Coarse-Grained Quantities in Aperiodic Systems

It is shown in a linear chain model that the value of the Green function Gn, n′(E) associated with n and n′th sites is almost determined by the environmental situations of sites up to the lth from these sites, where l is finite so far as Γ ≡ Im E > 0. Since many physical quantities coarse-grained over an energy (squared frequency) interval of the order 2Γ can be described in terms of such Green functions, the problem of obtaining such physical quantities in an aperiodic system is conceptually divided into the following two stages: (1) to obtain the contribution to the coarse-grained physical quantity from sites with various environmental situations, (2) to sum up these contributions according to the distribution law of such environmental situations which corresponds to the structure of the system. Since l is finite, the problem is essentially reduced to that of the system of finite size even if the aperiodic system is infinitely large. After estimating the order of magnitude of l, histograms of frequency spectra of isotopically disordered chains were calculated from this viewpoint, the result of which indicates the effectiveness of this approach.