  Mechanical disorder in solids, which is generated by a broad range of
physical processes and controls various material properties, appears in a wide
variety of forms. Defining unified and measurable dimensionless quantifiers,
allowing quantitative comparison of mechanical disorder across widely different
physical systems, is therefore an important goal. Two such coarse-grained
dimensionless quantifiers (among others) appear in the literature, one is
related to the spectral broadening of discrete phononic bands in finite-size
systems (accessible through computer simulations) and the other is related the
spatial fluctuations of the shear modulus in macroscopically large systems. The
latter has been recently shown to determine the amplitude of wave attenuation
rates in the low-frequency limit (accessible through laboratory experiments).
Here, using two alternative and complementary theoretical approaches linked to
the vibrational spectra of solids, we derive a basic scaling relation between
the two dimensionless quantifiers. This scaling relation, which is supported by
simulational data, shows that the two apparently distinct quantifiers are in
fact intrinsically related, giving rise to a unified quantifier of mechanical
disorder in solids. We further discuss the obtained results in the context of
the unjamming transition taking place in soft sphere packings at low confining
pressures, in addition to their implications for our understanding of the
low-frequency vibrational spectra of disordered solids in general, and in
particular those of glassy systems.
