This Physics question

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Banedon
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This Physics question

Unread postby Banedon » 10 Oct 2010, 09:50

Suppose there's a supermarket somewhere with items inside arranged nicely as in most supermarkets (e.g. all processed foods in one column, etc). Shopper A goes in, picks everything he wants to buy and puts it in a trolley, then leaves the supermarket. Considering only the items inside the supermarket (i.e. no humans), has the supermarket's entropy gone up or down?

Obvious answer would be that it has gone up, because the items in the trolley aren't nicely arranged in their respective rows anymore. For the supermarket it must have gone up, because now it has to delegate its employees to put the items back where they're supposed to be. However, for shopper A, the entropy has gone down - everything that he wants is in the trolley, while everything outside of the trolley doesn't concern him at all. If he goes shopping all he has to do is wheel the trolley out, and he's saved a lot of time finding the things he needs.

How can the entropy of a system go both down and up?
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Unread postby Pol » 10 Oct 2010, 11:59

I'm always surprised at your questions.

Are you sure that on one side you don't have x whereas on the other stays y? When at the same concluding that x is always equal to y.
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Unread postby Ethric » 10 Oct 2010, 15:59

There is no spoon.
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Unread postby ThunderTitan » 10 Oct 2010, 20:04

That's because it's a spork...


@Banedon

Unless that's an analogy i don't think entropy works at that level...
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Unread postby Pitsu » 11 Oct 2010, 12:02

Banedon wrote: However, for shopper A, the entropy has gone down - everything that he wants is in the trolley, while everything outside of the trolley doesn't concern him at all.
I do not follow this entropy going down logic here. Whether an observer finds a certain entropic state most beautiful or beneficial for himself, does not mean that his opinion defines the lowest entropy state of the system. Shopper A favors a certain high entropy state. Empty trolley and full selves are not in the state he likes (both have higher order) and thus he works in order to achieve his desired state, not to achieve lower entropy. Have i misunderstood something?
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Unread postby Corribus » 11 Oct 2010, 19:46

Full shelves and empty cart: low entropy.
Depleted shelves and full cart: higher entropy. (to a point)

Let's make this simple. Suppose the super market has only 1000 cans of tomato soup, and we can put some artificial numerical label on them. In the former case, all the 1000 cans are on the shelves. There's only one microstate available to the system at this point. [Really that's not true, because each can can occupy a different position on the shelves, but we'll just ignore that complicating factor here - in the end it doesn't change the result]. The entropy of a state is essentially a measure of the number of functionally identical microstates that can be used to describe the state.

Now a person grabs a can of soup (at random) and puts it in his basket. The entropy has increased because there are now a much higher number of functionally identical microstates that can describe a state defined as 999 cans on the shelves and 1 in the basket. For instance, if the cans are labelled 1-1000, we can have:

Shelf (cans 1-999), Basket (can 1000)
Shelf (cans 1-998, 1000), Basket (can 999)
Shelf (cans 1-997, 999-1000), Basket (can 998)
etc.

This assumes a can is grabbed at random - which isn't the case with soup cans but it is with molecular positions.

More cans = more available microstates because there are a higher number of functionally equivalent combination. Thus when I get more cans, the system's entropy increases more. This of course only works to a point. If I could put all the cans in my basket, that's also a low entropy state because I've limited the combinations again. I'm sure you can figure out the highest entropy state possible, then. Remember, entropy is essentially just statistics.

Increasing the variety of food doesn't change anything really - just the complexity of the problem.
Also note that in real thermodynamical considerations, other factors come into play to determine the ultimate (average) state of the system at equilibrium. For instance: the amount of energy required to pull the cans off the shelves (enthalpy, say). In supermarket systems, part of this energy takes the form of money - in fact many of the principles of thermodynamics work in economics.

In any case, while macroscopic systems can be used to illustrate thermodynamical concepts, keep in mind that these concepts really only truly applicable to ensembles of quantum particles Purchases in supermarkets are not random events, so really concepts like entropy become somewhat problematic in that case.
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Unread postby Pol » 11 Oct 2010, 20:47

Really liked that C. ;) If there would be a voting button I would hit your post. :P
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Unread postby Corribus » 11 Oct 2010, 22:03

Thank you. By the way, to address the original question: the cause of Banedon's confusion is his error in thinking that what's "good" for the shopper makes any difference. It doesn't. For that matter, "neatness" and "order" have nothing really to do with entropy either. Entropy is only concerned with how many ways you can arrange the components of the system in question while keeping the state of the system the same. The laws of thermodynamics state essentially two things when it comes to entropy:

(1) A system will tend to be found in a state that has a larger number of ways that its parts can be arranged identically. This is simply a matter of statistics. Flipping a coin ten times will tend to yield results where the number of heads results and the number of tales results are the same. This is because there are a greater number of combinations that will lead to this result. That doesn't mean you couldn't - by chance - flip ten heads in a row. It's just that when you repeat your experiment a large number of times, you'll tend to observe five heads and five tales more often than not because there are a larger number of equivalent cases that will lead to that result than the single case that leads to ten heads. The same logic works in molecular systems. You can find a molecular system in a low entropy state on some single observation just by chance. But on average, your system will tend to be in a higher entropy state because there are a larger number of ways that can happen. It's basic probability, really.

