On The Statistical Energy Analysis Hypothesis Of Coupling Power Proportionality And Some Implications Of Its Failure

Abstract The Statistical Energy Analysis hypothesis that the coupling power between two subsystems is proportional to their energy density difference is examined. A clear distinction is made between coupling loss factors defined when excitations are applied to different subsystems. For any specific excitation distribution it is seen that the subscripting relation n i η ij = n j η ji requires only that the coupling is conservative. However, if first subsystem i and then subsystem j is excited, the relation n i η ( i ) ij = n j η ( j ) ji holds only if coupling power proportionality (CPP) is exact. The case of a system comprising two subsystems is considered and a necessary and sufficient condition is found for CPP to hold. Some implications for coupling powers, energy densities and coupling loss factors are considered if CPP does not hold. It is seen that energy can flow from a subsystem with a lower to one with a higher energy density, and that coupling loss factors can be positive, negative or infinite. The estimation of coupling loss factors from experimental measurements is discussed. Then the case of a system comprising an arbitrary number of subsystems is considered. It is seen that the coupling power between two subsystems cannot be uniquely defined due to "power circulation", in which a group of three subsystems exchange energy. Necessary and sufficient conditions are found for CPP to be exact. In the general case these do not hold. Weak coupling is then defined to be that in which all coupling powers are small compared to the input powers. It is shown that CPP holds for two directly coupled subsystems in a weakly coupled system if it holds for these two subsystems in isolation, to a first approximation at least. Furthermore, the coupling powers between indirectly excited subsystems are very small compared to those between two subsystems, one of which is excited.