Chaos and Lyapunov exponents in classical and quantal distribution dynamics

We analytically establish the role of a spectrum of Lyapunov exponents in the evolution of phase-space distributions ρ(p,q). Of particular interest is λ₂, an exponent that quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and is generally larger than the maximal Lyapunov exponent {λ} for trajectories. The approach is trajectory independent and is therefore applicable to both classical and quantum mechanics. In the latter case we show that the →0 limit yields the classical, fully chaotic, result for the quantum cat map.