Information Technology Reference
In-Depth Information
7.6.4
Particle's Motion Is a Generalization of Gradient
Algorithms
As analyzed, the Wiener process describes the Brownian motion of a particle. In this
section, this motion is also stated in terms of equations of Langevin's equation. The
stochastic differential equation for the position of the particle is [
56
]:
dx
.t/ D
u
.x.t//
dt
C
dw
.t/
(7.62)
where
u
.x.t// is the so-called
drift function
, and is usually given in the form of a
spring force, i.e.
u
.x/ Dkx which tries to bring the particle to the equilibrium
x D 0 and is the result of a parabolic potential applied to the particle, i.e. V.x/ D
kx
2
.Theterm
w
.t/ denotes a random force (due to interaction with other particles,
e.g. collision) and follows a Wiener walk. For each continuous random path
w
.t/,a
continuous random path x.t/ is also generated, which can be written in the form
Z
t
x.t/ D x.0/C
u
.x.s//ds C
w
.t/
(7.63)
0
The integration of Langevin's equation and certain assumptions about the
noise
w
.t/, for instance white noise, dichotomic noise (also known as
Ornstein-Uhlenbeck noise), etc., enable the calculation of the mean position of
the particle Efxg and also to find its variance Efx Efxgg
2
[
66
].
Knowing that the QHO model imposes to the particle the spring force of
Eq. (
7.61
), the kinematic model of the diffusing particle becomes
dx
.t/ D!x.t/
dt
C
dw
.t/
(7.64)
with initial condition x.0/ D 0. The first term in the right part is the drift, i.e. the
spring forces that makes the particle return to x.0/, while the second term is the
random diffusion term (noise).
The extension of Langevin's equation gives a model of an harmonic oscillator,
driven by noise. Apart from the spring force, a friction force that depends on
the friction coefficient and on the velocity of the particle is considered. The
generalized model of motion of the particle can then be also written as [
13
,
66
]:
d
2
x
dt
2
C 2
dx
dt
C !
2
x D .t/
(7.65)
Thus an equation close to electrical or mechanical oscillators is obtained [
163
,
195
].
Equation (
7.64
) can be also written as
