  By extending Einstein's separation of wave and particle parts of the second
order thermal fluctuation to encompass "generalized fluctuations" in any Bose
field, P. E. Gordon has proposed alternative definitions for nth order
coherence and nth order coherent states. The main point of this paper is to
explore some of the physical insights to be gained by extending dualism to
higher orders. Recent experiments have examined aspects of the coherence of
Bose-Einstein condensates. It has been argued that the condensate state is
coherent to (at least) second or third order, but the coherence properties of
Bose-Einstein condensates remain somewhat controversial. Using probability
distributions developed by M. O. Scully and V. V. Kocharovsky et. al., we apply
Gordon's dualistic expression of the coherence conditions to investigate
coherence properties in Bose-Einstein condensation. Via numerical calculations,
we present a graphical survey of wave-like and particle-like fluctuations in
condensed and uncondensed fractions. Near the critical point, we find a very
marked peak in the ratio of nth order wave to nth order particle fluctuations
in the condensate. Not surprisingly, n-point correlations between the positions
of condensate atoms also peak near the critical temperature, and this
apparently mirrors, to higher orders, the well-known relation between the
integral of the 2-point correlation function over a certain volume and the rms
fluctuation in the number of particles in that volume.
