Clas Blomberg - Physics of life-Elsevier Science (2007)

386 Part IX. Going further
some examples which at the bottom are supposed to be deterministic, but at the same time
show important random features.
(1) I may consider the sequence of digits in the decimal expression of the number p, but
can also think of other calculable irrational numbers (calculable here means that there
are methods to calculate values with arbitrary accuracy). p is calculated with a number
of decimals, and the aim is just to see if there are any correlations in the sequence of
digits. Up till now, it is not clear whether there exist determinative patterns in this or if
the sequence can represent a true random sequence. Note now: there are methods of a
deterministic nature to calculate p by an arbitrary accuracy, and we in that way get an
expression in terms of a large number of decimals. But this decimal sequence may well be
a true random sequence. (Or that could be the case for any similar type of calculable number.)
Thus, a deterministically calculated sequence would go for a random sequence,
and there may not be any means to say it is not, neither that it is completely random.
I think that is a fascinating idea: a sequence calculated in some systematic way, producing
a number of digits and decimals which, seen as a sequence turns out to be completely random.
I cannot see any reason that this may not be the case.
(2) My next example is what may be the most relevant for this book and for the physics of
life. It means that there are mechanisms that are very complex involving very many
elements and what we refer to as “degrees of freedom” that their action is completely
out of control and is then best treated as random with probability rules. This applies to
the numbered balls in a lottery, which are mixed, and shaken in a way that there is no
control of which ball falls out. A strict theory might claim that the bouncing of balls
follows deterministic laws, but that cannot be inferred in the final result.
For us, the main accomplishment is the motion of a very large number of atoms and molecules
at a molecular, microscopic level. When we see objects around us and even the smallest
units in a living cell, these are much larger than the basic atomic parts, the motion of which
can be considered as a kind of random background. Again, it is not meaningful to take up all
details in the atomic motions to determine relevant mechanisms at a higher level. Ok, there are
projects, simulations, molecular dynamics or what they are called where one takes up quite
large units and determines complete motion. But the possible outcome there is again limited,
and they at best provide a background for the statistical treatment of larger motions.
(3) My third example is what is called “deterministic chaos”, and which has been exemplified
by the logistic equation and complete loss of accuracy in Section 33D. They are
described by relatively simple expressions, the sequence relation for the logistic equation
and the still simpler successive multiplication by 2 and dropping integers (and the baker’s
transformation) are among the simplest examples. In other cases, they can be obtained
from three or more coupled not too complicated ordinary differential equations. The
equations are deterministic in the sense that the solutions are completely determined
by initial values. Still, differences between solutions corresponding to slightly different
initial values diverge away from each other, which means that the outcome after some
initial time is not predictable and can be considered as random.