In an earlier work [S. A. Cheong and C. L. Henley, preceding paper], we derived an exact formula for the many-body density matrix ρ B of a block of B sites cut out from an infinite chain of noninteracting spinless fermions, and found that the many-particle eigenvalues and igenstates of ρ B can all be constructed out of the one-particle eigenvalues and one-particle eigenstates, respectively. In this paper we improved upon this understanding, and developed a statistical-mechanical analogy between the density-matrix eigenstates and the many-body states of a system of noninteracting fermions. Each density-matrix eigenstate corresponds to a particular set of occupation of single-particle pseudo-energy levels, and the density-matrix eigenstate with the largest weight, having the structure of a Fermi sea ground state, unambiguously defines a pseudo-Fermi level. Based on this analogy, we outlined the main ideas behind an operator-based truncation of the density-matrix eigenstates, where single-particle pseudo-energy levels far away from the pseudo-Fermi level are removed as degrees of freedom. We report numerical evidence for scaling behaviors in the single-particle pseudo-energy spectrum for different block sizes B and different filling fractions n. With the aid of these scaling relations, which tell us that the block size B plays the role of an inverse temperature in the statistical-mechanical description of the density-matrix eigenstates and eigenvalues, we looked into the performance of our operator-based truncation scheme in minimizing the discarded density-matrix weight and the error in calculating the dispersion relation for elementary excitations. This performance was compared against that of the traditional density-matrix-based truncation scheme, as well as against an operator-based plane-wave truncation scheme, and found to be very satisfactory.