Neutron Scattering

17 .2.2 Diffusion, microscopic approach : Langevin equation
For atomic distances and short times the above continuum theory has to be modified . A microscopie
model leads to the Langevin equation . A particle of mass M in a thermal bath is exposed
to stochastic kicks F(t) . After the kick it is slowed down by internal friction proportional to
its velocity with the viscosity p as proportionality factor .
~ll
dt
= - B + F(t)
dv
dt
-riv- + f (t) (17 .10)
To keep the energy of the system constant the two terms on the right hand side are related by
the fluctuation-dissipation theorem
( .f (t) .f (o)>
2k B T
= M
7js(t)
which means in words, that the stochastic force takes ils energy from friction losses . Integration
of (17 .10) yields
exp(77t') v(t) = exp(-Ot)
`
.f (t')dt' (17 .12)
~x
This result is used to calculate the velocity-velocity correlation function . For one component
it is
(v(t)v(o)>
t
= exp(-Ot)(f
and using (17 .11) (factor 3 for vectors)
0
dt' f-
" ~c. rr
dt " exp( 77t')f(t')exp(gt
) .f(t
)>
= exp(-Ot) dt,' dt,"(f(t')f (t ))exp(o(t' - J J t")
x x
z t z 0
3BT exp(-rit)
Integrating the velocity-velocity correlation function yields the mean square displacement [Il
1 e
3 ~ (t - tl)(p(tl)v(0))dtl
c
(t 3 T - ) e
3
tl xp(-rit)dtl
ih1
kBT
(1 77
( t - - exp(-~1t))
77 J
D(t-T r (1-exp(- t )) (17 .14)
Tr
The fig .
17 .1 summarizes the mean square displacements of varions translational motions
in Gaussian approximation . We consider the limits :
17- 4