马氏过程动态行为研究及应用

Until now, the temporally varying dynamic behavior of Markov processes has been rarely studied. However, for many interesting phenomena in statistical physics and biological physics, we mainly concern the dynamic behavior of the system, such as the fluctuations of nonequilibrium systems, the adaptation of receptors and the spontaneous oscillations of membrane potentials in cell signaling networks. In this project, the theory of Markov processes will be applied to research the fluctuations of non-equilibrium systems. We will generalize the well-known fluctuation-dissipation theorem to systems far from equilibrium obeying Markovian dynamics. We will study its relationship with entropy production, free energy dissipation and housekeeping heat, all of which are very important physical quantities in thermodynamics. We will prove fluctuation theorems for sample free energy dissipation and sample housekeeping heat. In this project, we will also use inhomogeneous Markov processes and coupled Markov processes, respectively, to model the adaptation of receptors and the spontaneous oscillations of membrane potentials in cell signaling networks. The dynamic behavior of these processes will be studied and then applied to explain the mechanisms of the phenomena of adaptation and spontaneous oscillations. We will emphasize the role played by the nonequilibrium characterizations, such as circulations, of the stochastic model. These problems will be studied in general situations, that is to say, we will research the dynamic behavior of general Markov processes and the stochastic resonance of coupled Markov processes.

至今人们对马氏过程随时间变化的动态行为的研究还相当缺乏，而许多统计物理、生物物理中的有趣现象所涉及的都是系统的动态行为，如非平衡系统的涨落、生物细胞信号网络中的受体适应性和膜电位的自发随机振荡。本项目将以马氏过程作为非平衡系统的数学模型，推导其涨落耗散定理，研究它与热力学中的熵产生、自由能耗散、"家务管理"热等物理量的关系；证明样本自由能耗散、样本"家务管理"热等物理量满足涨落定理。本项目还将利用非时齐马氏过程和耦合马氏过程（即随机切换的马氏过程）作为基本模型来研究细胞信号网络中的受体适应性和膜电位的自发振荡现象，建立这些现象的随机数学模型，分析相应马氏过程的动态行为，阐明受体适应性和膜电位自发振荡的发生机制，研究随机模型的非平衡特征（如环流）对这些现象发生与否所起的作用。我们还计划将上述两个具体问题抽象化，在尽可能一般的框架下研究马氏过程的动态行为和耦合马氏过程的随机共振行为。

到现在为止人们对马氏过程随时间变化的动态行为的研究还相当缺乏，而许多统计物理、生物物理中的有趣现象涉及的都是系统的动态行为，如非平衡系统的涨落、生物细胞信号网络中的受体适应性等。本项目以马氏过程作为非平衡系统的数学模型，研究其动态行为。对常返马氏链和圆周上的扩散过程，我们新发现并证明了一种环对称性，证明了样本环流满足大偏差原理以及多种形式的涨落定理。我们对非时齐马氏链建立了涨落耗散定理的严格数学理论。我们建立了钙离子通道IP3R的非平衡变构模型，模型能相当好地拟合实验数据，并能产生与实验相符的定态与动态行为。我们利用马氏链模型研究了生物系统的一种被称为“超调”（也叫生化适应性）的动力学行为，从理论上揭示了生命系统的超调(适应)现象是消耗能量的非平衡动态行为。我们在分子水平上提出了多稳态细菌系统的一种统一的随机模型，给出了相同基因型细菌群体内表型多样性、表型切换与对冲的数学解释。这些研究成果进一步丰富发展了非平衡系统的严格数学理论，并提供了一些应用这一理论解决生物物理等广泛邻域内实际问题可资借鉴的具体例子。