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