But, really, what is entropy?

Entropy (1): the measure of disorder. The increase in entropy is the decrease in information.

Entropy (2): the measure of the energy which is available for work.

Problem: Reconcile both definitions.

Some people tell me that there is no problem here… Yet… I have the feeling that we call entropy to many different things because we have the intuition that, in the end, they’re all the same. My main problem: entropy (1) is an epistemological magnitude, whilst entropy (2) is ontological. Confusion between these two planes have given rise to all sorts of problems.

I should explain better: entropy (1) refers to my knowledge of the world, and entropy (2) to its substance. Yet, we might be able to reconcile them. With care, of course. Let us give an example.

Imagine a box with particles bouncing inside. We have no information at all. All possible states are equally likely. With no information, there is no work we can extract from the particles in the box. But imagine that we’re given some information, such as the temperature. Then we can extract some work, if we’re clever. Now, even more: imagine that we’re given the exact position and velocity of all the particles at a given moment. Then, again if we’re clever, we can extract a lot of work from the system! The more information we have, the more work we can extract.

So that was a purely operational view on entropy. The information content –epistemological, entropy (1)– determines the amount of work we can get –entropy (2). But the ontological view fades away… The system has no intrinsic entropy. The amount of work which is available… available for whom?

Now a problem comes… the second law of thermodynamics, the most sacred of our laws of physics, states that the entropy of an isolated system tends to grow. “But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation”, as Arthur Eddington posed it.

Can the second law adapt itself to this view? Yes, it can, but the result is funny: For all observers, no matter their knowledge and their abilities, as time goes by, their information about an isolated physical system tends to reduce, and also the amount of work they can get from it.

Of course, isolated is key here. You’re supposed to do no more measurements at all! Then, evolution tends to decrease your information, increasing the initial uncertainties you might have. Is this statement non-trivial? I think it is, in the following sense: it excludes the possibility of some dynamical systems being physically realized.

Still, the operational point of view does not fully satisfy me yet. It states that, no matter how clever you are, the amount of work you can get from an isolated system decreases with time, since your information does. This maximization over the intelligence of people is disturbing… What do you think?


6 thoughts on “But, really, what is entropy?

  1. I see no problem to reconcile both definitions. Even if an amount of energy is available to work with, if you don’t have information about “where” it is, really it isn’t available.
    Lets say you are a cowboy and you need cows for milk. even if there is four cows “available” to milk, you need to know where they are, or else they are useless cause you can’t milk a cow that you don’t know where it is.
    Is there any milk? Yep
    is there any chance to get it? No
    So, in sum, really there’s no “efective” milk.

    According to Erynus’s Uncertainty principle: You can’t find anything if you don’t know where to search. So the only way to get all possible work from a system is to monitoring it forever, measuring every movement, wich is not possible.

  2. Yes, that’s somehow the point. The question is that cow evolution makes the milk more unavailable… That’s the second law.

    You know the quantum version of Zeno’s paradox: if you measure continuously the position of a particle, it won’t move!

    So, the second law can be formulated like this: (A) In order to get maximum work from a system you should be monitoring it all the time. (B) If you monitor a system all the time, it won’t move, so you get nothing from it. (C) Therefore…

  3. What do you mean with “more unavailable”? things are available or unavailable. When you don’t have that cow in the begining, you can’t know it evolution, so you can’t get more milk even if you could “potentially” have it.
    Another example without cows. Fruit trees. Lets say you have a group of 20 fruit trees. 10 of them are ready for collection, but you only can see only 5 of them ripe. Potentially you have 10 fruits, but really you just have 5, cause you don’t know the other 5 are ready. As time goes by, if you can’t check again for ripe fruit, you still have only 5 fruits. No more, but no less (untill that 5 fruits rot, of course). Does particles rot? Does particles rot in a isolated system?

    Should’nt be upside down the paradox? I mean, every measure takes time, on that time the particle moves a bit so you need to measure again, in the time you measure faster the particle moves a fraction, so again you need to measure, and so on. Why should a particle wait until you have measured its position to move?

  4. Hm… particles do not wait for you to move, that’s the point. And, even though you might be able to predict their motion solving some equations, they will have some uncertainty, so your information about where they are will decrease in time. Therefore, you get uncertainty about their position and they become “more unavailable”… You know there is a fruit tree… but you don’t know exactly where! :)

  5. But my point is that a fruit or particle that is where you don’t know where it is is just as “unavailable” if it is just around the corner or if is in a cute pond in a forgotten monastery in Nepal.

  6. No. Things can be “partly available”. There might be many apples, some of them you know how to reach them, some of them no. An example that I like. Imagine a well full of water that you want to extract with a bucket. But the bucket has a certain size, so you will never be able to get “all” the water with it!! Some of it is unavailable because your bucket is too large.

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