Tailoring the overlap distribution in driven mean-field spin models

In a statistical physics context, inverse problems consist in determining microscopic interactions such that a system reaches a predefined collective state. A complex collective state may be prescribed by specifying the overlap distribution between microscopic configurations, a notion originally introduced in the context of disordered systems like spin-glasses. We show that in spite of the absence of disorder, nonequilibrium spin models exhibiting spontaneous magnetization oscillations provide a benchmark to prescribe a non-trivial overlap distribution with continuous support, qualitatively analogous to the ones found in disordered systems with full replica symmetry breaking. The overlap distribution can be explicitly tailored to take a broad range of predefined shapes by monitoring the spin dynamics. The presence of a non-trivial overlap distribution is traced back to an average over infinitely many pure states, a feature shared with spin-glasses, although the structure of pure states is here much simpler.