Richardson’s four-thirds law is both analytically and numerically confirmed by applying the effective Hamiltonian method to relative diffusion of a pair of fluid particles subject to convection in steady, incompressible, statistically isotropic turbulent flow with a joint Gaussian distribution for velocity field when the spectrum satisfies Kolmogorov’s similarity law and the scaling law. In addition, the interesting phenomenon that the mean square value of the relative distance of a pair of particles increases faster than the cube of the time of diffusion at the part which connects with the range satisfying this law when the initial distance between them is small enough in comparison with the typical length of the flow is reported. This method is, further, applied to relative diffusion in the turbulent flow with multi-range spectrum. There it is qualitatively tried to reproduce the experimental data of Kellogg (1956). The possibility that two separate ranges, which satisfy Richardson’s law respectively, appear in practical experiment under the adequate conditions is showed, too.