11.E.1. Derivation of the Diffusion Equation
The difference between the tracer and normal particle densities is
 r, t   nN r, t   nT r, t    d 3 p1 f 0  p1  h   p1 , r, t 
  d 3 p1 f1 0 h1
(11.107)
which vanishes at equilibrium. Now,
d
3
p1 f10  11.105 gives

 h1 p1 h1 
3
0

d
p
f



d
p
f
C
1 h1
1
1
1




m r 
 t
3

0
1


2
   JD 
1, C h 
t
n0
where the diffusion current J D is defined as
p
p
J D  r, t    d 3 p1 1 f 0  p1  h   p1 , r, t    d 3 p1 1 f1 0 h1
m
m

Using the same technique as for C , we can show that C
Hence


2
   JD 
C 1, h   0
t
n0

(11.109)
is also hermitian.
(11.108)

where we've used C 1  0 , which follows directly from (11.106).
can write
J D   D
By Fick's law, we
(11.110)
where D is the self-diffusion coefficient.

 D 2
t
Combining (11.108,10), we have
(11.111)
which is a hydrodynamic equation for the self-diffusion process.
To find the
dispersion relation for propagating modes, we introduce the Fourier transform
  r, t   
d
d 3k
  k,   exp  i k  r  i t 
 2   2
3
so that (11.111) becomes
(11.113)
i   Dk 2
which can be satisfied for all  if
  iDk 2
(11.114)
(11.112)
Thus,  is imaginary so that all modes eventually die out.
becomes
  r, t   
d
d 3k
Eq(11.112) thus
  k,   exp  i k  r  Dk t 
 2   2
2
3
The decay time  for the k mode is therefore 
k  0 so that  can be quite large.
1
.
Dk 2
For long wavelength modes,