The Second Law fallacy

metherion · #**161** 4th August 2009, 02:05 PM

That's not Heisenberg's uncertainty principle is it?

No, that's only electrons. Microstates is more like... say you have two atoms. At 300K, they each vibrate, say, (pulls a random number out of thin air) 30 nm from where they would at absolute zero, in 1 nm increments. So they can both be anywhere from -15 to 0 to +15 nm from their absolute zero point. For the two atom system, then, there's then 30*30 microstates. But if we cut the temp by 10%, and instead of 30*30 microstates, there is now 27*27 microstates, because they don't vibrate as much due to the lower temp, you've lost information about the system.

Remember, I did just pull all the numbers out of thin air to show what I meant, none of the numbers represent an actual situation.

Metherion

chilehed · #**162** 8th December 2009, 07:18 PM

*bump*

chilehed · #**163** 31st August 2010, 09:35 PM

I might as well stick on the bit I posted in the other thread.

To outline my train of thought on information and its relationship to entropy:
We know that the entropy of a thermodynamic system is given by S = kB lnΩ, where Ω is the number of available microstates corresponding to the macrostate. The Second Law states that for every real process, dS >/= 0, which necessarily means that the number of microstates available to the system has increased.

I’m also given to understand that S = -kB ∑(ρi lnρi ), in which, in information theory, the term -∑(ρi lnρi ) is called the information function and the symbol rho denotes the probability of each microstate i.

This would seem to mean that lnΩ = -∑(ρi lnρi ), in which case for every real process the information function of the system increases.

metherion · #**164** 31st August 2010, 10:22 PM

Not quite.

Omega is our LACK of knowledge about the system, not our knowledge of it.

Yes, Omega is the number of microstates that are consistent with the macrostate. But the more possible microstates, the less we know of which one it actually is at that point in time.

Think of it like a variation on the powerball. In the powerball, your white ball numbers don't have to match in order, right? Now, say you KNOW 4 of the 5 white ball number ahead of time. Maybe you rigged the machine so those 4 would be picked, but still in a random order. You now have a 1 in 4! chance of picking them as coming out in the right order.

But now, we increase omega. You now have rigged the 5th ball. You know all 5 white balls that will come out. You know increased the number of microstates your pick of numbers could have (from 4! to 5!). But your odds of picking the order they come out in correctly have DECREASED from 1 in 4! to 1 in 5!.

I hope that makes sense.

ETA: In a flash of insight, I thought of some easier words to use.
Entropy is the randomness/chaos/UNCERTAINTY of the system.
So the more microstates, the MORE UNCERTAIN which one of them is the correct correlation of the macrostate is, the less information we have.

Metherion

chilehed · #**165** 1st September 2010, 12:01 AM

Originally Posted by metherion

Not quite.

Omega is our LACK of knowledge about the system, not our knowledge of it....So the more microstates, the MORE UNCERTAIN which one of them is the correct correlation of the macrostate is, the less information we have.

Metherion

Rather than get into verbal descriptions to quickly, let's first establish whether or not I have the math right.

Is it true that S = kB lnΩ = -kB ∑(ρi lnρi)?
If so, then the Second Law implies that for any real process the change in -∑(ρi lnρi) is >/= 0.
Right?

And is it true that -∑(ρi lnρi ) is called the information function?
If so, then the Second Law implies that for any real process the change in the information function is >/= 0.
Right?

metherion · #**166** 1st September 2010, 01:08 AM

Is it true that S = kB lnΩ = -kB ∑(ρi lnρi)?

Bah, this would be easier if I knew how to do subscripts.

It is true if and only if p(1)=p(2)=p(3)=...=p(i). Agreed.

If so, then the Second Law implies that for any real process the change in -∑(ρi lnρi) is >/= 0.

Yes. Delta S in an isolated system is >/=0, and kB is a constant, therefore delta -∑(ρi lnρi) is >/= 0 is correct.

And is it true that -∑(ρi lnρi ) is called the information function?

I have honestly never heard it called that before, so I cannot say. But I will grant it to you for purposes of the discussion.

If so, then the Second Law implies that for any real process the change in the information function is >/= 0.

Yes, it would.

HOWEVER, let us go a step further.

S = kB lnΩ is only true when the probability of all microstates is equal. So it is for MAXIMUM entropy.

