Ragged Boy on 05 Nov at 5:
The model could also be used to show that two different kinds of gasses would become thoroughly mixed. The reasoning he used for mixing is very similar to that for the diffusion of heat, but there is an important difference.
In the diffusion of heat, the entropy increase can be measured with the ratio of physical units, joules per degree. In the mixing of two kinds of gasses already at the same temperature, if no energy is dissipated, the ratio of joules per degree — thermodynamic entropy — is irrelevant.
The non-dissipative mixing process is related to the diffusion of heat only by analogy 5. Nevertheless, Boltzmann used a factor, k, now called Boltzmann's constant, to attach physical units to the latter situation.
Now the word entropy has come to be applied to the simple mixing process, too. Of course, Boltzmann's constant has a legitimate use — it relates the average kinetic energy of a molecule to its temperature.
Entropy in this latter sense has come to be used in the growing fields of information science, computer science, communications theory, etc. The story is often told that in the late s, John von Neumann, a pioneer of the computer age, advised communication-theorist Claude E.
Shannon to start using the term "entropy" when discussing information because "no one knows what entropy really is, so in a debate you will always have the advantage" 6.
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Richard Feynman knew there is a difference between the two meanings of entropy. He discussed thermodynamic entropy in the section called "Entropy" of his Lectures on Physics published in 7using physical units, joules per degree, and over a dozen equations vol I section He discussed the second meaning of entropy in a different section titled "Order and entropy" vol I section as follows: Suppose we divide the space into little volume elements.
If we have black and white molecules, how many ways could we distribute them among the volume elements so that white is on one side and black is on the other? On the other hand, how many ways could we distribute them with no restriction on which goes where? Clearly, there are many more ways to arrange them in the latter case.
We measure "disorder" by the number of ways that the insides can be arranged, so that from the outside it looks the same. The logarithm of that number of ways is the entropy. The number of ways in the separated case is less, so the entropy is less, or the "disorder" is less.
This is Boltzmann's model again. Notice that Feynman does not use Boltzmann's constant. He assigns no physical units to this kind of entropy, just a number a logarithm. And he uses not a single equation in this section of his Lectures.
The "number of ways" can only be established by first artificially dividing up the space into little volume elements. This is not a small point. In every real physical situation, counting the number of possible arrangements requires an arbitrary parceling. As Peter Coveney and Roger Highfield say 7.
There is, however, nothing to tell us how fine the [parceling] should be. Entropies calculated in this way depend on the size-scale decided upon, in direct contradiction with thermodynamics in which entropy changes are fully objective.
Shannon Claude Shannon himself seems to be aware of these differences in his famous paper, "A Mathematical Theory of Communcation" 8. With respect to the parcelling he writes, "In the continuous case the measurement is relative to the coordinate system. If we change coordinates the entropy will in general change" p 37, Shannon's italics.
In the same paper Shannon attaches no physical units to his entropy and never mentions Boltzmann's constant, k. At one point he briefly introduces K, saying tersely, "The constant K merely amounts to a choice of a unit of measure" p Although the the page paper contains more than equations, K appears only once again, in Appendix 2, which concludes, "The choice of coefficient K is a matter of convenience and amounts to the choice of a unit of measure" p Shannon never specifies the unit of measure.
This sort of entropy is clearly different. Physical units do not pertain to it, and except in the case of digital information an arbitrary convention must be imposed before it can be quantified.
To distinguish this kind of entropy from thermodynamic entropy, let's call it logical entropy.This Guide was created as a joint project of the Academic Resource Center and the William H.
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