’,Cambridge University Press(2002)
[51]C.Bustamante,J.Liphardt,and F.Ritort,‘The nonequilibrium thermodynamics of small systems’,Physics Today 58(7),43 (2005).
[52]F.Ritort,‘Nonequilibrium fluctuations in small systems:From physics to biology’,arXiv:0705.0455 (2007).
[53]E.H.Feng and G.E.Crooks,‘Length of Time's Arrow’,Phys.Rev.Lett.101,090602(2008)
[54]D.J.Evans,E.G.D.Cohen,and G.P.Morriss,‘Probability of second law violations in shearing steady states’,Phys.Rev.Lett.71,2401 (1993)
[55]D.J.Evans and D.J.Searles,‘Equilibrium microstates which generate second law violating steady states’,Phys.Rev.E 50,1645 (1994)
[56]G.Gallavotti and E.G.D.Cohen,‘Dynamical Ensembles in Nonequilibrium Statistical Mechanics’,Phys.Rev.Lett.74,2694 (1995).
[57]D.J.Evans and D.J.Searles,‘The Fluctuation Theorem’,Adv.Phys.51,1529 (2002)
[58]E.M.Sevick,R.Prabhakar,S.R.Williams,and D.J.Searles,‘Fluctuation Theorems’,Annu.Rev.Phys.Chem.59,603(2008).
[59]J.Kurchan,‘Fluctuation theorem for stochastic dynamics’,J.Phys.A:Math.Gen.31,3719(1998).
[60]J.L.Lebowitz and H.Spohn,‘A Gallavotti-Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics’,J.Stat.Phys.95,333 (1999).
[61]D.J.Searles and D.J.Evans,‘Fluctuation theorem for stochastic systems’,Phys.Rev.E.60,159(1999)
[62]V.Y.Chernyak,M.Chertkov,and C.Jarzynski,‘Path-integral analysis of fluctuation theorems for general Langevin processes’,J.Stat.Mech.P08001(2006).
[63]R.J.Harris and G.M.Schütz,‘Fluctuation theorems for stochastic dynamics’,J.Stat.Mech.:Theor.Exp.P07020 (2007).
[64]C.Jarzynski,‘Nonequilibrium Equality for Free Energy Differences’,Phys.Rev.Lett.78,2690(1997)
[65]C.Jarzynski,‘Equilibrium free-energy differences from nonequilibrium measurements:A master-equation approach’,Phys.Rev.E 56,5018 (1997).
[66]G.E.Crooks,‘Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences’,Phys.Rev.E 60,2721 (1999)
[67]G.E.Crooks,‘Path-ensemble averages in systems driven far from equilibrium’,Phys.Rev.E 61,2361 (2000).
[68]G.Hummer and A.Szabo,‘Free energy reconstruction from nonequilibrium single-molecule pulling experiments’,Proc.Natl.Acad.Sci.U.S.A.98,3658 (2001)
[69]C.Chatelain,‘A temperature-extended Jarzynski relation application to the numerical calculation of surface tension’,J.Stat.Mech.P04011(2007)
[70]O.Braun,A.Hanke,and U.Seifert,‘Probing Molecular Free Energy Landscapes by Periodic Loading’,Phys.Rev.Lett.93,158105 (2004).
[71]H.Oberhofer,C.Dellago,and P.L.Geissler,‘Biased sampling of nonequilibrium trajectories:Can fast switching simulations outperform conventional free energy calculation methods?’,J.Phys.Chem.B 109,6902 (2005).
[72]W.Lechner,H.Oberhofer,and C.Dellago,‘Equilibrium free energies from fasts witching trajectories with large time steps’,J.Chem.Phys.124,044113 (2006).
[73]I.Kosztin,B.Barz,and L.Janosi 'Calculating potentials of mean force and diffusion coefficients from nonequilibrium processes without Jarzynski's equality’,J.Chem.Phys.124,064106(2006)
[74]U.Seifert,‘Entropy production along a stochastic trajectory and an integral fluctuation theorem’,Phys.Rev.Lett.95,040602 (2005)
[75]T.Hatano and S.Sasa,‘Steady-State Thermodynamics of Langevin Systems’,Phys.Rev.Lett.86,3463(2001).
