  A 2D temperature-dependent effective potential is calculated for the
interacting longitudinal and transverse $L-$phonons of $\beta$ zirconium in the
frozen-phonon model. The effective potentials obtained for different
temperatures are used for the numerical solution of a set of stochastic
differential equations with a thermostat of the white-noise type. Analysis of
the spectral density of transverse vibrations allows one to determine the
temperature at which $\beta$-Zr becomes unstable with respect to the
longitudinal $L-$vibrations. The obtained temperature value practically
coincides with the experimental temperature of the $\beta \to \alpha$
structural transition in zirconium. The role of electronic entropy in the
$\beta-$Zr stability is discussed.
