Detail:
Abstract: We present a review of the recently developed landscape and flux theory for non-equilibrium dynamical systems. We point out that the global natures of the associated dynamics for non-equilibrium system are determined by two key factors: the underlying landscape and, importantly, a curl probability flux. The landscape (U) reflects the probability of states (P) (U = − ln P) and provides a global characterization and a stability measure of the system. The curl flux term measures how much detailed balance is broken and is one of the two main driving forces for the non-equilibrium dynamics in addition to the landscape gradient. The landscape and flux theory has many interesting consequences including the fact that irreversible kinetic paths do not necessarily pass through the landscape saddles; non-equilibrium transition state theory at the new saddle on the optimal paths for small but finite fluctuations; a generalized fluctuation–dissipation relationship for non-equilibrium dynamical systems; non-equilibrium thermodynamics; gauge theory and a geometrical connection; coupled landscapes; stochastic spatial dynamics. We provide concrete examples of biological systems to demonstrate the new insights from the landscape and flux theory. These include models of the cell cycle; stem cell differentiation; cancer biology; evolution; neural networks; chaotic strange attractor; development in space. We also give the philosophical implications of the theory and the outlook for future studies.