And also, lnΩ = -∑(ρi lnρi). Which makes sense, because ln (0<n<1) is negative, and probabilities must be 0<n<1.

So, if ln(number of microstates) INCREASES, then -∑(ρi lnρi) increases, which means that ∑(ρi lnρi) itself DECREASES. So if lnΩ=17, then -∑(ρi lnρi)=17, and ∑(ρi lnρi)= -17. And if the number of microstates increases, then lnΩ =18 (for example), then -∑(ρi lnρi)=18 and ∑(ρi lnρi)=-18, so it has DECREASED.

Metherion

chilehed · #**167** 1st September 2010, 10:30 AM

Originally Posted by metherion

Bah, this would be easier if I knew how to do subscripts.

Use the code "sub /sub"

Originally Posted by metherion

It is true if and only if p(1)=p(2)=p(3)=...=p(i). Agreed.

Well, yeah, that's always one of the base assumptions.

Originally Posted by metherion

Yes. Delta S in an isolated system is >/=0, and kB is a constant, therefore delta -∑(ρi lnρi) is >/= 0 is correct.

If your analysis ever finds that dS > 0 , you can bet the farm that you drew the system boundaries incorrectly. But let's not head out on a tangent quite yet.

Originally Posted by metherion

S = kB lnΩ is only true when the probability of all microstates is equal. So it is for MAXIMUM entropy.

And also, lnΩ = -∑(ρi lnρi). Which makes sense, because ln (0<n<1) is negative, and probabilities must be 0<n<1.

So, if ln(number of microstates) INCREASES, then -∑(ρi lnρi) increases, which means that ∑(ρi lnρi) itself DECREASES. So if lnΩ=17, then -∑(ρi lnρi)=17, and ∑(ρi lnρi)= -17. And if the number of microstates increases, then lnΩ =18 (for example), then -∑(ρi lnρi)=18 and ∑(ρi lnρi)=-18, so it has DECREASED.

As far as I can tell the information function includes the negative sign, and you can’t arbitrarily remove it. So it's correct to say that in all real processes the delta of the information function is >/= 0.

I understand the idea that you can look at it as information that you can’t quantify, but it seems to me that to say that there’s been an increase in the amount of information that you don’t know is to admit that there’s more of it. To illustrate, if I have a closed warehouse full of marbles and I know that a dumptruck full has been added to it, the fact that I don’t know how many are which color doesn’t change the fact that there are now more marbles in the building.

This whole "it's less information because there's more of it that we don't know" thing seems perfectly designed to cause a lot of confusion among the general public; it's just as bad as the common explanation that entropy is disorder.

I emailed one of the profs at my alma mater, hopefully she'll get back with me. There's another one at MSU that I want to ask as well, as soon as I can remember his name.

metherion · #**168** 1st September 2010, 07:01 PM

As far as I can tell the information function includes the negative sign, and you can’t arbitrarily remove it. So it's correct to say that in all real processes the delta of the information function is >/= 0.

I didn’t arbitrarily remove it. I did include the value of the full -∑(ρi lnρi) in my explanations., as well as just the pi lnpi part.

I understand the idea that you can look at it as information that you can’t quantify, but it seems to me that to say that there’s been an increase in the amount of information that you don’t know is to admit that there’s more of it. To illustrate, if I have a closed warehouse full of marbles and I know that a dumptruck full has been added to it, the fact that I don’t know how many are which color doesn’t change the fact that there are now more marbles in the building.

But that is only valid if teh number of marbles (microstates) IS the information. It isn’t. The information is what we know about WHICH microstate it is.

So, with your marble analogy, in the warehouse is one specifically patterned marble and p(i) other patterened marbles. You can’t look in it before you select your one marble to draw out. So, if you add a dump truck more marbles, your probability of drawing the one marble you want has DECREASED, though the number of marbles that are not the one you want has INCREASED. That particular marble is the information we want, the microstate that currently describes the system. And even though alllllllll the marbles in the warehouse are microstates that CAN describe the system, only one does. And the more marbles that are NOT the one we want in the warehouse, the less we know where the one we want is. So we know less precisely BECAUSE there are more possibilities to cloud the issue.

And to make it worse, because the system changed, you also just got handed a notecard that has a NEW unique pattern on it that may or may not be in the dumptruck that just pulled up, since if the entropy has changed, the microstate describing the state of the system has changed too. So now you don’t know if the new marble you’re looking for after the entropy-increasing reaction was already in the warehouse, or is in the dumptruck that just pulled up. So you know less.

