Chaotic Fields Behave Universally out of Equilibrium

Chaotic dynamics is always characterized by swarms of unstable trajectories,unpredictable individually, and thus generally studied statistically. It isoften the case that such phase-space densities relax exponentially fast to alimiting distribution, that rules the long-time average of every observable ofinterest. Before that asymptotic timescale, the statistics of chaos isgenerally believed to depend on both the initial conditions and the chosenobservable. I show that this is not the case for a widely applicable class ofmodels, that feature a phase-space (`field') distribution common to allpushed-forward or integrated observables, while the system is still relaxingtowards statistical equilibrium or a steady state. This universal profile isdetermined by both leading and first subleading eigenfunctions of the transportoperator (Koopman or Perron-Frobenius) that maps phase-space densities forwardor backwards in time.