Statistical mechanics with non-integrable topological constraints: Self-organization in knotted phase space

The object of this study is the statistical mechanics of dynamical systems lacking a Hamiltonian structure due to the presence of non-integrable topological constraints that limit the accessible regions of the phase space. Focusing on the simplest three dimensional case, we develop a procedure (Poissonization) that assigns to any three dimensional non-Hamiltonian system an equivalent four dimensional Hamiltonian system endowed with a proper time. The statistical distribution is then constructed in the recovered four dimensional canonical phase space. Projecting in the original reference frame, we show that the statistical distribution departs from standard Maxwell–Boltzmann statistics. The deviation is a function of the knottedness of the phase space, which is measured by the helicity density of the topological constraint. The theory is then generalized to a class of non-Hamiltonian systems in higher dimensions.