Environmental Engineering Reference
In-Depth Information
ξ
dn
(
t
)
1
k
2
k
1
t
2
Figure 2.1. Parameters of the dichotomous noise and representation of a typical
realization.
rate
k
1
for the transition
1
. A realization of the process
is shown in Fig.
2.1
. The path of the noise is a step function with instantaneous jumps
between
→
2
and
k
2
for
→
1
2
2
and random permanence times,
t
1
and
t
2
, in these two states. The
mean permanence times in the two states are
1
and
/
k
2
.
Moreover, when the transition rates
k
1
and
k
2
are constant in time, the permanence
times are exponentially distributed random variables (e.g.,
Bena
,
2006
). If
t
1
=
τ
1
=
1
/
k
1
and
t
2
=
τ
2
=
1
,
the noise is called
symmetric
DMN; otherwise it is called
asymmetric
DMN. This
type of noise was first introduced in information theory under the name of
random
telegraph noise
or
Poisson square wave
(e.g.,
McFadden
,
1959
;
Pawula
,
1967
); this
process, studied in detail by physicists (
Hongler
,
1979
;
Kitahara et al.
,
1980
), is called
a two-state Markov process or DMN.
The probability
P
1
(
t
) that the process is in the state
1
=|
2
|
1
at time
t
obeys the kinetic
equation
d
P
1
(
t
)
d
t
=
−
,
k
2
P
2
(
t
)
k
1
P
1
(
t
)
(2.2)
which includes a gain term
k
2
P
2
(
t
) accounting for the probability of being in
ξ
dn
=
2
and jumping to
ξ
dn
=
1
and a loss term
−
k
1
P
1
(
t
) that accounts for the probability
of escaping from the state
ξ
dn
=
1
. Analogously, for the probability
P
2
(
t
) that the
process is in the state
ξ
=
2
at time
t
,wehave
dn
d
P
2
(
t
)
d
t
=
k
1
P
1
(
t
)
−
k
2
P
2
(
t
)
.
(2.3)
We obtain the steady solutions by neglecting the temporal derivatives on the left-
hand side of Eqs. (
2.2
)and(
2.3
):
k
2
k
1
+
k
1
k
1
+
P
1
=
k
2
,
P
2
=
k
2
.
(2.4)
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