Diffusion on the torus for Hamiltonian maps (Q1897061)

The authors propose and investigate a definition of the diffusion coefficient for a mapping on the torus \(\mathbb{T}^2\) suggested by the solution of the diffusion equation on \(\mathbb{T}^2\). This definition is based on the limit of moments of the invariant measure, and depends on the set \(\Omega\), where an initial uniform distribution is assigned. When the mapping of the torus is the algebraic automorphism known as Arnold's cat mapping, the limit is proved to exist and have the same value for almost all initial sets \(\Omega\) in the family of parallelograms on \(\mathbb{T}^2\). The main result of this paper is that, for the Arnold cat mapping, the diffusion coefficient exists, and for the second moment and a wide choice of sets \(\Omega\subset \mathbb{T}^2\) has the value \[ D= 2\log \lambda[\mu_L(\mathbb{T}^2)/2\pi]^2, \] where \(\lambda\) is the positive Lyapunov exponent and \(\mu_L\) is Lebesgue measure. The authors also offer numerical evidence that \(D\) has the same value for arbitrary polygons \(\Omega\) and arbitrary moments.