Anomalous Size Dependence of Inverse Participation Ratio of Eigenfunctions in Graded Elastic Lattices

Abstract

We have studied the inverse participation ratio (IPR) denoted by P -1 of vibrational modes in one-dimensional graded elastic chains. The size dependence of IPR can be shown to derive from a quantum interpretation of vibrational modes. The quantum analogue of peculiar vibrational modes (gradons) to graded systems is established for the hump structure of the gradon front via the fact that the probability of a quantum particle is inversely proportional to its velocity. In this way, the envelope function can be determined analytically, and matches the mode pattern quite well. We find that the IPR exhibits an anomalous size ( N ) dependence and can be captured accurately by the relation: N P -1 = C 1 log N + C 2 , where C 1 and C 2 are constants. This interpretation is important in understanding a wide variety of properties of graded systems.