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independent events (in this example, 8 bits corresponds to 8 independent events).
Furthermore it is a continuous and monotonic function of probability.
Given that the likelihood of two independent events is the product of their
individual probabilities, it follows that information as a function must be of the
entropic type:
Info(p i ) = log n (p i ) = kln(p i )
(9.14)
where n is the base of the decision.
In the case of a binary decision, the smallest amount of information, a bit,
becomes log 2 (2). Drawing a parallel, any ensemble of molecules in a given ther-
modynamic state has a probability distribution, P = (p 1 ;p 2 :::p n ) and hence its
entropy is given by:
X
S(P) = R
p i lnp i
(9.15)
i=1
The above formula (Eq. (9.15)) is Shannon's entropy (Shannon, 1948) and ap-
plies to thermodynamic systems (Eq. (9.5)) as well as to any set of events defined
by a probability distribution.
9.4 An entropic vision of mining and smelting
The entropic concepts shown in the previous section can be used to describe the
whole mining and metallurgical cycle, from exploration to exploitation through to
smelting. The conventional energy perspective of the product life cycle looks at the
amount of energy needed to manufacture a product from the cradle to the grave.
However, with this approach, there is a lack of a sense of “evolution”, since it favours
the idea that technology will increase e ciency forevermore. On the contrary, an
entropic vision gives an evolving explanation in which any action in the future will
require a greater amount of energy than it does at present. Such increase is in
response to an inherent exponential behaviour which technological improvements
will sooner or later be unable to offset. In what follows the different stages of
mining and metallurgy described in Chap. 7 are explained from an entropic point
of view.
9.4.1 Mining exploration
As stated throughout this topic, the continental crust is not homogeneous, since it is
neither liquid nor gaseous as can occur on other planets. Instead, the Earth contains
enriched mineral deposits, which appear as “lumps” dotted around the crust. The
probability, p i , of finding a relevant mineral accumulation (lump) is understood to
lie somewhere between 0 and 1. The greater a mineral's scarcity, the smaller this
figure will be, although it will never fall below zero.
 
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