Dynamic phase diagram of the Number Partitioning Problem

Abstract

We study the dynamic phase diagram of a spin model associated with the Number Partitioning Problem, as a function of temperature and of the fraction $K/N$ of spins allowed to flip simultaneously. The case K=1 reproduces the activated behavior of Bouchaud's trap model, whereas the opposite limit $K=N$ can be mapped onto the entropic trap model proposed by Barrat and Mézard. In the intermediate case $1 \ll K \ll N$, the dynamics corresponds to a modified version of the Barrat and Mézard model, which includes a slow (rather than instantaneous) decorrelation at each step. A transition from an activated regime to an entropic one is observed at temperature $T_g/2$ in agreement with recent work on this model. Ergodicity breaking occurs for $T<T_g/2$ in the thermodynamic limit, if $K/N \to 0$. In this temperature range, the model exhibits a non trivial fluctuation-dissipation relation leading for $K \ll N$ to a single effective temperature equal to $T_g/2$. These results give new insights on the relevance and limitations of the picture proposed by simple trap models.