Choosing an approach
CriticalTransitions.jl provides several complementary methods that all answer the same underlying question: how, where, and how often does a system undergo a rare transition between metastable states? They differ in the regime where they are valid (noise strength, state-space dimension) and in what they return (a scalar rate, the most probable path, a spatial flux field). This page is the map; each method has its own manual page with the details.
The method menu
| Your goal | Reach for | Regime / output |
|---|---|---|
| Observe transitions directly | Direct sampling | any noise level; brute-force trajectories |
| Most probable path and exponential rate | Large deviation theory | weak noise ($\varepsilon \to 0$); instanton & quasipotential |
| Escape rate, mean first-passage time, (quasi-)stationary density | Rates, distributions & the generator | finite noise; spectral observables |
| Where reactive paths concentrate, transition channels, committor | Transition path theory | finite noise; spatial flux |
| Transition driven by parameter drift rather than noise | Rate-induced transitions | non-autonomous forcing |
Picking by what you need
- Only a rate or waiting time? Use the spectral observables (
mean_first_passage_time, quasi-stationary distribution) on the generator, or Kramers asymptotics for high barriers. - The transition path itself? Use large deviation theory (the instanton via MAM / gMAM / sgMAM / shooting).
- Spatial flux and channels? Use transition path theory.
- High barrier, weak noise? Direct sampling becomes exponentially expensive; prefer LDT or the spectral observables.
- Shallow barrier? Just simulate with direct sampling.
- Parameter drift, not noise? Use rate-induced transitions.
Composing methods
The methods are not mutually exclusive. Transition path theory and the spectral rate observables are both built on the same discretised generator. In large deviation theory, a fast gMAM / sgMAM solve makes a good warm start for the more delicate multiple-shooting boundary-value problem; see Large deviation theory.