  The jamming transition of non-spherical particles is fundamentally different
from the spherical case. Non-spherical particles are hypostatic at their
jamming points, while isostaticity is ensured in the case of the jamming of
spherical particles. This structural difference implies that the presence of
asphericity affects the critical exponents related to the contact number and
the vibrational density of states. Moreover, while the force and gap
distributions of isostatic jamming present power-law behaviors, even an
infinitesimal asphericity is enough to smooth out these singularities. In a
recent work [PNAS 115(46), 11736], we have used a combination of marginal
stability arguments and the replica method to explain these observations. We
argued that systems with internal degrees of freedom, like the rotations in
ellipsoids, or the variation of the radii in the case of the \textit{breathing}
particles fall in the same universality class. In this paper, we review
comprehensively the results about the jamming with internal degrees of freedom
in addition to the translational degrees of freedom. We use a variational
argument to derive the critical exponents of the contact number, shear modulus,
and the characteristic frequencies of the density of states. Moreover, we
present additional numerical data supporting the theoretical results, which
were not shown in the previous work.
