The full solution of the independent oscillator model and the properties of the environment

By using Laplace transformation, a set of analytic solutions of the coordinates and velocities for a system plus an environment are given. In the absence of system potential, based on the solutions, we rigorously prove that if the initial state of the composite system lies in its thermal equilibrium, then the environment is equipartition of energy, which, however, is moving with the system. At the same time, we discuss the time development of the mean square displacements, square velocities and the mean energies of both the system and the environment when the initial state is in partial thermal equilibrium or far away from thermal equilibrium. In addition, we also pay attention to the environmental behaviour when the initial state is deterministic. It is shown that the properties of the environment are quite different from those of the collection of oscillators discussed by Ford et al. (G.W. Ford, M. Kac and P Mazur, J. Math. Phys. 6 (1965) 504).