# Brownian Control and the Test

[Hans Blom, 960222c]

(Bill Powers (960220.1130 MST))

Your proposed [Brownian] control system makes r depend on d. You
don't mention any reference level for d, but I presume there is one
in your simulation (even if it's just zero). The reference level of
d is defined for your situation as that value of d at which there is
no tendency to increase or decrease r.

No, I didn't set my system up as a control system. I did the
following:

1. define the chemical's concentration c as position dependent:
c = f (x, y) if in two-space

2. define the particle's radius as concentration dependent:
r = g (c)

3. compute the particle's next position change d based on Einstein's
formula: d = k / sqrt (r) with a suitably chosen k; note that this
disregards that in Einstein's formula d is an _average_ value
only; since I do not know d's probability density function, I
cannot simulate how it would vary

4. make the particle move a distance d in a randomly chosen direction
yielding new x and y coordinates.

5. loop to 1.

My observations showed that the simulated particle's movements depend
upon the choice of the functions f and g. In particular, if f is
constant (no concentration gradient) or if g is constant (no change
of the particle's radius) we have the standard Brownian movement. But
with suitably chosen f and g we observe that the particle's position
tends to remain in the vicinity of the maximum concentration. This
led me to the hypothesis that we might have something resembling a
control system here, where the particle's radius functions both as
the sensor _and_ as the actuator. Kind of funny, maybe, and certainly
different from the standard control setups that we normally consider.
On the other hand, the explanation might be that the particle cannot
escape from a gravitation-like "field", which would indicate that we
have an equilibrium system. Now, which is the "better" explanation?

I didn't program a control system; I iterated a number of (more or
less realistic) physics-like equations in order to see what kind of
movement ("behavior") would result for the particle.

So basically we want to see if the partial derivative of (d^2) with
respect to T is less when your control system is acting than when it
is not acting. And of course it is of interest to see _how much
less_ the effect is when control is present.

This statement indicates that the description of my experiment was
unclear. Sorry. In my simulation, I cannot discriminate between "a
control system acting" or not, because the simulation wasn't set up
that way. I hope the above makes clearer why I cannot do The Test as
you propose it.

Greetings,

Hans

···

================================================================
Eindhoven University of Technology Eindhoven, the Netherlands
Dept. of Electrical Engineering Medical Engineering Group
email: j.a.blom@ele.tue.nl

Great man achieves harmony by maintaining differences; small man
achieves harmony by maintaining the commonality. Confucius