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Ta b l e 4 . 6
The Pythagorean recombination game
Randomly choose two numbers
a
,
b
of the given number population;
Randomly choose a number
a
1
,such
a
1
≤
a
, and split
a
into
a
1
and
a
2
=
a
2
−
a
1
;
b
2
−
b
1
;
Randomly choose a number
b
1
,such
b
1
≤
b
, and split
b
into
b
1
and
b
2
=
a
1
+
b
1
+
b
with the pair
a
=
b
2
,
b
=
a
2
.
Replace the pair
a
,
collision, exchange the components of their velocities along the collision line, by
leaving unchanged the components orthogonal to that line.
Now if we apply the Pythagorean recombination game to a given distribution,
along a number of steps we observe two facts: 1) the
H
function decreases (does not
increase), and 2) the distribution approximates to a
2
distribution, which is typical
of sum of squares of stochastic variables following normal distributions.
By using a simple MATLAB function, a series of numerical experiments were
performed, some of which are reported in Figs. 4.10, 4.11, 4.12, 4.13, and 4.14.
We claim the validity of the following statement which corresponds to an abstract
formulation of the H theorem.
χ
Boltzmann's
function
H
decreases,
or
does
not
increase,
during
the
Pythagorean recombination game.
In particular, the
H
theorem follows from the Maxwell's velocity distribution
theorem, by means of statistical and informational arguments. In fact, the normal
distribution is the distribution holding for casual phenomena where a great num-
ber of independent causes influence their evolution. Therefore, it is reasonable to
expect that after a great number of collisions the velocity distribution of each com-
ponent follows a normal distribution. Moreover, given the nature of the collisions
which follow the conservation principles of elastic collisions, we can assume that
the variance of the distributions remains constant during the evolution. In fact, the
sum of squares of two colliding “values” does not change after the application of a
Pythagorean recombination (the velocities they represent exchange one component,
see Table 4.6).
Therefore, starting from an initial value distribution, by applying the Pythagorean
recombination game, we get distributions which keep the same variance of the ini-
tial probability distribution. Moreover, these distributions are the sum of the distri-
butions of the two Pythagorean components, which for reasons of symmetry have
the same variance.
From information theory, we can deduce that, given a random variable
Z
such
that:
X
2
Y
2
Z
=
+
