  We address the question on how weak perturbations, that are quite ineffective
in small many-body systems, can lead to decoherence and hence to
irreversibility when they proliferate as the system size increases. This
question is at the heart of solid state NMR. There, an initially local
polarization spreads all over due to spin-spin interactions that conserve the
total spin projection, leading to an equilibration of the polarization. In
principle, this quantum dynamics can be reversed by changing the sign of the
Hamiltonian. However, the reversal is usually perturbed by non reversible
interactions that act as a decoherence source. The fraction of the local
excitation recovered defines the Loschmidt echo (LE), here evaluated in a
series of closed $N$ spin systems with all-to-all interactions. The most
remarkable regime of the LE decay occurs when the perturbation induces
proliferated effective interactions. We show that if this perturbation exceeds
some lower bound, the decay is ruled by an effective Fermi golden rule (FGR).
Such a lower bound shrinks as $ N $ increases, becoming the leading mechanism
for LE decay in the thermodynamic limit. Once the polarization stayed
equilibrated longer than the FGR time, it remains equilibrated in spite of the
reversal procedure.
