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In-Depth Information
Equation (
7.128
) is formally identical to the linear Langevin equation considered
in Section 3.2.2. Thus, for a vanishing initial condition we are immediately led to the
average response
t
t
)
]
η(
t
)
dt
.
)
=
Q
(
t
exp
[−
γ(
t
−
(7.132)
0
Apparently, this equation is different from (
7.70
) and (
7.73
) because it involves the
autocorrelation function
t
)
=
t
)
]
s
(
t
−
exp
[−
γ(
t
−
(7.133)
rather than the derivative of the autocorrelation,
d
dt
s
(
t
)
=
γ
t
)
]
.
t
−
exp
[−
γ(
t
−
(7.134)
However, for a proper comparison we must take into account that
ξ
s
and
ξ
p
are dimen-
η
sionless variables, whereas
Q
has the dimensions of space and
has the dimensions of
velocity. We are assuming that the particle's mass is unity. Therefore we introduce the
transformations
1
Q
2
ξ
s
(
t
)
=
Q
(
t
)
(7.135)
and
1
ξ
p
(
t
)
=
Q
2
η(
t
),
(7.136)
γ
η
and, by expressing
Q
and
in terms of the dimensionless variables and cancelling out
the common factor
Q
2
,weturn(
7.132
)into
t
e
−
γ(
t
−
t
)
ξ
p
(
t
)
dt
,
ξ
s
(
t
)
=
γ
(7.137)
0
which coincides with the theoretical prediction of (
7.70
) together with (
7.73
). Note that
the Langevin equation is a form of
linear
stochastic physics that makes the LRT valid
with no restriction that the perturbation has to be extremely small in intensity. This
explains the lack of the parameter
ε
in both (
7.137
) and (
7.132
).
7.3.3
The experimental approach
The result (
7.97
), which is widely accepted by the community studying stochastic reso-
nance, suggests how to extend LRT using conjectures associated with the non-stationary
case. Let us assume that the number of events per unit time, rather than being constant,
for physical reasons becomes smaller and smaller, until asymptotically this rate reaches
averysmallvalue
g
min
that is much smaller than the frequency
of the periodic driver.
Let us imagine that after achieving this very small value the rate of event generation
ω
