  Verlinde's proposal on the entropic origin of gravity is based strongly on
the assumption that the equipartition law of energy holds on the holographic
screen induced by the mass distribution of the system. However, from the theory
of statistical mechanics we know that the equipartition law of energy does not
hold in the limit of very low temperature. Inspired by the Debye model for the
equipartition law of energy in statistical thermodynamics and adopting the
viewpoint that gravitational systems can be regarded as a thermodynamical
system, we modify Einstein field equations. We also perform the study for
Poisson equation and modified Newtonian dynamics (MOND). Interestingly enough,
we find that the origin of the MOND theory can be understood from Debye
entropic gravity perspective. Thus our study may fill in the gap existing in
the literature understanding the theoretical origin of MOND theory. In the
limit of high temperature our results reduce to their respective standard
gravitational equations.
