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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
 
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