[76]T Speck and U Seifert,‘Integral fluctuation theorem for the housekeeping heat’,J.Phys.A:Math.Gen.38,L581(2005)
[77]D.Andrieux and P.Gaspard,‘Fluctuation theorem and Onsager reciprocity relations’,J.Chem.Phys.121,6167(2004)
[78]R.van Zon and E.G.D.Cohen,‘Stationary and transient work-fluctuation theorems for a dragged Brownian particle’,Phys.Rev.E 67,046102(2003)
[79]T Speck and U Seifert,‘The Jarzynski relation,fluctuation theorems,and stochastic thermodynamics for non-Markovian processes’,J.Stat.Mech.L09002(2007)
[80]T.Ohkuma and T.Ohta,‘Fluctuation theorems for non-linear generalized Langevin systems’,J.Stat.Mech.PI0010(2007)
[81]W.De Roeck,C.Maes,‘A quantum version of free energy—irreversible work relations’,Phys.Rev.E 69,026115(2004)
[82]W.D.Roeck,‘Quantum fluctuation theorem:can we go from micro to meso?’,C.R.Physique 8,674 (2007)
[83]K.Saito and A.Dhar,‘Fluctuation Theorem in Quantum Heat Conduction’,Phys.Rev.Lett.99,180601 (2007)
[84]D.Andrieux and P.Gaspard,‘QuantumWork Relations and Response Theory’,Phys.Rev.Lett.100,230404(2008)
[85]A.Porporato,J.R.Rigby,and E.Daly,‘Irreversibility and Fluctuation Theorem in Stationary Time Series’,Phys.Rev.Lett.98,094101(2007)
[86]F.Ritort,‘Single-molecule experiments in biological physics:methods and applications’,J.Phys.:Condens.Matter 18,R531 (2006).
[87]J.Liphardt,et.al,‘Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of Jarzynski's Equality’,Science 296,1832 (2002).
[88]D.Collin,et.al,‘Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies’,Nature 437,231 (2005).
[89]E.H.Trepagnier,et.al,‘Experimental test of Hatano and Sasa's nonequilibrium steady-state equality’,Proc.Natl.Acad.Sci.U.S.A.101,15038 (2004).
[90]G.M.Wang,et.al,‘Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales’,Phys.Rev.Lett.89,050601(2002).
[91]D.M.Carberry,et.al,‘Fluctuations and Irreversibility:An Experimental Demonstration of a Second-Law-Like Theorem Using a Colloidal Particle Held in an Optical Trap’,Phys.Rev.Lett.92,140601 (2004).
[92]G.M.Wang,et.al,‘Experimental study of the fluctuation theorem in a nonequilibrium steady state’,Phys.Rev.E 71,046142 (2005).
[93]V.Blickle,et.al,‘Thermodynamics of a Colloidal Particle in a Time-Dependent Nonharmonic Potential’,Phys.Rev.Lett.96,070603(2006)
[94]S.Schuler,et.al,‘Experimental Test of the Fluctuation Theorem for a Driven Two-Level System with Time-Dependent Rates’,Phys.Rev.Lett.94,180602 (2005)
[95]C.Tietz,et.al,‘Measurement of Stochastic Entropy Production’,Phys.Rev.Lett.97,050602(2006)
[96]S.Ciliberto,et.al,‘Experimental test of the Gallavotti-Cohen fluctuation theorem in turbulent flows’,Physica A,340,240(2004)
[97]N.Gamier and S.Ciliberto,‘Nonequilibrium fluctuations in a resistor’,Phys.Rev.E 71,060101(R)(2005)
[98]S.Joubaud,N.B.Gamier and S.Ciliberto,‘Fluctuation theorems for harmonic oscillators’,J.Stat.Mech.P09018(2007)
[99]P.Gaspard,‘Fluctuation theorem for nonequilibrium reactions’,J.Chem.Phys.120,8898(2004).
[100]D.Andrieux and P.Gaspard,‘Fluctuation theorems and the nonequilibrium thermodynamics of molecular motors’,Phys.Rev.E 74,011906 (2006)
[101]D.Andrieux and P.Gaspard,‘Fluctuation theorem and mesoscopic chemical clocks’,J.Chem.Phys.128,154506(2008)
[102]U.Seifert,‘Fluctuation theorem for birth-death or chemical master equations with time-dependent rates’,J.Phys.A:Math.Gen.37,L517 (2004).
[103]U.Seifert,‘Fluctuation theorem for a single enzymer or molecular motor’,Europhys.Lett.70,036(2005)
[104]T.Schmiedl and U.Seifert,‘Stochastic thermodynamics of chemical reaction networks’,J.Chem.Phys.126,044101 (2007).
[105]T.Schmiedl,T.Speck and U.Seifert,'Entropy Production for Mechanically or Chemically Driven Biomolecules',J.Stat.Phys.128,77(2007)
[106]W.Min et.al,‘Nonequilibrium Steady State of a Nanometric Biochemical System:Determining the Thermodynamic Driving Force from Single Enzyme Turnover Time Traces’,Nano lett.5,2373(2005)
[107]H.Qian and X.S.Xie,‘Generalized Haldane equation and fluctuation theorem in the steady-state cycle kinetics of single enzymes’ Phys.Rev.E 74,010902(R)(2006)
[108]U.Seifert,‘Stochastic thermodynamics:Principles and perspectives’ Eur.Phys.J.B 64,423(2008)
[1]D.T.Gillespie,'Chemical Langevin Equation',J.Chem.Phys.113,297(2000)
[2]J.Guckenheimer and P.Holmes,'Nonlinear Oscillations,Dynamical Systems,and Bifurcations of Vertor Fields',Springer-Verlag,New York(1983)
[3]R.Seydel,'Practical Bifurcation and Stability Analysis:From Equilibrium to Chaos'(Second Edition),Springer-Verlag,New York(1994)
[4]C.W.Gardiner,Handbook of Stochastic Methods for Physics,Chemistry and the Natural Sciences,Springer,Heidelberg(1983).
