A simple entropy quiz:

[chilehed]

I have 900 tiles.
197 of them are at a single hot temperature (the red ones), the rest are at a single cold temperature (the blue ones).
They are otherwise identical in every way.

Below you will find two pictures of them arranged in different manners.

Question: which arrangement has a lower thermodynamic entropy?

I'll provide the answer within the next day or two, depending on how many people have answered.

 

[Loudmouth]

I have 900 tiles.
197 of them are at a single hot temperature (the red ones), the rest are at a single cold temperature (the blue ones).
They are otherwise identical in every way.

Below you will find two pictures of them arranged in different manners.

Question: which arrangement has a lower thermodynamic entropy?

I'll provide the answer within the next day or two, depending on how many people have answered.

I am going to say both pictures are equally ordered. Entropy has to do with the distribution in heat, and in both diagrams you have the same amount of disequilibrium.

 

[essentialsaltes]

Question: which arrangement has a lower thermodynamic entropy?

I think the question is a little ill-posed. The statistical definition of entropy relates to the number of microstates that produce the same macro-state. The two things you've depicted are both microstates of a system.

But to skip to the bottom line, I'd say that statistically, assuming a random distribution of individual tiles, both states are equally likely microstates for the macro-state that has that same total given amount of energy.

 

[leftrightleftrightleft]

They are both the same. There's the exact same amount of energy, Q, in the system and the exact same average temperature, T, in the system. Entropy is Q/T.

Both states are equally probable to occur. One just happens to be in a pattern that our brains recognize.

 

[chilehed]

I think the question is a little ill-posed. The statistical definition of entropy relates to the number of microstates that produce the same macro-state. The two things you've depicted are both microstates of a system.

Allow me to clarify: it's not a collection of 900 atoms, it's a collection of 900 tiles, each of them containing many quadrillions of atoms.

But to skip to the bottom line, I'd say that statistically, assuming a random distribution of individual tiles, both states are equally likely microstates for the macro-state that has that same total given amount of energy.

The arrangement on the top is far from random.
The one at the bottom is truly random, I generated it from a source of atmospheric noise.

 

[essentialsaltes]

Allow me to clarify: it's not a collection of 900 atoms, it's a collection of 900 tiles, each of them containing many quadrillions of atoms.

If the relevant macrostate is a state in which 197 tiles have one temperature, and the rest have another, they should, I think, have the same entropy.

The arrangement on the top is far from random.

Nevertheless, it is just as likely as the other, in the space of all equivalent situations.

 

[Strathos]

The arrangement on the top is far from random.
The one at the bottom is truly random, I generated it from a source of atmospheric noise.

I see a cross in the middle

 

[metherion]

I'm going to say they're both the same.

According to statistical thermodynamics, S=-(Boltzmann's constant) * (summation till the ith term) P[sub]i[/sub] ln P[sub]i[/sub], where P[sub]i[/sub] is the probability of being the ith microstate. Sorry, I'm not sure how to do the symbols.

Since we're dealing with statistical thermodynamics, and we are assuming the possible placement of these squares is truly random, all microstates have an equal possibility of occuring. All possible combinations of the 197 hot blocks spatially in relation to each other are equal. This assumption is justified by the statement in the OP

They are otherwise identical in every way.

So all the blocks are the same except for temperature, so every block has an identical probability of being in a single given space.

This assumption means S= (Boltzmann's constant) * ln (# of microstates)

Since both pictures are 197 hot blocks, 900 total blocks, and 703 cool blocks, the number of possible combinations of the hot blocks should be (900 choose 197). I might be misremembering, and it is instead (900 choose 703), but I think it's the former. So, since the number of microstates in both tile systems is the same, the natural log of the number of microstates is the same, and boltzmann's constant is... well... a constant, it will be the same, so the entropy will also be the same.

So, when you type in
"ln (900 choose 197) * boltzmann's constant" into google, it gives you
6.48*10[sup]-21[/sup] J/K... except with several more significant digits and it gives you the units as m[sup]2[/sup] kg s[sup]-2[/sup] K[sup]-1[/sup]. Well, that would be more for 900 molecules than 900 tiles, but still, it's just one conversion away.

Metherion

Okay, it's good now, dealt with super- and subscripts.

 

[Tomk80]

Allow me to clarify: it's not a collection of 900 atoms, it's a collection of 900 tiles, each of them containing many quadrillions of atoms.


The arrangement on the top is far from random.
The one at the bottom is truly random, I generated it from a source of atmospheric noise.

Does not matter for the probability of both occurring. The probability of the specific patterns placement of tiles you have depicted is exactly the same for both.

