Quasipotentials as Thinness Margins: Large- Deviation Diagnostics for Thin, Transitional, and Thick Chaos

This paper establishes a precise link between Freidlin–Wentzell large-deviation theory and the geometric framework of thin, transitional, and thick chaos. Classical diagnostics such as positive Lyapunov exponents detect local instability, yet they do not determine whether that instability actually reaches the variables that govern operational coherence or failure. The thin/thick framework addresses the distinction by means of an operational projection, a designated failure set in the projected space, and the projected reachable set over a finite horizon. Chaos is thin when local instability exists while the reachable set remains at a positive distance from the failure set; it is thick when unstable directions couple to the projected variables and intersect the failure set; and it is transitional when the distance, though still positive, is small relative to the initial margin. Large-deviation theory supplies the exponential cost of rare transitions in the small-noise limit of a stochastic differential equation, or the large-volume limit of a Markov jump process. The Freidlin–Wentzell action functional yields a Hamiltonian whose characteristics are the instantons; the quasipotential is the infimal action of paths connecting a coherent region to a target. When the target is chosen to be the operational failure set, the quasipotential height becomes an asymptotic expression for the thinness margin: the probability that the projected trajectory reaches the failure set before a fixed horizon decays exponentially with that height. The associated instanton identifies the dominant path along which local instability becomes transport-effective. The correspondence is stated under standard regularity assumptions that guarantee a large-deviation principle with a good rate function. It is illustrated on the Schlögl model, where barrier heights computed from the zero-energy Hamiltonian surface are reinterpreted as thinness margins between a low-density coherent regime and a high-density failure state. A computational protocol based on geometric minimum-action methods converts the infinite-horizon problem into a finite-dimensional optimization, after which the thinness index is obtained by comparing the barrier height with the deterministic distance to failure. The statements hold only in the joint limit of small noise and observation times short compared with the mean transition time. Outside this regime the approximation ceases to be accurate. The numerical values also inherit the projection dependence of the original framework, and high-dimensional or spatially extended systems require dimension reduction or manifold-adapted solvers. Within these limits, the quasipotential and its instanton furnish quantitative diagnostics for the operational reach of instability without altering the classical large-deviation calculus.