E.1 Stochastic Dynamics
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probability T (ω, t)|ω 0 , t 0 ), given by Eq. E.4, is a solution of the diffusion
equation in ω, with the diffusion coefficient D = 1,
∂T (ω, t|ω 0 , t 0 )
∂t
=
1
2
∂ 2 T (ω, t|ω 0 , t 0 )
∂ω 2
(E.6)
with the initial condition
T (ω, t 0 |ω 0 , t 0 ) = δ(ω − ω 0 )
(E.7)
This means the matter which is spread is at the initial time t 0 , all concentrated
at the origin ω 0 .
E.1.4 Stochastic Process Derivative
As an stochastic process , X(t), is not a function, in the usual sense, we have to say
what we understand for theirderivative,
˙
X(t) ≡
X(t)
dt
(E.8)
“ ˙
X(t) means the stochastic process whose realizations are the derivatives of the
realizations of the SP X(t)”.
E.1.5 SDE with Aditive Noise
We will integrate the Eq. 2.4 from t to t + t and after we will take the limit t →
dt,
t+t
0
dX
t
= −
1
γ
t+t
0
dV (X(t ))
dX(t )
dt
+
2k B T
γ
1/2 t+t
0
ξ(t
)dt
(E.9)
or
X(t) ≡ X(t + t) − X(t)
= A(X))t + B (W (t + Δt) − W (t))
≡ A(X))t + BBW (t),
(E.10)
we have used the definition of Wiener’s process, Eq. E.2 and A(X) = −
1
γ
dV (x(t ))
dx(t ) ,
this last term means the mean valor in the interval t. In the limit t → dt