Some may have wondered why me (a creationist) has taken the side of the ID-haters with regards to the 2nd law. It is because I am concerned for the ability of college science students in the disciplines of physics, chemistry and engineering understanding the 2nd law. The calculations I’ve provided are textbook calculations as would be expected of these students.
The fundamental problem is 2LOT is concerned with energy (or position/momentum) microstates, whereas IDists are concerened with “design space” microstates. The number of microstates can both be expressed in information bits, but it does not mean we are dealing with the same microstates. I’m providing sample calculations to prove the point that it is disastrous for IDists to invoke textbook 2LOT for the simple reason 2LOT is concerened with energy (or position/momentum) microstates which has little or nothing to do with “design space” microstates of interest to ID.
I’m going through textbook thermodynamics here. If we have 500 fair copper pennies, how many “design space” microstates are there? Standard ID answer:
2^500
since there are 500 coins and each coin has 2 states, a system of 500 coins then has 2^500 possible symbolic configurational states or microstates. This can also be expressed in bits:
I_design_space = – log2( 1/ (2^500) ) = 500 bits
What is the design space entropy?
I_design_space = S_design_space = 500 bits
IN CONTRAST, how many thermodynamic energy microstates are there in this system of 500 pure copper pennies at standard “room” temperature (298 Kelvin). The textbook style calculation is as follows:
Mass of a copper penny 3.11 grams.
Molar weight of copper 65.546.
Standard molar entropy of copper 33.2 J/K/mol.
Thermodynamic entropy of 500 copper pennies is therefore:
S_thermodynamic = 500 * 33.2 Jolues/Kelvin/Mol * 3.11 grams 65.546 grams/ mol = 826.68 J/K
The thermodynamic entropy in J/K can be converted to bits by simply dividing by Boltzman’s constant and then converting the natural log measure to log-base-2 measure.
Boltzmann’s constant is 1.381x 10-23 J/K).
The natural log to log-base-2 conversion is ln(2) = .693147.
Thermodyamic entropy in bits is computed as follows:
S_thermodynamic = I_thermodynmic =826.86 J/K = 826.68 J/K / (1.381x 10^-23 J/K) / .693147 = 8.636 x 10^25 bits
The number of thermodynamic microstates is simply taking 2 raised to the power of I_thermodynmic
2^(8.636 x 10^25)
which is a GIGANTIC number.
Clearly the design space entropy is not the same as the thermodynamic entropy because the design space microstate is not the same as the thermodynamic microstate.
Now let us heat the coins from room temperature to near boiling of water (373 Kelvin). What is the change in entropy or the number of microstates?
At 373 Kelvin the “design space” entropy is still 500 bits since the possible number heads tails microstates does not change with this increase in temperature.
However the thermodynamic entropy and thermodynamic microstates change. What is the change in entropy? Again using standard textbook thermodynamics.
Specific heat of copper 0.39 J/gram
Heat capcity C of 500 copper pennies:
C = 0.39 J/gram/K * 500 pennies * 3.11 grams/penny/K = 606 J/K
T_initial = 298 K
T_final = 373 K
To calculate the change in entropy I used the formulas from:
http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node41.html
delta-S_thermodynamic = C ln ( T_final/T_initial) = 606 J/K ln (373/298) = 136.13 J/K
Total thermodynamic entropy is calculated as follows:
S_thermodynamic_initial = 826.86 J/K
S_thermodyanmic_final = S_thermodyanmic_initial + delta-S_thermodynamic = 826.86 J/K + 136.13 J/K = 963.0 J/K
Again we can convert this to bits using procedures similar to the above conversions:
S_thermodyanmic_final = 963.0 J/K = 963.0 J/K / (1.381x 10-23 J/K) / .693147 = 1.01 x 10^26 bits
The ADDED number of microstates due to the increase in temperature is calculated as follows:
delta-S_thermodynamic = 136.13 J/K = 136.13 J/K / (1.381x 10^-23 J/K) / .693147 = 1.42 x 10^25 bits
Thus the number of thermodynamic microstates added by heating is simply found by rasing 2 to the power of delta-S_thermodynamic
2^delta-S_thermodyanmic = 2^(1.42 x 10^25)
Adding heat can be said to make the copper molecules bounce around more chaotically (disorderly if you will), and hence increase the thermodynamic entropy and microstates, but it says nothing of the change in design space entropy or microstates.
BOTTOM LINE:
Increasing heat increases the thermodynamic entropy and the individual copper molecules look more chaotic (disorderly if you will) because they are vibrating faster from the added heat, but it does nothing to change the design space entropy.
