“…In this point, however, we adopt a more recent development in statistical mechanics, in that we keep the number of oscillators large, , but finite [ 8 , 9 , 10 , 11 , 12 ], so that the dynamics of the total system can be treated in the framework of the time-reversal invariant Hamiltonian mechanics of closed systems. It has been demonstrated for classical as well as for quantum systems [ 13 , 14 ], and is corroborated by the present work, that despite its time-reversal symmetry, this approach reproduces irreversible behaviour on all relevant timescales. Poincaré recurrences, which prevent true irreversibility in systems with a finite number of freedoms, occur only on timescales that diverge geometrically with N [ 15 ].…”