Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors

TL;DR

This paper explores the mathematical landscape of non-gradient dynamics without detailed balance, focusing on stable limit cycles and multiple attractors, using large deviation theory to analyze their properties and implications.

Contribution

It explicitly analyzes singularly perturbed diffusion processes with stable limit cycles and multiple attractors, revealing the structure of local and global landscapes in nonequilibrium systems.

Findings

Stationary solutions show exponential and algebraic separation of time scales.

Large deviation rate function is zero on entire stable limit cycles.

Nondifferentiable points can occur in the global landscape with cycle flux.

Abstract

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.