8.1 The Quantum Langevin Equation
129
8.1.2 The Quantum Overdamped Langevin Equation with
Colored Noise
We follow the formalism, except for the noise, given by [10] and used by [11],
namely
˙
x = −
U (x)
0
+
F
0
− 0.5
λ
0
U
(x) + (t)
(8.40)
U(x) is the internal energy generated by the motor, F is an external force. See
bellow for the noise (t). Then from the fitting of Fig. 8.1, we can write,
I ntegral =
3
k=1
D k
τ k
exp
−
t
τ k
(8.41)
or,
˜
(˜ t)˜ (˜ t
)
=
D eff ( ˜
x)
τ
3
k=1
D k
τ k
exp
−
t s
τ k τ
(8.42)
where we have considered that the system itself is which perceives the time scale
of the noise, consequently, we must measure time in units of τ s , Equation 8.42, was
equivalently written by [6],
Fig. 8.1 Integral from Eq. 8.32 fitted with 3 exponentials
129
8.1.2 The Quantum Overdamped Langevin Equation with
Colored Noise
We follow the formalism, except for the noise, given by [10] and used by [11],
namely
˙
x = −
U (x)
0
+
F
0
− 0.5
λ
0
U
(x) + (t)
(8.40)
U(x) is the internal energy generated by the motor, F is an external force. See
bellow for the noise (t). Then from the fitting of Fig. 8.1, we can write,
I ntegral =
3
k=1
D k
τ k
exp
−
t
τ k
(8.41)
or,
˜
(˜ t)˜ (˜ t
)
=
D eff ( ˜
x)
τ
3
k=1
D k
τ k
exp
−
t s
τ k τ
(8.42)
where we have considered that the system itself is which perceives the time scale
of the noise, consequently, we must measure time in units of τ s , Equation 8.42, was
equivalently written by [6],
Fig. 8.1 Integral from Eq. 8.32 fitted with 3 exponentials