  Can certain degrees of freedom of a closed physical system, described by a
time-independent Hamiltonian, become more and more classical as they evolve
from some state? This question is important because our universe seems to have
done just that! We construct an explicit, simple, example of such a system with
just two degrees of freedom, of which one achieves `spontaneous
classicalization'. It starts from a quantum state and under the usual
Hamiltonian evolution, becomes more and more classical (in a well-defined
manner in terms of the Wigner function) as time progresses. This is achieved
without the usual procedures of integrating out a large number of environmental
degrees of freedom or conventional decoherence. We consider different ranges of
parameter space and identify the conditions under which spontaneous
classicalization occurs in our model. The mutual interaction between the
sub-systems of a larger system can indeed drive some of the subsystems to a
classical configuration, with a phase space trajectory of evolution. We also
argue that the results of our toy model may well be general characteristics of
certain class of interacting systems. Several implications are discussed.
