Trial modes account for excess slow modes
For system size L (in 3D) there are
N(L) ~ L2  trial modes
Their frequency w0(L) ~ 1/L as in a normal solid.
The density of lowest modes
D0(w) ~ N   / (wL3)  ~ L0    É. as observed
N(L) ~ Ld-1  trial modes; D(w) ~ N / (wLd)  ~ L0
Lowest frequency modes:
Higher-frequency modes:
For subsystem of size L/2, this argument Þ D(w) Å D0(w)  up to w(L/2) Å 2 w0
Extending to L Å few x particle size Þ D(w) Å D0(w) for nonzero fraction of modes.
N(L) = 3   fundamental acoustic modes
D0(w) ~ N   / (wL3)  ~ L2
D0(w) ~ d / (wLd)  ~ Ld-1
cf normal solid
In d dimensions, there are