We numerically investigated deterministic methods for constructing the Boltzmann-Gibbs (BG) distributions used in molecular dynamics simulations. Tsallis dynamics (TD) is a deterministic dynamical equation that can generate the Tsallis distribution and reconstruct the BG distributions by a reweighting method. This method is applied to the 1-dimensional harmonic oscillator and 1-dimensional double-well potential, which are fundamental interaction models and have been typical examples yielding problematic results in conventional molecular dynamics simulations. We examined the ability of TD to produce the BG distributions using these low-dimensional models and the alanine tripeptide system, comparing the results with those of a conventional deterministic method that directly produces the BG distribution. We found the TD produced efficient sampling, including full phase-space covering to generate the correct BG distribution for the 1-dimensional harmonic oscillator, powerful barrier crossing against the traps at the local minima of the double-well potential, and a wide-region sampling in the potential energy space for the alanine system. Problems for the method are also discussed.