Mapping out the equation of state (EOS) of nuclear matter is a long standing problem in nuclear physics. Both experimentalists and theoretical physicists spare no effort in improving understanding of the EOS. In this thesis, we examine observables sensitive to the EOS within the pBUU transport model based on the Boltzmann equation. By comparing theoretical predictions with experimental data, we arrive at new constraints for the EOS. Further we propose novel promising observables for analysis of future experimental data.One set of observables that we examine within the pBUU model are pion yields. First, we find that net pion yields in central heavy-ion collisions (HIC) are strongly sensitive to the momentum dependence of the isoscalar nuclear mean field. We reexamine the momentum dependence that is assumed in the Boltzmann equation model for the collisions and optimize that dependence to describe the FOPI measurements of pion yields from the Au+Au collisions at different beam energies. Alas such optimized dependence yields a somewhat weaker baryonic elliptic flow than seen in measurements.Subsequently, we use the same pBUU model to generate predictions for baryonic elliptic flow observable in HIC, while varying the incompressibility of nuclear matter. In parallel, we test the sensitivity of pion multiplicity to the density dependence of EOS, and in particular to incompressibility, and optimize that dependence to describe both the elliptic flow and pion yields. Upon arriving at acceptable regions of density dependence of pressure and energy, we compare our constraints on EOS with those recently arrived at by the joint experiment and theory effort FOPI-IQMD. We should mention that, for the more advanced observables from HIC, there remain discrepancies of up to 30\%, depending on energy, between the theory and experiment, indicating the limitations of the transport theory.Next, we explore the impact of the density dependence of the symmetry energy on observables, motivated by experiments aiming at constraining the symmetry energy. In contradiction to IBUU and ImIQMD models in the literature, that claim sensitivity of net charged pion yields to the density dependence of the symmetry energy, albeit in direction opposite from each other, we find practically no such sensitivity in pBUU. However, we find a rather dramatic sensitivity of differential high-energy charged-pion yield ratio to that density dependence, which can be qualitatively understood, and we propose that that differential ratio be used in future experiments to constrain the symmetry energy.Finally, we present Gaussian phase-space representation method for studying strongly correlated systems. This approach allows to follow time evolution of quantum many-body systems with large Hilbert spaces through stochastic sampling, provided the interactions are two-body in nature. We demonstrate the advantage of the Gaussian phase-space representation method in coping with the notorious numerical sign problem for fermion systems. Lastly, we discuss the difficulty in trying to stabilize the system during its time evolution, within the Gaussian phase-space method.
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