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Homework 8 (due December 16)
Problem 81 (5 pts.)
Starting with Langevin equation for a 1D Brownian particle,
f
nd the mean square displacement as a function of
time
τ
:
D
(
x
(
t
+
τ
)
−
x
(
t
))
2
E
t
. Neglect the inertial e
f
ects, and take the particle mobility to be
b
. Consider the
following cases:
a) Free particle,
U
(
x
)=0
.
b) Particle subjected to a uniform force,
U
(
x
)=
−
Fx
.
c) Particle in a Harmonic potential,
U
(
x
)=
−
kx
2
/
2
.
d) By comparing the result of (a) with the solution of the di
f
usion equation, express di
f
usion coe
ﬃ
cient
D
in terms
of mobility
b
.
Solution:
a)
˙
x
=
u
(
t
)
,
h
u
(
t
+
t
0
)
u
(
t
)
i
=
2
T
b
δ
(
t
0
)
D
(
x
(
t
+
τ
)
−
x
(
t
))
2
E
=
*
τ
Z
0
τ
Z
0
u
(
t
+
t
00
)
u
(
t
+
t
0
)
dt
0
dt
00
+
=
2
T
b
τ
Z
0
τ
Z
0
δ
(
t
0
−
t
00
)
dt
0
dt
00
=
2
T
b
τ
b)
˙
x
=
F
b
+
u
(
t
)
,
D
(
x
(
t
+
τ
)
−
x
(
t
))
2
E
=
μ
Fτ
b
¶
2
+
*
τ
Z
0
τ
Z
0
u
(
t
+
t
00
)
u
(
t
+
t
0
)
dt
0
dt
00
+
=
2
T
b
τ
+
μ
Fτ
b
¶
2
c)
˙
x
−
k
b
x
=
u
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 Winter '04
 ANON
 mechanics, Inertia, Work, Statistical Mechanics

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