(2) You can change those results by putting work into the system. Going back to flipping coins, you could force your average result to favor more heads flips than tales flips, but this would require modification of the coins, which in turn would require energy. In other words, you can cause a system to favor a state with a smaller number of ways that its parts can be arranged identically, but to force the system into the lower entropy state, you must use energy. This is why entropy, a purely statistical concept, has units of energy (well, energy divided by temperature). In thermodynamical problems, we don't speak of entropy per se - we speak of the amount of energy required to overcome the statistical preference for large numbers of equivalent microstates. It is entropically favorable for gas molecules to spread out, not because (to an approximation) there's some fundamental force that causes them to spread out - but because it's statistically favorable for them to explore the widest space possible. You can cram them all back into a smaller space, using a vacuum pump, but that takes a hell of a lot of energy to do, and the better you want to cram them, the more energy it takes.

Entropy is a concept that most people are familiar with but few people really understand. Nevertheless, it's amazing how much of the universe thernmodynamical concepts can explain: from chemical reactions to biological evolution to human behavior and beyond.
Last edited by Corribus on 14 Oct 2010, 20:26, edited 1 time in total.
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Unread postby Corribus » 11 Oct 2010, 22:19

By the way, there is this thread from the last time Banedon was having trouble with entropy, which might be useful.
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Unread postby Kalah » 12 Oct 2010, 11:59

Whenever people start to move away from the basic principles and onto mathematical formulas to explain physics, I tend to black out and wake up on the floor immersed in my own drool a few hours later ... :jester:

No complex formulas for me; I think I'll just go back to my gaming, thank you. :)
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Unread postby Corribus » 12 Oct 2010, 13:35

I see no formulas in this thread.
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Unread postby Qurqirish Dragon » 12 Oct 2010, 14:13

Corribus wrote: (1) A system will tend to be found in a state that has a larger number of ways that its parts can be arranged identically. This is simply a matter of statistics. Flipping a coin ten times will tend to yield results where the number of heads results and the number of tales results are the same. This is because there are a greater number of combinations that will lead to this result. That doesn't mean you couldn't - by chance - flip ten heads in a row. It's just that when you repeat your experiment a large number of times, you'll tend to observe five heads and five tales more often than not because there are a larger number of equivalent cases that will lead to that result than the single case that leads to ten heads.
Careful with your wording here- there are more cases where you have 5 heads and 5 tails than any other individual outcome (252 ways out of 1024 possible flip sequences), but there are not more ways of getting 5/5 than not.

Entropy is a concept that most people are familiar with but few people - even many scientists - really understand. Nevertheless, it's amazing how much of the universe thernmodynamical concepts can explain: from chemical reactions to biological evolution to human behavior and beyond.
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Unread postby Corribus » 12 Oct 2010, 15:17

Sorry, yes, that was a figure of speech not meant to be taken literally.
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Unread postby Qurqirish Dragon » 13 Oct 2010, 13:16

Corribus wrote:Sorry, yes, that was a figure of speech not meant to be taken literally.
And if it weren't a mathematical discussion, I would have let it slide. Unfortunately, I got pounded on enough for being sloppy in my proofs in this fashion that I am a bit sensitive to it.
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Unread postby Banedon » 14 Oct 2010, 10:46

I get it. Thanks, Corribus. On hindsight it's actually really simple.

Although someone else I asked mentioned something about Jaynes entropy and how there's supposed to be a summation or something ...
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Unread postby Tress » 15 Oct 2010, 14:09

How can the entropy of a system go both down and up?
Because it happened in two separate place. Buyers bag's entropy have gone down at expense of shops. Also minus 1 bottle from shops shelf is only arguably increasment of entropy.
Also entropy is somewhat subjective term for human made things and it is hard to actually measure it, so it doesnt matter that much.

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Unread postby ThunderTitan » 26 Oct 2010, 09:27

tress wrote:
How can the entropy of a system go both down and up?
Because it happened in two separate place. Buyers bag's entropy have gone down at expense of shops. Also minus 1 bottle from shops shelf is only arguably increasment of entropy.
And that's why i wondered if the example in the OP actually applies...
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Unread postby Banedon » 14 Mar 2011, 12:33

Here's something else that's been bothering me ...

Which of the following has higher entropy:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

or

0, 2, 3, 6, 7, 1, 9, 4, 5, 8?
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Unread postby Corribus » 25 Mar 2011, 22:03

Why don't you tell us what you think, and I'll tell you if and why you're wrong.
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Unread postby Banedon » 27 Mar 2011, 10:48

Sure, why not.

I think both sequences have the same entropy - zero - because there's only one way to arrange both sequences. Moving any number from the first sequence for example breaks the order (no longer in increasing numerical order), while moving any from the second also breaks the order (no longer in reverse alphabetical order).

Problem if this is right is, intuitively, it's easy to suppose that the first sequence has lower entropy because its underlying sequence is much easier to recognize. Certainly it took me longer to piece together the second sequence than the first. The second sequence won't be in reverse alphabetical order if we consider it in a different language as well. To say the second sequence has zero entropy because it's in reverse alphabetical order of one specific language, English, is arbitrary and sounds nonsensical.

I don't trust the answer, but it's the only one that remotely makes sense, unfortunately.
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