At 298 Kelvin:
Design Space Entropy: 500 bits
number of Design Space microstates: 2^500
Thermodyamic Entropy: 8.636 x 10^25 bits
number Thermodynamic microstates: 2^(8.636 x 10^25)
At 373 Kelvin by adding heat :
Design Space Entropy: 500 bits
number of Design Space microstates: 2^500
change in Design Space entropy due to heat change : 0 bits
change in number of Design Space microstate due to heat change: 0 microstates
Thermodyamic Entropy: 1.01 x 10^26 bits
number Thermodynamic microstates: 2^(1.01 x 10^26)
change in thermodynamic entropy due to heat change : 1.42 x 10^25 bits
change in number of thermodynamic microstates due to heat change: 2^(1.42 x 10^25) microstates
Moral of the story: don’t use 2lot to argue for design space entropy change. Besides, as pointed out earlier, increasing design complexity usually entails increase of both design and thermodynamic entropy.
Why all this obsession with reducing entropy to increase design complexity? I hope one can see it can be desirable to INCREASE entropy (both design and thermodynamic) in order to increase design complexity. A warm living complex human has more thermodynamic and design space entropy than a dead lifeless ice cube.
A warm living human has substantially more thermodynamic entropy than a lifeless ice cube. This can be demonstrated by taking the standard molar entropies of water and ice and estimating the entropy of water in a warm living human vs entropy of water in a lifeless ice cube.
http://en.wikipedia.org/wiki/Water_(data_page)
Std Molar Entropy liquid water: 69.95 J/mol/K
Std Molar Entropy ice: 41 J/mol/K
A human has more liquid water, say 30 liters, than an ice cube (12 milliliters).
Let S_humum be the entropy of a human, and S_ice_cube the entropy of an ice cube.
Order of magnitude entropy numbers:
S_human > 30 liters * 55.6 mol/liter * 69.95 J/K = 116,677 J/K
S_ice_cube ~= 0.012 liters * 55.6 mol/liter * 41 J/K = 27 J/K approximately (ice is a little less dense than liquid water, but this is inconsequential for the question at hand).
Thus warm living human has more entropy than a lifeless cube of ice.
So why do creationists worry about entropy increasing in the universe as precluding evolution? Given that a warm living human has more entropy than an ice cube, then it would seem there are lots of cases where MORE entropy is beneficial.
Ergo, the 2nd law does not preclude evolution. Other lines of reasoning should be used by ID proponents to criticize evolution, not the 2nd law.
Would it not ultimately be more productive for ID proponents to work on developing ID rather than attacking anything? Would that be an idea to mention to your fellow ID supporters perhaps?
Why attack something you already know is broken anyway? I’m sure an OP here from you on this, if you have some thoughts on how that development could potentially go, would be very interesting.
Hi Sal. Where’ve you been?
But that’s not the important question.
The important question is whether a block of ice (having the appropriate mix of chemicals, as per Barry) can spontaneously self-assemble itself into a warm, fuzzy human being.
Just add microwaves.
This seems to be Sewell’s understanding of naturalism.
I’ve assumed it is because you actually know something about thermodynamics and want to be correct.
My question is, does an IQ above room temperature have more entropy than a bag of hammers.
That would depend on the temperature of the hammers.
It’s well known that a cluster of small blocks of ice, having the appropriate mix of chemicals, can increase the fuzziness of a warm human being.
Sal, have you read this?
http://www.angelfire.com/pro/kairosfocus/resources/Info_design_and_science.htm#thermod
I’ll look forward to you pointing out anything wrong in that to gordo (kairosfocus) at UD.
Hi Creodont2,
I actually don’t read most anything Kairosfocus writes. I can’t figure out what he’s trying to communicate.
Sal
It is as far as I can tell a variation of the Gish Gallup: http://www.antievolution.org/cgi-bin/ikonboard/ikonboard.cgi?s=5522ddec5abc84c9;act=ST;f=14;t=8960;st=0
stcordova,
Hey sal, gordo says you’re wrong:
KF says Hoyle had a point. I tend to agree that any pre-specified outcome of evolution is junkyard 747 unlikely. For that reason, I think SETI is doomed, except that we may find microbes elsewhere.
KF cannot accept the possibility that humans were not pre-ordained, and are just an accident.
