MadSci Network: Physics Query:

### Re: How do you measure entropy?

Date: Wed Oct 28 08:59:06 1998
Posted By: Georg Hager, Grad student, Theoretical Particle Physics
Area of science: Physics
ID: 907721055.Ph
Message:

Greetings!

Entropy is likely to be one of the most misunderstood and misused terms from physics. So I will first try to explain what entropy is, in simple terms. Unfortunately I will not be able to leave mathematics completely out of the game.

Entropy is, roughly speaking, a measure for our missing knowledge, or ignorance, about a system. There are numerous ways to translate this stetement into a more rigorous, mathematical form which can be useful in practice, and I will give two of those possibilities here. The first one is actually the most fundamental one, and all other definitions can be shown to be compatible with it.

1. Entropy of a system can be defined as the number of yes/no-questions one must ask to gain complete knowledge about the system. For example, take a chessboard and throw a coin onto it. In the worst case, another person (who didn't see the throw) has to ask you six questions in order to know on which tile the coin has landed. There is a formula which can tell you this number, and in which you have to insert all the probabilities for the experiment to take different ways (64 times 1/64, in our example, if the coin can occupy any tile with equal probability).

This specific definition of entropy comes from information theory, but it was not the first one that had been conceived. In fact, physicists have discovered entropy first, which must be regarded as one of their greatest achievements. The following definition gives you the physicist's view about entropy, in the language of thermodynamics. This can be shown to be equivalent to the first definition, apart from numerical factors and an arbitrary choice of the point of zero entropy.

2. Entropy of a thermodynamical system (like a gas at a certain temperature) is defined as the logarithm of the number of states the system may take while having a fixed energy. In the example of the gas, imagine the gas being inside a box of volume V. Each molecule can take a certain amount of space, and a lot of rearrangements can be made among the molecules without altering the overall energy. Now take the same gas, at the same temperature, but in a volume that is just V/2. Now there is much less space left for each molecule, and there are far less possibilities for rearrangement. In other words, our knowledge of the system has increased. Now reduce the volume further and further, and at zero volume we will know exactly where each molecule is, namely at one certain point. To reduce entropy we could as well lower the temperature, because `knowledge' does not only cover the positions but also the velocities of all the molecules. At absolute zero all molecular motion will have stopped (apart from quantum effects, which I will leave out of the game here). At the point where we know exactly where each molecule is and how fast it is moving, entropy is at its minimum.
The second definition is not very well suited for measuring entropy, and in fact one can translate it into a more usable formula which is able to yield changes in entropy of a system, which is all one is usually interested in: The change in entropy is equal to the amount of heat energy transported into a system divided by its temperature. This is valid only for small changes, as temperature may of course change when the system is heated. So in order to get the overall change in entropy for a finite amount of heat energy transferred you would have to sum up all the small contributions `along the way'.

Hope that helps,
Georg.

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