Singular invariant probability densities in dissipative dynamical systems

Abstract

Dynamical systems are considered dissipative when they asymptotically produce a net contraction of volume in phase space, so that V(t) → 0 as , where V(t) is the volume at time t into which the dynamics transforms a nonzero initial volume V0. The trajectories are thereby forced to asymptotically converge toward time-invariant subsets of measure zero, which are typically strange attractors with fractal structure. Since a probability density is a probability per unit volume in phase space, any initially smooth probability density must become singular as , so invariant probability densities do not exist as proper mathematical functions in dissipative systems. For this reason, the statistical behaviour of such systems is generally analysed in terms of invariant probability measures, which remain well defined even when densities do not. The purpose of this paper is to point out that invariant probability densities in dissipative systems can nevertheless be rigorously defined as singular generalised functions which (a) are natural generalisations of the Dirac delta function, (b) are easier to work with than invariant measures, (c) satisfy the same stationary Liouville equation as continuous invariant densities, and (d) can therefore be formally manipulated in the same way. These simplifying features facilitate the analysis of the statistical properties of dissipative dynamical systems, thereby making abstract measure-theoretical theories of such systems more accessible to students and nonspecialists.