  In this study we try to answer the qustion : What happens when explicit
constraints are introduced such that the low energy, long wavelength modes of a
system are unavailable ? This question has assumed some importance in recent
years due to the advent of nano technology and the growing use of nanometer
scale devices and structures. In a small system, the size limits the scale of
the fluctuations and makes it imperative for us to understand how the response
of the system is altered in such a situation. In this thesis, this question is
answered for the special case of interfacial fluctuations in two dimensions
(2d). The energy of an interface between two phases in equilibrium is invariant
with respect to translations perpendicular to the plane (or line in 2d) of the
interface. We study the consequence of breaking this symmetry explicity using
an external field gradient. One expects that since low energy excitations are
suppressed, the interface would be flat and inert at all times. We show that
surprisingly there are novel fluctuations and phenomena associated with such
constrained interfaces which have static as well as dynamic consequences. The
Ising interface on a square lattice is shown to undergo a multitude of
structural transitions as a function of velocity and the orientation. Liquid
solid interfaces show coherent addition and removal of atomic layers providing
novel mechanisms of stress relaxation in a nanosized single crystal without
defects. We study momentum and energy transfer across the liquid solid
interface in the presence of this ``layering'' transition.
