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We investigate early-time equilibration rates of observables in closedmany-body quantum systems and compare them to those of two correlationfunctions, first introduced by Kubo and Srednicki. We explore whether thesedifferent rates coincide at a universal value that sets the timescales ofprocesses at a finite energy density. We find evidence for this coincidencewhen the initial conditions are sufficiently generic, or typical. We quantifythis with the effective dimension of the state and with a state-observableeffective dimension, which estimate the number of energy levels thatparticipate in the dynamics. Our findings are confirmed by proving that thesedifferent timescales coincide for dynamics generated by Haar-randomHamiltonians. This also allows to quantitatively understand the scope ofprevious theoretical results on equilibration timescales and on random matrixformalisms. We approach this problem with exact, full spectrum diagonalization.The numerics are carried out in a non-integrable Heisenberg-like Hamiltonian,and the dynamics are investigated for several pairs of observables and states.

Relaxation of non-integrable systems and correlation functions