If you want to ask the question what the probability is for the tiles to occur in a formation that has the exact same meaning to us as the top picture, versus the tiles occurring in a pattern that looks more random to us, the answer would be different. But that is not the question you asked.

 

[essentialsaltes]

This assumption means S= (Boltzmann's constant) * ln (# of microstates)

Since both pictures are 197 hot blocks, 900 total blocks, and 703 cool blocks, the number of possible combinations of the hot blocks should be (900 choose 197).

See, that's what was bothering me about the set-up. You're calculating (very reasonably -- it's what I wanted to do, too) the entropy of the state "197 hot blocks out of 900, arranged any which way," but the original question is to measure the entropy of "this particular arrangement of 197 hot blocks out of 900" and "that particular arrangement of 197 hot blocks out of 900". There is only one arrangement that is like this arrangement, and only one arrangement that is like that one. So again, I think the entropy would be the same, given that the tiles are otherwise identical.

I might be misremembering, and it is instead (900 choose 703)

Fortunately, they're the same. If you've chosen 197 of them, you've not-chosen the remaining 703. Mathematically, in "n choose k", k and (n-k) both appear in the denominator.

 

[Loudmouth]

So again, I think the entropy would be the same, given that the tiles are otherwise identical.

You may want to reread the opening post.

 

[chilehed]

I see a cross in the middle

Christ is the center of all things.

 

[chilehed]

I'm surprised that there aren't more responses. Some of the folks who usually post heavily in 2nd Law threads seem to be MIA.

I'll give it another day.

 

[keith99]

It has been far too long since I did the math,

But I'd argue the sates are different. And the top is more ordered.

BUT NOT because the tiles happen to form letters, rather because in hte top the hot tiles are more concentrated into groups. (look at the upper left with almost no hot tiles.

A grid with all the hot tiles on hte left side would be more ordered still.

 

[essentialsaltes]

It has been far too long since I did the math,

But I'd argue the sates are different. And the top is more ordered.

BUT NOT because the tiles happen to form letters, rather because in hte top the hot tiles are more concentrated into groups. (look at the upper left with almost no hot tiles.

A grid with all the hot tiles on hte left side would be more ordered still.

I guess looking at it from an information theory standpoint (another way to quantify a sort of entropy) it would take less information to specify the first one (even if we didn't include 'words', but just said 'the next five tiles in a straight line' and so on). By that measure, the first would have a lower entropy.

 

[AECellini]

the first has lower entropy.

 

[Subduction Zone]

I'm surprised that there aren't more responses. Some of the folks who usually post heavily in 2nd Law threads seem to be MIA.

I'll give it another day.

Oh, I know the answer then.

The answer is whichever answer disproves the theory of evolution. After all that is the only time creationists care about the 2nd Law

 

[chilehed]

There’s been some interesting discussion. I’d like to jump in with this post as a starting point:

I guess looking at it from an information theory standpoint (another way to quantify a sort of entropyhttp://en.wikipedia.org/wiki/Entropy_(information_theory)#Relationship_to_thermodynamic_entropy) it would take less information to specify the first one (even if we didn't include 'words', but just said 'the next five tiles in a straight line' and so on). By that measure, the first would have a lower entropy.

“A sort of entropy” is a very astute observation. Entropy is a function of the probabilites of the microstates associated with a macrostate, and the macrostate is defined by specific macrovariables.

In thermodynamics, the macrovariables that define a macrostate are composition, mass, temperature, and pressure. There are no other variables that determine a thermodynamic macrostate, and so by definition variations in other variables have no influence on thermodynamic entropy. E. T. Jaynes (Washington University, St. Louis, Missouri) wrote a very interesting paper on the Gibbs Paradox that touches on this. It’s well worth reading.

In both cases the composition and pressure are unchanged, and the masses at the two temperatures are identical as well as the temperatures themselves. Because the thermodynamic macrovariables are identical between the two cases, the thermodynamic entropies are as well... and the question specifically asked about thermodynamic entropy.

There are other macrovariables which can be used to define entropy. The distribution of the colors of the tiles or of letters in a string of English language test are examples: there is an entropy of color distribution, and an entropy of letter distribution. However, neither color nor letter are thermodynamic macrovariables, therefore those entropies are entirely independent of thermodynamic entropy. You can extract work from a substance by manipulating temperature and pressure, but you can’t do it by manipulating color or letters.

Oh, I know the answer then.

The answer is whichever answer disproves the theory of evolution. After all that is the only time creationists care about the 2nd Law

*grin*
Those guys are conspicuously absent, but if they show up I’m sure they’ll have some rationalization or another.

 

[essentialsaltes]

“A sort of entropy” is a very astute observation.

I cannot deny a certain pride at having made it!

 

[GoldenBoy89]

I thought this was supposed to be easy