In this Letter, I propose that a properly rescaled spatiotemporal correlation function of the energy density fluctuations may be applied to characterize the equilibrium diffusion processes in lattice systems with finite temperature. Applying this function, the diffusion processes in three one-dimensional nonlinear lattices are studied. The diffusion exponent is shown to be related to the diverging exponent of the thermal conductivity of a lattice through the relation , as has been proved based on the Lévy walk assumption. The diffusion behavior is explained in terms of solitons and phonons.