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History-Dependence and Ageing in a Periodic Long-Range Josephson Array

History-dependence and ageing are studied in the low-temperature glass phase of a long-range periodic Josephson array. This model is characterized by two parameters, the number of wires ($2N$) and the flux per unit strip ($α$); in the limit $N \to \infty$ and fixed $α\ll 1$ the dynamics of the model are described by the set of coupled integral equations, which coincide with those for the $p=4$ disordered spherical model. Below the glass transition we have solved these equations numerically in a number of different regimes. We observe power-law ageing after a fast quench with an exponent that decreases rapidly with temperature. After slow cooling to a not-too-low temperature, we see ageing characterized by the appearance of a new time scale which has a power law dependence on the cooling rate. By contrast if the array is cooled slowly to very low temperatures the ageing disappears. The physical consequences of these results in different cooling regimes are discussed for future experiment. We also study the structure of the phase space in the low temperature glassy regime. Analytically we expect an exponential number of metastable states just below the glass transition temperature with vanishing mutual overlap, and numerical results indicate that this scenario remains valid down to zero temperature. Thus in this array there is no further subdivision of metastable states. We also investigate the probability to evolve to different states given a starting overlap, and our results suggest a broad distribution of barriers.