And I am looking forward to hearing what the profs say

Metherion

SkyWriting · #**169** 1st September 2010, 07:32 PM

Originally Posted by chilehed

The state of clutter in your bedroom impacts entropy only insofar as it reflects the distribution of heat energy. Anything sitting on a shelf stores potential energy because you had to do work to it to pick it up, but since all of the potential energy stored in the raised object can be recovered when you lower it back down to the floor the process is by definition reversible and is thus isentropic. If you seal the room and come back a thousand years later nothing will have moved. If you lay down a mosaic of loose floor tiles they will remain in place forever, provided no one disturbs them and they don't rot away.
Rooms do not spontaneously clutter themselves up.

I'll just stop at this point and comment.

Yes they do.

Every item in the room began to oxidize the moment it was created.
Come back in a Billion years or 100 Billion and every item in the room will be a randomized pile of dust similar to it original state. Another 100 Billion and the action of air in the seal room will have randomized every molecule so each pile of dust will have no unique characteristic.

Adding a small child to the room will reduce this waiting time to a couple of hours.
Make it a teenager with a match and you can accomplish most of this in a few minutes.

For abriogenesis to occur just add water and sunshine.
But the second law of thermodynamics says that you'll just get dust.

Let's assume that I'm wrong, and that entropy is indeed generically dependent on spatial arrangement. Frozen water at 32 degrees F, being arranged in a very ordered crystal structure, would then have a higher entropy than liquid water at the same temperature. If this were true, then heat would flow out of the ice into the water and your drink would boil away! We could create an engine that has no effect but to make heat flow from a cold object to a warmer object, which is a Second Law violation.

So your claiming that as a glass of water looses energy, it becomes more "orderly" as it crystallizes, and this is how life would occur in your dreams. From frozen ice crystals. Let's consider the ability of these "crystals" to perform tasks. I say they are less able. Let's consider the amount of useful information found in these crystals. If you dip your finder in one side of a glass, the oils will slide across the serface and reach the other side of the glass. Chemicals will enter the water and the diffusion speed and concentration of these chemicals can tell you exactly when the water was touched and by who. The speed of this and waves will tell you the exact water temperature. All this information in "disordered" water. Actually, the disordered water holds and transmits a ton of valuable information. The ice cube expands as it freezes then looses all these abilities. When referring to "order" we are saying "useful" order, or information at best. The "order" you refer to has zero available energy to offer. You say the world becomes more orderly as is looses energy. As time goes forward, I don't find anyone agreeing with you about the growth and expansion of Life-Cubes.
"In this case the Universe will also die a cold static death"

“act”, “atc”, “cat”, “cta”, “tac”, and “tca” all have the same amount of information, and all have the same odds of coming up in a random draw if the letters. Thus, by this theory of information all combination's have the same “information entropy” and are equally allowable.

By your odd definition. But when my robot/surgeon built each combination for me, only the "CAT" version lived long enough for me to show off to my friends. The other versions had less value and I had to toss them.

chilehed · #**170** 1st September 2010, 10:42 PM

Originally Posted by metherion

But that is only valid if teh number of marbles (microstates) IS the information. It isn’t. The information is what we know about WHICH microstate it is.

So, with your marble analogy, in the warehouse is one specifically patterned marble and p(i) other patterened marbles. You can’t look in it before you select your one marble to draw out. So, if you add a dump truck more marbles, your probability of drawing the one marble you want has DECREASED, though the number of marbles that are not the one you want has INCREASED. That particular marble is the information we want, the microstate that currently describes the system. And even though alllllllll the marbles in the warehouse are microstates that CAN describe the system, only one does. And the more marbles that are NOT the one we want in the warehouse, the less we know where the one we want is. So we know less precisely BECAUSE there are more possibilities to cloud the issue.

I don't see anything in the information function about "the microstate that you want". All I see is that it's the negative of the sum, from 1 to i, of the product of the probability and the natural logarithm of the probability of each available microstate, for every available microstate.

Originally Posted by metherion

And I am looking forward to hearing what the profs say

Yeah, but I expect it'll take a while, both for the reply and my comprehension of it. Also, I'll be occupied for the next week and probably won't be posting much in the meantime. In case I'm not back soon, have a blessed Labor Day weekend.