Page 89 - Coincidences in the Bible and in Biblical Hebrew

COINCIDENCES IN THE BIBLE AND IN BIBLICAL HEBREW
68 68 COINCIDENCES IN THE BIBLE AND IN BIBLICAL HEBREW

Scenario C: We are completely ignorant about the fortune location.
Therefore, we have no alternative but assign equal probabilities to each box—
which, in this case, since the probabilities have to sum up to 1, means that each
box is assigned a probability of 1/7.
Shannon entropy is

H = –Σ (1/7) log 2(1/7) = –log 2(1/7) = 2.807 bits.

Shannon’s entropy has increased to 2.807, which is the maximum entropy that
the defined random variable (the box number where the fortune is hidden) can

assume. The reason is that it can be easily shown that when the distribution of the
probabilities is uniform—the entropy reaches its maximum possible value.
The decision in Scenario C will be taken under conditions of ignorance, and it
will be completely random.

From this simple example, we realize that Shannon entropy is in fact a measure
of the amount of randomness that is inherent in the distribution of probabilities,
associated with the values of the random variable (in our case, the box num-
ber containing the fortune). That this indeed is the case may be demonstrated
by increasing the number of boxes for Scenario C, from 7 to 20. Intuitively, we
would assert that the amount of randomness associated with our decision has

increased. Shannon entropy reflects this:
H = –Σ (1/20) log 2(1/20) = –log 2(1/20) = 4.322 bits
(compared to 2.807 for 7 boxes)

3.2.2 Thermodynamics (Boltzmann Entropy)

Boltzmann provided a complete theory for his concept of entropy, which resulted
in a formula to calculate entropy of complex physical systems. A good demonstra-
tion for the derivation of Boltzmann’s entropy is given in Penrose (2004, 690),

and the more scientifically and mathematically oriented reader may refer to this
source for clear derivation and explanation of Boltzmann’s entropy. Here, we show
how a simple process of transfer of a quantity of heat, Q, from a hot object to a
cold object, changes the total entropy of the two objects so that this total increases
(in accordance with the Second Law of Thermodynamics) .
Suppose that heat flows in response to a temperature gradient between two

objects that are in thermal contact. The temperatures of these objects are T 1 and
T 2 (assume that T 1>T 2), and energy is transferred in the form of heat, Q. To