  For free particles in a simple harmonic potential plus a weak anharmonicity,
characterized by a set of anharmonic parameters, Newtonian mechanics asserts
that there is a renormalization of the natural frequency of the periodic
motion; and statistical mechanics claims that the anharmonicity causes a
correction to the heat capacity of an ideal gas in the anharmonic potential.
The orbital motion and thermal motion depend on the same anharmonic parameters,
but in different combinations. These two manners of combinations are
fundamentally different, demonstrating that statistical law can not emerge from
the many-body limit of deterministic law for one-body.
