Information Technology Reference
In-Depth Information
dQ
(
t
)
=−
∂(
Q
)
Q
+
η(
t
),
(3.124)
dt
∂
the associated phase-space variable is denoted
q
and the dynamics of the variable are
given by the potential
U
The path from the Langevin equation to the phase-space
equation for the probability density can be very long and technically complicated. It
depends on whether the random fluctuations are assumed to be delta-correlated in time
or to have memory; whether the statistics of the fluctuations are Gaussian or have a more
exotic nature; and whether the change in the probability density is local in space and
time or is dependent on a heterogeneous, non-isotropic phase space. Each condition
further restricts the phase-space equation for zero-centered, delta-correlated-in-time
fluctuations with Gaussian statistics
(
q
).
η(
t
)
=
2
D
λ
t
),
η(
t
)
=
0
and
t
)η(
2
δ(
t
−
(3.125)
with the final result being the form of the Fokker-Planck equation (FPE)
∂(
P
∂
P
(
q
,
t
|
q
0
)
=
∂
∂
q
)
D
λ
2
∂
+
(
q
,
t
|
q
0
).
(3.126)
∂
∂
∂
t
q
q
q
The FPE describes the phase-space evolution of complex webs having individual tra-
jectories described by (
3.124
) and an ensemble of realizations described by
P
(
q
,
t
|
q
0
).
The steady-state solution to the FPE is independent of the initial condition,
∂
P
ss
(
q
,
t
|
q
0
)
=
∂
P
ss
(
q
)
=
0
,
(3.127)
∂
t
∂
t
and is given by the probability density
Z
exp
Z
exp
−
λ
U
(
q
)
U
(
q
)
P
ss
(
q
)
=
=
−
,
(3.128)
D
k
B
T
where we have used the Einstein fluctuation-dissipation relation (FDR)
D
λ
=
k
B
T
(3.129)
to express the strength of the fluctuations
D
in terms of the bath temperature
T
;
Z
is the
partition function
exp
dq
U
(
q
)
Z
=
−
(3.130)
k
B
T
and the distribution has the canonical form given in equilibrium statistical mechanics.
The FPE can also be expressed as
∂
P
(
q
,
t
|
q
0
)
=
L
FP
P
(
q
,
t
|
q
0
)
(3.131)
∂
t
in terms of the Fokker-Planck operator
∂(
L
FP
≡
∂
∂
q
)
D
λ
2
∂
+
.
(3.132)
q
∂
q
∂
q