By using Laplace transformation, a set of analytic solutions of the co
ordinates and velocities for a system plus an environment are given. I
n the absence of system potential, based on the solutions, we rigorous
ly 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 discu
ss the time development of the mean square displacements, square veloc
ities and the mean energies of both the system and the environment whe
n the initial state is in partial thermal equilibrium or far away from
thermal equilibrium. In addition, we also pay attention to the enviro
nmental 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).