Flow Equation Approach to the Statistics of Nonlinear Dynamical Systems

The probability distribution function of non-linear dynamical systems is governed by a linear framework that resembles quantum many-body theory, in which stochastic forcing and/or averaging over initial conditions play the role of non-zero $\hbar$. Besides the well-known Fokker-Planck approach, there is a related Hopf functional method\footnote{Uriel Frisch, {\it Turbulence: The Legacy of A. N. Kolmogorov} (Cambridge University Press, 1995) chapter 9.5.}; in both formalisms, zero modes of linear operators describe the stationary non-equilibrium statistics. To access the statistics, we investigate the method of continuous unitary transformations\footnote{S. D. Glazek and K. G. Wilson, Phys. Rev. D {\bf 48}, 5863 (1993); Phys. Rev. D {\bf 49}, 4214 (1994).} (also known as the flow equation approach\footnote{F. Wegner, Ann. Phys. {\bf 3}, 77 (1994).}), suitably generalized to the diagonalization of non-Hermitian matrices. Comparison to the more traditional cumulant expansion method is illustrated with low-dimensional attractors. The treatment of high-dimensional dynamical systems is also discussed.