  We have used a toy model to study the behaviour of confined binary systems in
2$D$ and compare it with previously known result in 3$D$. In the case of 2$D$,
in which canonical distribution exists only above a critical temperature, we
evaluate the exact form of partition function for this system and compare the
exact partition function with the mean field partition function for the case of
two-particle system. In contrast of its 3$D$ counterpart, there is no phase
transition here. If this system, however, studied in microcanonical ensemble,
it shows two different phases of kinetic energy dominated with positive
specific heat and potential energy dominated with negative specific heat in
presence of short distance cutoff. In absence of short distance cutoff,
surprisingly, the negative specific heat region will be replaced by region of
large specific heat. This feature is completely new and there is no such a case
in 3$D$.
