侯中怀 2009年4月 成都 中国科技大学化学物理系 合肥微尺度科学国家实验室 Nonequilibrium Thermodynamics in Mesoscopic Oscillation Systems (介观化学振荡体系的非平衡热力学) 侯中怀 2009年4月 成都 中国科技大学化学物理系 合肥微尺度科学国家实验室

非平衡 非线性 复杂性 Our Research Interest 非平衡非线性化学动力学 介观化学体系非平衡统计力学 复杂化学体系多尺度理论方法 复杂网络动力学 非平衡 非线性 复杂性

? 不可逆性佯谬 小体系热力学性质！ 宏观体系 微观运动 时间箭头 时间可逆 Solid Clusters 1~10nm Magnetic Domains < 300nm 小体系热力学性质！ Quantum Dots 2~100nm Ion channels Molecular Motors 2~100nm Subcellular reactions

小体系非平衡热力学 非平衡涨落~均值 , 宏观热力学失效 热力学量是随机量, 分布决定性质 实例：拉伸大分子 Protocol：X(t) 不同次实验，W,Q与U有显著涨落 分布P(W)和P(Q)决定体系性质 Physics Today, 58, 43, July 2005;

涨落定理(Fluctuation Theorem) Nonequilibrium Steady States Valid beyond linear response Second Law: More likely to deliver heat than absorb P(+)/P(-) grows exponentially with size and time For small system and short time, capture heat from bath is possible (Molecular motor) Adv. In Phys. 51, 1529(2002); Annu. Rev. Phys. Chem. 59, 603(2008); ……

随机热力学(Stochastic thermodynamics) Stochastic process(Single path based) A Random Trajectory Trajectory Entropy Total Entropy Change Fluctuation Theorems 第一定律 Second Law

许多应用…… Prof. Udo Seifert Probing molecular free energy landscapes by periodic loading PRL(2004) Entropy production along a stochastic trajectory and an integral fluctuation theorem , PRL (2005) Experimental test of the fluctuation theorem for a driven two-level system with time-dependent rates, PRL (2005) Thermodynamics of a colloidal particle in a time-dependent non-harmonic potential, PRL(2006) Measurement of stochastic entropy production, PRL(2006) Optimal Finite-Time Processes （最小功）In Stochastic Thermodynamics, PRL(2007) Stochastic thermodynamics of chemical reaction networks, JCP(2007) Role of external flow and frame invariance in stochastic thermodynamics, PRL(2008) Recent Review: EPJB(2008) Prof. Udo Seifert

Nonequilibrium Statistical Mechanics 化学振荡：非平衡自组织行为 Oscillation：Temporally Periodic Variations of Concentrations/Numbers Synthetic Gene Oscillator CO+O2 Rate Oscillation 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

Hopf bifurcation leads to oscillation 宏观体系：确定性唯象方程 N Species, M reaction channels, well-stirred in V Reaction j: Rate: Macro- Kinetics: Deterministic, Cont. Hopf bifurcation leads to oscillation

介观体系：随机过程理论 Mesoscopic Level: Stochastic, Discrete Master Equation Kinetic Monte Carlo Simulation (KMC) Gillespie’s algorithm Exactly Approximately Fluctuation ! Deterministic kinetic equation

内涨落效应：噪声诱导振荡 A model system: The Brusselator Stochastic Deterministic FFT Noisy Oscillation

最佳尺度效应 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. Brusselator: CPC 2004; Circadian clock: JCP 2003; Calcium Oscillation: CPC 2004, CPC 2005, PRE 2005, PCCP 2006; Gene network: CPL ; Surface Catalytic Reaction: JCP 2005; JPCA 2005, JPCA 2007; Neuron Network: CPC 2004, CPC 2006; ……

问题：介观振荡体系非平衡热力学？ 小体系非平衡热力学？ 涨落定理？ 振荡特性？ 分岔行为？ …… N, V (Small) 小体系 远离平衡 涨落显著 随机过程 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.

不可逆Brusselator体系 宏观：Hopf分岔 介观：随机振荡 微观：态空间随机行走

随机轨线及轨线熵 主方程： 随机轨线： 轨线熵： 熵变： 动力学不可逆性（动力学耗散）

随机振荡的熵变及分布 熵变的分布： 随机振荡（闭轨）： Hopf 分岔对总熵变的分布影响不大 存在总熵变小于0 的轨线(违反第二定律的事件) 总熵变的平均值大于0,第二定律满足

涨落定理

熵产生的标度律 Hopf 分岔的影响？ 熵产生 介观动力学分岔的随机热力学特征？ 分岔后 分岔前 Entropy production and fluctuation theorem along a stochastic limit cycle T Xiao, Z. Hou, H. Xin. J. Chem. Phys. 129, 114508(Sep 2008)

任意介观振荡体系 化学朗之万方程(CLE)描述 相应Fokker-Planck方程

轨线熵和熵产生 轨线熵 熵产生动力学耗散 总熵变 系统熵变 路径积分 介质熵变 平均熵产生

Hopf分岔点附近：随机范式理论 Effect of internal noise in mesoscopic chemical systems near Hopf bifurcation T Xiao, Z. Hou, H. Xin. New J. Phys. 9, 407(Nov 2007)

熵产生：理论结果

标度律：理论解释

普适物理图像 FT Holds Stochastic Thermodynamics in mesoscopic chemical oscillation systems T Xiao, Z. Hou, H. Xin. J. Phys. Chem. B (In press)

普适性：介观Hopf分岔的随机热力学特征！ 小结 Brusselator体系随机热力学 1. 微观态可逆涨落定理(FT) 2. 分岔点：熵产生分布和FT无定性变化 3. 分岔点前后：熵产生标度律突变 任意介观振荡体系 1. CLE水平：熵产生表达式 2. 分岔点：随机范式理论标度律 普适性：介观Hopf分岔的随机热力学特征！

Thank you ! 致谢 Detail work: Dr. Tiejun Xiao (肖铁军), USTC Discussions: Prof. Yijin Yan, HKUST Funding: National Science Foundation of China Thank you !