Hartmut-Loewen.pdf

idea of Archer and Evans JCP 121, 4246 (2004)
integrate Smoluchowski equation
N dr dr d ˆ ω ... d ˆ ω
... 2
∫ 2 ∫ N ∫ ∫ N
v
v v
∂ρρ ( r , ˆ ω , t ) v t ⎡ v v 1 v v v F ( r , ˆ ω , t ) ⎤
= ∇ v • D(
ˆ)
• ⎢ ∇ ( ˆ t)
+ ( ˆ t)
∇ V ( ˆ
r D(
ω)
v
r ρ(
r,
ω,
t)
ρ(
r,
ω,
t)
v
rVext
( r,
ω,
t ) − ⎥
∂t ⎣
kBT
kBT
⎦
⎡
⎤
+ ˆ • ˆ v 1 v
v 1 v v
D ⎢ ( , , ) + ( , ˆ,
) ∇ ( , ˆ,
) − T ( r,
ˆ
rR
Rρ
r ω t ρ r ω t v
⎢
rVext
r ω t
ω,
t)
⎥
⎣
k kB T
k kB
T ⎦
with average force and torque
v r
F(
r,
ˆ ω,
t)
v v
T ( r,
ˆ ω,
t)
=
3
= −∫dr′
∫
−
2
d ω′
ρ
( 2)
v v v v v v v v v
( r,
r′
, ˆ ω,
ˆ ω′
, t)
∇ vV ( r,
r′
, ω,
ω′
, t)
= ρ ( r,
ˆ ω)
∇
v v
( r,
r′
, ˆ ω,
ˆ ω′
, t)
Rˆ
[ ρ]
δFexc
δρ ( r,
ˆ ω)
r 2
0
v
r v
0
v v v v
( r,
r′
, ω,
ω′
, t)
[ ρ]
v ˆ δFexc
ρ ( , ˆ 0 r ω)
R
δρ ( r,
ˆ ω)
3 2 ( 2)
∫dr′ ∫ d ω′
ρ
v
rV2 =
v
0
in general unknown
in equilibrium (Gubbins CPL 76, 329 (1980))