Physical meaning of the probability theory
Yurii Kosovtsov
The probability theory is very nice logically consistent theory. The
hallmark of the theory is the notion of (statistical) independence.
It is very essential that for each problem in the framework of the
probability theory the initial (probability) measure is specified.
But in the physical statement of probabilistic problems we face some
fundamental difficulties.
1. One of the basic concepts of physics is the use only measurable
variables. In the sense, that there are instrumentation and procedure
to obtain a "number" for given variable. Alas, there is not any
procedure to obtain the "probabilistic number" without hiring a priori
the (statistical) independence for series of mensurations. That is we
obviously fall into a circle. It is not surprising that very often the
probabilistic characteristics of input processes of some physical
systems are considered as uncertain.
2. This forces to formulate "physical" ("intuitive") versions of
(statistical) independence, which dramatically differs from its
original mathematical meaning. In most popular version two random
variables are considered as (statistically) independent if they are
affected by many causally independent disturbances. The more or less
rigorous consideration of physical systems from the probability theory
point of view makes it clear that the probability to meet
(statistically) independent physical random variables is equal to
zero.
3. Spurious way out of points 1 and 2 commonly is considered as
follows. Yes, the probabilistic characteristics of input processes of
some physical systems are uncertain. But the probability theory has
the extremely helpful proposition - the Central Limit Theorem (CLT).
The output processes are usually a result of an inertial
transformation of initial random functions, therefore very often they
may be regarded as a sum of a large number of random items. For this
reason and owing to CLT we as though come to a conclusion that for
many practically important cases the distribution function of the
output of the linear (or even non-linear) inertial system is
approaches to Gaussian one. Hence it follows very powerful
consequences: despite the fact that a probability as a measure on a
certain set of elementary events is a priori unknown, the profound
conclusions are possible, since for many cases the functions
interested us have been determined on this set are weakly depend on
its exact probability measure. In a different way, as a result of a
transformation the probability measure is unified and practically
"forgets" about own origin. This position is extremely attractive
since it causes an impression that it is possible to extract "free of
charge" the probability laws from anything. It is unlikely possible
to embrace all fields of the probability theory applications where
Gaussian distribution of a random function is led "naturally" only on
reason of inertial character of a dependence of a "output" random
function from "input" one or when the outcome of a chance experiment
is determined by a large number of random factors. All of these had
created the ground for deifying of Gaussian distribution.
However, the main feature of totality of statements are known under
the name CLT of the probability theory is that under increasing of
number of items the distribution of sums of (statistically)
independent items is tending to Gaussian one IF AND ONLY IF the sum is
NORMALIZED by STRICTLY definite way.
But in the nature there is not this specific however necessary
normalization. There is not proper counter of numbers N of independent
items in any physical systems and there is not proper divider on (N)^1/2. That
is why there are not grounds for CLT application here.
So, what is the physical meaning of the probability theory?
Yurii Kosovtsov
Lviv Radio Engineering Research Institute, Ukraine