  Discovery of electron hydrodynamics in graphene system has opened a new scope
of analytic calculations in condensed matter physics, which was traditionally
well cultivated in science and engineering as a non-relativistic hydrodynamics
and in high energy nuclear and astro physics as relativistic hydrodynamics.
Electrons in graphene follow neither non-relativistic nor relativistic
hydrodynamics and thermodynamics. Present article has gone through systematic
microscopic calculations of thermodynamical quantities like pressure, energy
density, etc. of electron-fluid in graphene and compared with corresponding
estimations for non-relativistic and ultra-relativistic cases. Identifying the
Dirac fluid and Fermi liquid domains, we have sketched the transition of
temperature and Fermi energy dependency of electron thermodynamics for graphene
and other cases. An equivalent transition for quark matter is also discussed.
The most exciting part is the general expression of specific heat, whose Fermi
to Dirac fluid domain transition can be realized as a transition from a
solid-based to a fluid-based picture. This understanding may be connected to
the experimentally observed Wiedemann-Franz Law violation in the Dirac fluid
domain of graphene system.
