Department of Physics and Astronomy and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794, USA.
J Chem Phys. 2013 Sep 28;139(12):121920. doi: 10.1063/1.4816376.
We established a potential and flux field landscape theory to quantify the global stability and dynamics of general spatially dependent non-equilibrium deterministic and stochastic systems. We extended our potential and flux landscape theory for spatially independent non-equilibrium stochastic systems described by Fokker-Planck equations to spatially dependent stochastic systems governed by general functional Fokker-Planck equations as well as functional Kramers-Moyal equations derived from master equations. Our general theory is applied to reaction-diffusion systems. For equilibrium spatially dependent systems with detailed balance, the potential field landscape alone, defined in terms of the steady state probability distribution functional, determines the global stability and dynamics of the system. The global stability of the system is closely related to the topography of the potential field landscape in terms of the basins of attraction and barrier heights in the field configuration state space. The effective driving force of the system is generated by the functional gradient of the potential field alone. For non-equilibrium spatially dependent systems, the curl probability flux field is indispensable in breaking detailed balance and creating non-equilibrium condition for the system. A complete characterization of the non-equilibrium dynamics of the spatially dependent system requires both the potential field and the curl probability flux field. While the non-equilibrium potential field landscape attracts the system down along the functional gradient similar to an electron moving in an electric field, the non-equilibrium flux field drives the system in a curly way similar to an electron moving in a magnetic field. In the small fluctuation limit, the intrinsic potential field as the small fluctuation limit of the potential field for spatially dependent non-equilibrium systems, which is closely related to the steady state probability distribution functional, is found to be a Lyapunov functional of the deterministic spatially dependent system. Therefore, the intrinsic potential landscape can characterize the global stability of the deterministic system. The relative entropy functional of the stochastic spatially dependent non-equilibrium system is found to be the Lyapunov functional of the stochastic dynamics of the system. Therefore, the relative entropy functional quantifies the global stability of the stochastic system with finite fluctuations. Our theory offers an alternative general approach to other field-theoretic techniques, to study the global stability and dynamics of spatially dependent non-equilibrium field systems. It can be applied to many physical, chemical, and biological spatially dependent non-equilibrium systems.
我们建立了一个势能和通量场景观理论，用于量化一般空间相关非平衡确定性和随机系统的全局稳定性和动力学。我们将我们的势能和通量景观理论从描述福克-普朗克方程的空间独立非平衡随机系统扩展到由一般功能福克-普朗克方程以及从主方程导出的功能克拉默斯-莫亚尔方程控制的空间相关随机系统。我们的一般理论适用于反应扩散系统。对于具有详细平衡的平衡空间相关系统，仅势能场景观，根据稳态概率分布泛函定义，就可以确定系统的全局稳定性和动力学。系统的全局稳定性与势能场景观的地形密切相关，体现在场构型状态空间中的吸引盆地和势垒高度上。系统的有效驱动力仅由势能场的功能梯度产生。对于非平衡空间相关系统，旋度概率通量场在打破详细平衡和为系统创造非平衡条件方面是不可或缺的。空间相关系统的非平衡动力学的完整特征需要同时考虑势能场和旋度概率通量场。虽然非平衡势能场景观沿着功能梯度吸引系统，类似于电子在电场中运动，但非平衡通量场以卷曲的方式驱动系统，类似于电子在磁场中运动。在小波动极限下，作为空间相关非平衡系统小波动极限的固有势能场与稳态概率分布泛函密切相关，被发现是空间相关确定性系统的李雅普诺夫函数。因此，固有势能景观可以表征确定性系统的全局稳定性。随机空间相关非平衡系统的相对熵泛函被发现是系统随机动力学的李雅普诺夫函数。因此，相对熵泛函量化了具有有限波动的随机系统的全局稳定性。我们的理论为研究空间相关非平衡场系统的全局稳定性和动力学提供了一种替代的一般方法，可以应用于许多物理、化学和生物空间相关非平衡系统。
