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or where it was between the recorded points. The lines might be thought of as “possible”
or “average” paths of the heavy particle rather than the actual path.
The random motion depicted in Figure
3.2
can be modeled by the dynamics of the
heavy particle using Newton's force law
M
d
V
dt
=
F
(
t
),
(3.49)
where
M
is the mass of the heavy particle,
V
is its velocity and
F
is the vector sum
of all the forces acting on it. For simplicity we set
M
1, reduce the problem to one
dimension and restrict all the forces to be those generated by collision with the lighter
fluid particles, so that (
3.49
) reduces to
=
dV
dt
=
F
(
t
).
(3.50)
This is of the form (
3.39
) for a free particle being buffeted about by the fluid.
For additional simplicity we require the average of the random force to vanish, but
physical arguments establish that for Brownian motion the average force is given by the
hydrodynamic back-reaction of the fluid to the motion of the heavy particle. This fluid
back-reaction is the Stokes drag produced by the heavy particle pulling ambient fluid
along with it as it moves through the fluid, yielding an average force proportional to the
velocity of the heavy particle:
F
(
t
)
=−
λ
V
(
t
).
(3.51)
On replacing the force in (
3.50
) by its average value (
3.51
) plus random fluctuations,
the equation for the Brownian motion of a heavy particle becomes
dV
dt
=−
λ
V
(
t
)
+
η(
t
),
(3.52)
where
is a zero-centered random force and is of the form (
3.48
). This mode of argu-
ment was introduced into the physics literature by Langevin [
29
] and resulted in phys-
ical stochastic differential equations. These equations were the first indication that the
continuous nature of physical phenomena might not be as pervasive as had been believed
in traditional theoretical physics. But then physicists as a group never let rigorous math-
ematics stand in the way of a well-argued physical theory. A new calculus involving
stochastic processes had to be developed in order to rigorously interpret (
3.52
).
It is probably time to step back and consider what we have. The source of the above
Langevin equation is undoubtedly physics and the attempt to model complex physical
interactions of a simple web with the environment. But the form of the equation is actu-
ally a great deal more. Suppose that
V
is not the velocity of a Brownian particle, but is
the variable describing the dynamics of the web aggregating information from a num-
ber of sources. If the sources are incoherent, such as the collisions, then the incremental
information is random. The term linear in the dynamical variable
V
in an information
context may be considered to be a filter on the noise. Consider the Fourier transform of
a function
η(
t
)
