32
3 Biased Brownian Motion
where P (a, t|x, t 0 ) is the conditional probability that the variable η (t) will assume
the value a at some time t given that it was x at earlier time t 0 . P (b, t|x, t 0 ) can be
defined similarly. The condition for the conservation of the total probability is
P (a, t|x, t 0 ) + P (b, t|x, t 0 ) = 1
(3.25)
The initial condition for Eqs. (3.13) and (3.14) are given by:
P (xt’t|x, t 0 ) = δ xt’x at t = t 0
(3.26)
From Eqs. (3.13) and (3.14) we obtain the steady state solutions
P
s (a) = P (a, ∞|x, t 0 ) =
μ b
μ a + μ b
(3.27)
P
s (b) = P (b, ∞|x, t 0 ) =
μ a
μ a + μ b
The conditional probabilities P (a, t 1 |x, t) and P (b, t 1 |x, t) with x = a, b can be
obtained by solving the Eqs. (3.13) and (3.14) subject to the conditions (3.15), (3.16)
and (3.17):
P =
P (a, t 1 |a, t) P (b, t 1 |a, t)
P (a, t 1 |b, t) P (b, t 1 |b, t)
= τ c
μ b + μ a e −t/τ c μ b (1 − e −t/τ c )
μ a (1 − e −t/τ c ) μ a + μ b e −t/τ c
(3.28)
where
τ c =
1
μ a + μ b
(3.29)
is the characteristic relaxation time to the stationary state of the DMN . In the
foregoing we consider stationary DMN, for which the stationary probabilities of
the two states are
P robability(η = b) = μ a τ c , P robability(η = a) = μ b τ c
(3.30)
We will use the symmetric DMN, for which a = 1 and b = −1, and μ a = μ b = μ,
the corresponding mean value
η(t) = μ a τ c × 1 + μ b τ c × −1 = (μ a − μ b )τ c = 0.
(3.31)
It does not exist a dichotomous noise with η(t) = 0 if one of the states is
zero. Generally is considered η(t) = 0, in order to avoid any systematic bias
introduced by the noise in the dynamics of the driven system. The stationary
temporal autocorrelation function of DMN is exponentially-decaying:
3 Biased Brownian Motion
where P (a, t|x, t 0 ) is the conditional probability that the variable η (t) will assume
the value a at some time t given that it was x at earlier time t 0 . P (b, t|x, t 0 ) can be
defined similarly. The condition for the conservation of the total probability is
P (a, t|x, t 0 ) + P (b, t|x, t 0 ) = 1
(3.25)
The initial condition for Eqs. (3.13) and (3.14) are given by:
P (xt’t|x, t 0 ) = δ xt’x at t = t 0
(3.26)
From Eqs. (3.13) and (3.14) we obtain the steady state solutions
P
s (a) = P (a, ∞|x, t 0 ) =
μ b
μ a + μ b
(3.27)
P
s (b) = P (b, ∞|x, t 0 ) =
μ a
μ a + μ b
The conditional probabilities P (a, t 1 |x, t) and P (b, t 1 |x, t) with x = a, b can be
obtained by solving the Eqs. (3.13) and (3.14) subject to the conditions (3.15), (3.16)
and (3.17):
P =
P (a, t 1 |a, t) P (b, t 1 |a, t)
P (a, t 1 |b, t) P (b, t 1 |b, t)
= τ c
μ b + μ a e −t/τ c μ b (1 − e −t/τ c )
μ a (1 − e −t/τ c ) μ a + μ b e −t/τ c
(3.28)
where
τ c =
1
μ a + μ b
(3.29)
is the characteristic relaxation time to the stationary state of the DMN . In the
foregoing we consider stationary DMN, for which the stationary probabilities of
the two states are
P robability(η = b) = μ a τ c , P robability(η = a) = μ b τ c
(3.30)
We will use the symmetric DMN, for which a = 1 and b = −1, and μ a = μ b = μ,
the corresponding mean value
η(t) = μ a τ c × 1 + μ b τ c × −1 = (μ a − μ b )τ c = 0.
(3.31)
It does not exist a dichotomous noise with η(t) = 0 if one of the states is
zero. Generally is considered η(t) = 0, in order to avoid any systematic bias
introduced by the noise in the dynamics of the driven system. The stationary
temporal autocorrelation function of DMN is exponentially-decaying: