How do
marginal modes transmit energy, momentum

from V. Vitelli et al
PRE 2009

Excite modes of
frequency È w in a finite region

How does the energy
spread out in time?

distance2 È **d(****w****)**
(time)

as with particle
diffusion, we can infer **d(****w****)** from a correlation function

particles: **d(****w****) **=
(const) ˜0 ‡v(0) v(t)– dt

´

energy: **d(****w****) **=
(const) ˜0 ‡Je(0) Je(t)– dt

´

particle current

energy current

What is the energy
current for our system?

eg: independent steps
of length b every time t: ˜
‡v(0) v(t)– ¨ b2/t