[5]L.Arnold,N.S.Namachchivaya and K.R.Schenk-HoppR,'Toward an understanding of stochastic Hopf bifurcation:A case study',Int.J.Bifur.Chaos 6,1947(1996)
[6]J.B.Roberts and P.D.Spanos,'Stochastic averaging:An approximate method of solving random vibration problems',Int.J.Non-linear.Mech.21,111-134(1986)
[7]A.D.Wentzell and M.I.Freidlin,Random Perturbations of Dynamical Systems(New York:Springer,1998).
[8]N.S.Namachchivaya,and S.T.Ariaratnam,'Stochastically perturbed Hopf bifurcation.'Int.J.Nonlin.Mech.22,363(1987)
[9]W.Vance and J.Ross,'Fluctuations near limit cycles in chemical reaction systems',J.Chem.Phys.105,479(1996)
[10]P.Gaspard,'The correlation time of mesoscopic chemical clocks',J.Chem.Phys.117,8905(2002)
[11]N.Barkai and S.Leibler,'Circadian clocks limited by noise',Nature 403,267(2000)
[12]J.Ma,T.Xiao,Z.Hou and H.Xin,'Coherence resonance induced by colored noise near Hopf bifurcation',Chaos 18,043116(2008)
[13]陆启韶,‘分岔与奇异性',上海科技出版社(1995)
[14]胡岗,‘随机力与非线性系统',上海科技教育出版社(1994)
[15]P.Jung,A.Neiman,M.K N.Afghan,S.Nadkarni and G.Ullah,'Thermal activation by power-limited coloured noise',New Journal of Physics 7,17(2005).
[1]D.T.Gillespie,‘Exact stochastic simulation of coupled chemical reactions’,J.Phys.Chem.81,2340(1977).
[2]U.Seifert,‘Entropy production along a stochastic trajectory and an integral fluctuation theorem’,Phys.Rev.Lett.95,040602(2005);U.Seifert,‘Stochastic thermodynamics:Principles and perspectives’,Eur.Phys.J.B 64,423(2008)
[3]D.Andrieux and P.Gaspard,‘Fluctuation theorem and mesoscopic chemical clocks’,J.Chem.Phys.128,154506(2008)
[4]Hong Qian,Saveez Saffarian and Elliot L.Elson,‘Fluorescence correlation spectroscopy with high-order and dual-color correlation to probe nonequilibrium steady states’,Proc.Natl Acad.Sci.USA.99,10376(2002)
[5]Zhonghuai Hou and Houwen Xin,‘Internal noise stochastic resonance in a circadian clock system’,J.Chem.Phys.119,11508(2003);Zhonghuai Hou and Houwen Xin,‘Optimal System Size for Mesoscopic Chemical Oscillation’,ChemPhysChem.5,407(2004)
[6]D.T.Gillespie,‘Chemical Langevin Equation’,J.Chem.Phys.113,297(2000)
[7]D.Andrieux,et.al,‘Entropy Production and Time Asymmetry in Nonequilibrium Fluctuations’,Phys.Rev.Lett.98,150601(2007)
[8]V.Y.Chernyak,M.Chertkov,and C.Jarzynski,‘Path-integral analysis of fluctuation theorems for general Langevin processes’,J.Stat.Mech.P08001(2006)
[9]Tiejun Xiao,Juan Ma,Zhonghuai Hou,and Houwen Xin,‘Effects of internal noise in mesoscopic chemical systems near Hopf bifurcation’,New J.Phys.9,403(2007)
[10]B.D.Hassard,N.D.Kazarinoff and Y.H.Wan,‘Theory and Applications of Hopf bifurcation’,London Mathematical Society Lecture Note Series vol 4(Cambridge:Cambridge University Press 1981)
[11]L.Arnold,N.S.Namachchivaya and K.R.Schenk-Hoppe,‘Toward an understanding of stochastic Hopf bifurcation:a case study’,Int.J.Bifurcat.Chaos 6.1947(1996)
[12]J.B.Roberts and P.D.Spanos,‘Stochastic averaging:An approximate method of solving random vibration problems’,J.Nonlinear Mech.21,111(1986)
[13]D.Gonze,J.Halloy and A.Goldbeter,‘Robustness of circadian rhythms with respect to molecular noise’,Proc.Natl Acad.Sci.USA.99,673(2002);D.Gonze,J.Halloy,P.Gaspard,‘Biochemical clocks and molecular noise:Theoretical study of robustness factors’,J.Chem.Phys.116,10997(2002)
[14]T.Schmiedl and U.Seifert,‘Stochastic thermodynamics of chemical reaction networks’,J.Chem.Phys.126,044101(2007).
[15]D.T.Gillespie,‘Approximate accelerated stochastic simulation of chemically reacting systems’,J.Chem.Phys.115,1716(2001)
[16]J.L.Lebowitz and H.Spohn,‘A Gallavotti-Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics’,J.Stat.Phys.95,333(1999)