  We introduce a simple linear equation relating the line-of-sight peculiar
velocity and density contrast correlation functions. The relation, which we
call the "Gaussian cell two-point energy-like equation", is valid in the
distant-observer-limit and requires Gaussian smoothed fields. In the variance
case, i.e., at zero lag, the equation is similar in its mathematical form to
the Layzer-Irvine cosmic energy equation. \beta estimation with this equation
from the PSC$z$ redshift galaxy survey and the SEcat catalogue of peculiar
velocities is carried out, returning a value of \beta = 0.44+-0.08. The
applicability of the method for the 6dF galaxy redshift and peculiar motions
survey is demonstrated with mock data where it is shown that beta could be
determined with ~5% accuracy. The prospects for constraining the dark energy
equation of state with this method from the kinematic and thermal
Sunyaev-Zel'dovich cluster surveys are discussed. The equation is also used to
construct a nonparametric mass density power spectrum estimator from peculiar
velocity data.
