Navier-Stokes Equations and Turbulence. Encyclopedia of Math and its Applications, Vol. 83

Approximating stationary statistical properties

“…Even the simple logistic map T (x) = 4x(1 − x) on the unit interval, the Lorenz 63 and Lorenz 96 model possess intrinsic chaotic behavior which renders approximation of single trajectory extremely difficult over a long time. On the other hand, it is well-known that the statistical properties of these kind of systems are much more important, physically relevant and stable than single trajectories (see [10,18,20,24,25,36]). Indeed, much of the classical turbulence theories are formulated in statistical forms (via spatial and temporal averages), for instance, the famous Kolmogorov U 3 L scaling law of the energy dissipation rate per unit mass as well as the Kolmogorov k − 5 3 energy spectrum in the inertial range in three dimensional homogeneous isotropic turbulence (see [10,11,25]).…”

“…Therefore, for complex physical processes, due to the intrinsic stochasticity, it is necessary to consider statistical properties (averaged quantities) of the system instead of properties of individual orbit (see, for instance, [10,11,20,24,25,36]). Then it is obvious that we need probability measures on the phase space that respects the dynamics in order to discuss statistical properties (statistical averages).…”

“…Recall that a compact set A is called the global attractor of the dynamical system if it is invariant and attracts all bounded sets in the phase space H (see [10,13,33]). …”

“…However, computing the invariant measure (or the associated pdf in the finite dimensional case) is usually very hard and costly if the spatial dimension is high. One of the commonly used alternative methods in calculating the statistical quantity is to substitute spatial average by long time average under Boltzmann's assumption of ergodicity (see [10,20,24,37]),…”

“…For a given abstract autonomous continuous in time dynamical system determined by a semi-group {S(t), t ≥ 0} on a separable metric space H, we recall that if the system reaches a statistical equilibrium in the sense that the statistics are time independent (stationary statistical properties), the probability measure µ on H that describes the stationary statistical properties can be characterized via either the strong (pull-back) or weak (push-forward) formulation (see [10,20,24,36,37]). for all bounded continuous test functionals Φ. Invariant measure (stationary statistical solution) for a discrete dynamical system generated by a map S discrete on a metric space H is defined in a similar fashion with the continuous time t replaced by discrete time n = 0, 1, 2, · · · .…”