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<<K<<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-->0 . In this temperature range, the model exhibits a nontrivial fluctuation-dissipation relation leading for K<<N to a single effective temperature equal to T(g) /2 . These results give insights into the relevance and limitations of the picture proposed by simple trap models.