This dissertation is divided in two parts. In the first part, we study the dynamics of a classical particle system. We calculate the efficiency of a rejectionree dynamic Monte Carlo method in the limit of low temperatures and/or high densities for d-dimensional off-lattice homogeneous particles interacting through a repulsive power-law potential r−p. Theoretically we find the algorithmic efficiency is asymptotically proportional to ρ (p+2)/2 T-d/2 with the particle density ρ and the temperature T. Dynamic Monte Carlo simulations are performed in 1-, 2- and 3-dimensional systems with different powers p, and the results agree with the theoretical predictions. In the second part, we study the dynamics of a quantum spin 1/2 system. We calculate the average of the z component of the spin operator when one spin 1/2 particle is interacting with one, two, and three baths of 1/2-integer spins. For the cases of one and two baths we find the asymptotic behavior of and the quantum purity P(t) using the density of states of the system. In both cases and P(t) decay, exponentially for a single bath and algebraically for two baths. We present simulations in real time for one spin 1/2 particle coupled to one or two baths of 1/2-integer spins. The simulations were performed using the algorithm and code of Prof. De Raedt. We first simulated one spin coupled to one or two spin-baths with no interactions between the bath spins to compare with the theoretical calculations. We extend these simulations by introducing random interactions between the bath spins, in an attempt to reach the asymptotic decay rate at earlier times and for fewer spins in the baths. We also perform preliminary studies of the correlations of two and six spin 1/2 particles coupled to one, or two bath of spins, as a step to investigate quantum synchronization. We introduce a preliminary result for the spectral density of a spin-bath system, a promising quantity to measure the synchronization of spin systems.