  We review the behavior of the entropy per particle in various two-dimensional
electronic systems. The entropy per particle is an important characteristic of
any many body system that tells how the entropy of the ensemble of electrons
changes if one adds one more electron. Recently, it has been demonstrated how
the entropy per particle of a two-dimensional electron gas can be extracted
from the recharging current dynamics in a planar capacitor geometry. These
experiments pave the way to the systematic studies of entropy in various
crystal systems including novel two-dimensional crystals such as gapped
graphene, germanene and silicene. Theoretically, the entropy per particle is
linked to the temperature derivative of the chemical potential of the electron
gas by the Maxwell relation. Using this relation, we calculate the entropy per
particle in the vicinity of topological transitions in various two-dimensional
electronic systems. We show that the entropy experiences quantized steps at the
points of Lifshitz transitions in a two-dimensional electronic gas with a
parabolic energy spectrum. In contrast, in doubled-gapped Dirac materials, the
entropy per particles demonstrates characteristic spikes once the chemical
potential passes through the band edges. The transition from a topological to
trivial insulator phase in germanene is manifested by the disappearance of a
strong zero-energy resonance in the entropy per particle dependence on the
chemical potential. We conclude that studies of the entropy per particle shed
light on multiple otherwise hidden peculiarities of the electronic band
structure of novel two-dimensional crystals.
