In applied mathematics, Arnold diffusion is the phenomenon of instability of nearly-integrable Hamiltonian systems.
The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964.
[1][2] More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly-integrable Hamiltonian systems that exhibit a significant change in the action variables.
Arnold diffusion describes the diffusion of trajectories due to the ergodic theorem in a portion of phase space unbound by any constraints (i.e. unbounded by Lagrangian tori arising from constants of motion) in Hamiltonian systems.
It occurs in systems with more than N=2 degrees of freedom, since the N-dimensional invariant tori do not separate the 2N-1 dimensional phase space any more.
Thus, an arbitrarily small perturbation may cause a number of trajectories to wander pseudo-randomly through the whole portion of phase space left by the destroyed tori.
However, as first noted in Arnold's paper,[1] there are nearly integrable systems for which there exist solutions that exhibit arbitrarily large growth in the action variables.
Arnold conjectured that "the mechanism of 'transition chains' which guarantees that nonstability in our example is also applicable to the general case (for example, to the problem of three bodies).
"[1] The KAM theorem and Arnold diffusion has led to a compendium of rigorous mathematical results, with insights from physics.
The general case of Arnold's diffusion problem concerns Hamiltonian systems of one of the forms where
For systems as in (1), the unperturbed Hamiltonian possesses smooth families of invariant tori that have hyperbolic stable and unstable manifolds; such systems are referred to as a priori unstable.
[5] In either case, the Arnold diffusion problem asserts that, for `generic' systems, there exists
Informally, the Arnold diffusion problem says that small perturbations can accumulate to large effects.
[13][14] In the context of the restricted three-body problem, Arnold diffusion can be interpreted in the sense that, for all sufficiently small, non-zero values of the eccentricity of the elliptic orbits of the massive bodies, there are solutions along which the energy of the negligible mass changes by a quantity that is independent of eccentricity.