Finite time scaling of energy in simulated annealing

The energy of a local minimum obtained by the simulated annealing generally depends on a time tau in which a complex system has been immersed in a heat bath. How the resultant energy E( tau ) scales with a time tau is an interesting question. The diffusion process of a point in a wiggly parabola is analysed to discuss the scaling. The model is exactly solvable and the energy is found to scale as E( tau )= in +c(ln tau )-1. This scaling is considered rather common to general complex systems. However, the limit in obtained from practical data is not necessarily the ground state energy of a system.