The steady distribution function for homogeneous turbulence is studied starting from Liouville's equation, modified by the introduction of an instantaneously fluctuating external force, which acts as a random source of energy. A new technique for solving Liouville's equation is presented giving a systematic development of the concepts of turbulent diffusion and turbulent viscosity. It amounts to a consistent generalization of the random phase approximation. When the rate of input of energy into the kth Fourier component uk has a power form h|k|−α, the functional form of the mean value 〈 uku−k 〉 can be determined exactly in the limit of large Reynolds number; it is $Ah^{\frac{2}{3}}|\bf K|^ {-{\frac {1}{3}}(5+2 \alpha)}$. Liouville's equation proves an inadequate basis for the steady time-dependent mean $\langle u_k(t)u_{-k}(t^ \prime) \rangle $ and a more general equation is derived. The new equation can be solved in a similar way and shows that the time-dependent correlation starts like a Gaussian in time, then passes through an exponentially decaying state, then eventually has a power dependence $|t-t^\prime|- \gamma^ {|k|}$.