Landscape Framework and Global Stability for Stochastic Reaction Diffusion and General Spatially Extended Systems with Intrinsic Fluctuations

Abstract

Spatially extended systems are widely encountered in physics, chemistry, and biology for studying many important natural phenomena. In this work, we established a landscape framework for studying general stochastic spatially extended systems with intrinsic statistical fluctuations that can be applied to both equilibrium systems with detailed balance and nonequilibrium systems without detailed balance. We set up the master equation for general stochastic spatially extended systems (functional master equation) from two fundamental dynamical ingredients that characterize the elementary state transitions of spatially extended systems. We explored the entire spectrum of the various approximations of the functional master equation under certain conditions, from the functional Kramers-Moyal equation to the functional Fokker-Planck equation and its equivalent, the functional Langevin equation, to the macroscopic deterministic equation. We uncovered the Lyapunov functionals which are required to quantify the global stability and function of the system for both the deterministic and stochastic spatially extended systems. The global potential landscape functional for stochastic spatially extended systems can be quantified by the steady state probability distribution functional. In the small fluctuation limit, the potential landscape functional becomes the Lyapunov functional for the corresponding deterministic spatially extended system. The relative entropy functional (or free energy functional) proves to be the Lyapunov functional for the stochastic spatially extended systems. The potential landscape functional and the relative entropy functional quantify the global stability and function of the deterministic and stochastic spatially extended systems for both equilibrium and nonequilibrium conditions. The chemical reaction-diffusion systems as a typical and important class of spatially extended systems is explored on its own terms as well as used as a direct application of the general framework to derive more specific results for reaction-diffusion systems. A biological system, the bicoid protein concentration distribution in fruit fly embryo development, which can be modeled as a specific type of reaction-diffusion dynamics, is studied using the proposed framework. We found both the bicoid concentration and its fluctuation decay from anterior to posterior when the source producing the bicoid protein is located at the anterior point. The corresponding local funneled landscape basin of the bicoid concentration becomes narrower and steeper from anterior to posterior in such a case.