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entropy
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DEFINING THE ENTROPY

We shall define the entropy in three different yet equivalent ways.

WHAT IS A THEORY :

"A theory is the more impressive the greater the simplicity of its premises, the more varied the kinds of things that it relates and the more extended the area of its applicability. Therefore classical thermodynamics has made a deep impression on me. It is the only physical theory of universal content which I am convinced, within the areas of the applicability of its basic concepts, will never be overthrown." -- Einstein (1949)

ENTROPY : IN TERMS OF PROBABILITY

First we consider the probabilities for dice that every good backgammon players knows. Below, the left column indicates the sum of the two die; the next column lists all the possible combinations that give that sum; the third column counts the number of combinations for that sum. As you can see, there are a total of 36 different combinations for the two die, and each are equally probable to occur for "honest" dice. Thus the probability of getting a particular sum, as shown in the last column, is just the number of combinations divided by 36.
The most probable result, occurring one-sixth of the time, is to get seven.
Surprisingly, the first known successful calculation of the above probabilities didn't occur until the sixteenth century, by Girolamo Cardano. An exploration of why this should have occurred so late, plus a great deal more about probability, is the wonderful book by Peter L. Bernstein, Against the Gods: the remarkable story of risk (Wiley, 1996).

Surprisingly, the first known successful calculation of the above probabilities didn't occur until the sixteenth century, by Girolamo Cardano. An exploration of why this should have occurred so late, plus a great deal more about probability, is the wonderful book by Peter L. Bernstein, Against the Gods: the remarkable story of risk (Wiley, 1996), ISBN 0 471 29563 9.

The probabilities for dice lead us to our first definition of the entropy:

The entropy is a measure of the probability of a particular result.

Here,...
entropy

Now we consider a "box" with two white marbles and two black marbles inside it. The box is made so that exactly two of the marbles are always on the left hand side and two are always on the right hand side of the box. In the case shown to the right, both white marbles are on the left side of the box and both black marbles are on the right side of the box. There is only one combinations of marbles that gives this arrangement. Imagine that we shake the box, and how the marbles distribute themselves is random. Now we have a white marble and a black marble on the left, and a white and black marble on the right. Call the black marbles B1 and B2, and the white ones W1 and W2. Then for this arrangement we could have B1,W1 on the left and B2,W2 on the right. We could also have B1,W2 on the left and B2,W1 on the right; or B2W1 on the left and B1,W2 on the right; or B2,W2 on the left and B1,W1 on the right. Thus there are four combinations that give this arrangement of the marbles. Finally, we show the single arrangement with both black marbles on the left and both white ones on the right.

picture by fantafabulous
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