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Entropy, Information, and Probability
For over sixty years, since I first read Arthur Stanley Eddington's Even more challenging has been to find the best way to teach the mathematical and philosophical connections between entropy and information. A great deal of the literature complains that these concepts are too difficult to understand and may have nothing to do with one another.
Finally, there is the important concept of Determinist philosophers (perhaps a majority) and scientists (a strong minority) say that we use probability and statistics only because our finite minds are incapable of understanding reality. Their idea of the universe is that it contains infinite information which only an infinite mind can comprehend. Our observable universe contains a very large but finite amount of information. Entropy is a measure of that lost energy.
A very strong connection between entropy and probability is obvious because Ludwig Boltzmann's formula for entropy
is mathematically identical to Claude Shannon's expression for information
Boltzmann entropy: I = - ∑ p.
_{i} ln p_{i}
The first entropy is material, the latter mathematical - indeed purely But they have deeply important connections which information philosophy must sort out and explain.
First, both Boltzmann and Shannon expressions contain
Today many thinkers propose chaos and complexity theories (both theories are completely
In statistical mechanics, the summation ∑ is over all the possible distributions of gas particles in a container. If the number of distributions is
In the communication of information,
On the other hand, if there is only one possible message, its probability is unity, and the information content is If there is only one possible message, no new information is communicated. This is the case in a deterministic universe, where past events completely cause present events. The information in a deterministic universe is a constant of nature. Religions that include an omniscient god often believe all that information is in God's mind.
Note that if there are no alternative possibilities in messages, Shannon (following his Bell Labs colleague Ralph Hartley) says there can be no new information. We conclude that the creation of new information structures in the universe is only possible because the universe is in fact indeterministic and our futures are
Thermodynamic entropy involves matter and energy, Shannon entropy is entirely mathematical, on one level purely It is true that information is neither matter nor energy, which are conserved constants of nature (the first law of thermodynamics). But information needs matter to be embodied in an "information structure." And it needs ("free") energy to be communicated over Shannon's information channels. Boltzmann entropy is intrinsically related to "negative entropy." Without pockets of negative entropy in the universe (and out-of-equilibrium free-energy flows), there would no "information structures" anywhere.
Pockets of negative entropy are involved in the
Visualizing Information
There is a mistaken idea in statistical physics that any particular distribution or arrangement of material particles has exactly the same information content as any other distribution. This is an anachronism from nineteenth-century deterministic statistical physics.
If we measure the positions in
This led many statistical physicists to claim that information in a gas is the same wherever the particles are,
But the information in the hemoglobin is much higher and the disorder (entropy) near zero. A human being is not just a "bag of chemicals," despite plant biologist Anthony Cashmore. Each atom in hemoglobin is not merely in some volume limited by the uncertainty principle ℏ Where an information structure is present, the entropy is low and Gibbs free energy is high.
When gas particles can go anywhere in a container, the number of possible distributions is enormous and entropy is maximal. When atoms are bound to others in the hemoglobin structure, the number of possible distributions is essentially
Even more important, the parts of every living thing are Normal | Teacher | Scholar |