  We propose that the properties of the capacity of entanglement (COE) in
gapless systems can efficiently be investigated through the use of the
distribution of eigenvalues of the reduced density matrix (RDM). The COE is
defined as the fictitious heat capacity calculated from the entanglement
spectrum. Its dependence on the fictitious temperature can reflect the
low-temperature behavior of the physical heat capacity, and thus provide a
useful probe of gapless bulk or edge excitations of the system. Assuming a
power-law scaling of the COE with an exponent $\alpha$ at low fictitious
temperatures, we derive an analytical formula for the distribution function of
the RDM eigenvalues. We numerically test the effectiveness of the formula in
relativistic free scalar boson in two spatial dimensions, and find that the
distribution function can detect the expected $\alpha=3$ scaling of the COE
much more efficiently than the raw data of the COE. We also calculate the
distribution function in the ground state of the half-filled Landau level with
short-range interactions, and find a better agreement with the $\alpha=2/3$
formula than with the $\alpha=1$ one, which indicates a non-Fermi-liquid nature
of the system.