Hi Sal, I applaud the effort. I’ve spent a little time at UD in the past trying to get somewhere on the clear distinction between thermodynamic entropy and informational ‘entropy’, but feel it was ultimately an exercise in bashing my head against a brick wall (namely, Kairosfocus). A small amount of physics knowledge and none of chemistry makes for a pretty feeble grasp of the application of entropy to physicochemical systems, of which the electron transport chains are a central part. Life doesn’t violate the 2nd Law, it lives off it – it taps into it as an energy source, just as we do with wind power, hydro etc. If there is a lower-energy, higher-entropy state available, allowing it to be reached (equilibration), can drive a motor – indefinitely, provided there is a continuing source of the non-equilibrium state. Try driving a motor from an all-heads row of fair coins.
There are some to whom there is simply no such thing as a bad pro-Creation argument. Meanwhile KF pens another 10,000 words, liberally sprinkled with Gibbs and Boltzmann and those all-too-familiar tics.
Allan,
Thank you for the kind words.
Thermodynamic entropy is a subset of possible information entropies. 500 fair coins can be said to have:
1. 500 bits of “heads/tails” Shannon entropy
2. 8.636 x 10^25 bits of thermodynamic Shannon entropy
To be fair it was only 4,458 words and 17 pages with lots of pictures and diagrams. 🙂
UD’s arguments are pre CSI. They’re ID circa 1985.
stcordova,
Hmmm. Not so sure about that one. Thermodynamics, as I understand it, relates to energy and work. I don’t think any kind of information entropy is capable of doing work. The information available from a ‘true’ thermodynamic system, meanwhile, varies depending on the fineness with which you can divide the macrostate, a feature absent from Shannon entropy where the ‘microstate’ is the indivisible digital bit.
Did you misspeak? “Thermodynamic Shannon entropy”? Isn’t this tendency to conflation the whole point?
Allan,
See
http://en.wikipedia.org/wiki/Boltzmann_constant
The bits are Shannon information entropy bits. Recall, that entropy in the Boltzman definition is the logarithm of the number of microstates.
S = kb ln W
S = entropy
kb = Boltzmann’s consant
W = number microstates
In the Clausius view entropy is expressed in J/K. Using Boltzman’s constant, we can, with a little math give S, solve for W.
I showed the S amount for 500 copper pennies using standard tables is 826.86 J/K.
I then could solve for W.
826.86 J/K = kb ln W
826.86 J/K / kb = ln W
exponentiating both sides
e^ [ (826.86 J/K / (1.381x 10^-23 J/K) ) ] = W
insanely large dimensionless number = W
I can then rescale this number as follows given W above
S_shannon = log2 W
log2 (W ) = log 2 [e^ (826.86 J/K / kb ) ] =
= 826.68 J/K / (1.381x 10^-23 J/K) / ln(2)
826.68 J/K / (1.381x 10^-23 J/K) / .693147
8.636 x 10^25 bits
Which was the result of my calculation above.
It doesn’t have to be as mysterious as some ID proponents try to make it. At the root, we’re just counting microstates! The Shannon entropy is merely a logarithmic count of the possible microstates in a system. For computer memory, 1 Giga bit of memory is 1 giga bit of Shannon entropy. 1 giga bit of Shannon entropy represents
2 ^(1,000,000,000) microstates for computer memory.
In analogous manner 8.636 x 10^25 bits of thermodynamic entropy represents 2^(8.636 x 10^25) possible thermodynamic micrsotates. Which is an insanely gigantic number. Boggles the mind.
Allan,
Someone quite well respected at Talk Origins agreed with my conversion of J/K entropy into Shannon bits. See:
Sal
PS
I probably really made myself persona non-grata in certain ID circles by publicly agreeing with someone from Talk Origins of Gordon Davisson’s caliber. But facts are facts.
Shannon information entropy bits are abstract states, while Boltzman microstates are physical states.
A mathematical platonist might say that information entropy exists in a platonist world, while thermal energy exists in a material world.
I’m inclined to agree with Allan.
Sal,
You sowed quite a bit of confusion at UD by claiming that thermodynamic entropy is dimensionless:
That’s not true. The Kelvin is an SI base unit, so J/K is not dimensionless.
It could have been dimensionless, as Arieh Ben-Naim points out, if the temperature scale had been introduced after the atomic hypothesis was widely accepted, because then temperature could have been defined in terms of the average kinetic energy per molecule. In that case the Kelvin would not have been a base unit, J/K would have been dimensionless, and it would have made sense to define S as ln W, leaving Boltzmann’s constant out of the picture. History unfolded otherwise.
So Allan is right to question your use of the phrase “thermodynamic Shannon entropy”. Thermodynamic entropy can be converted to a Shannon entropy, but it’s not the same thing.
Keiths,
J is dimensional (kg m^2/s^2), and so is K, but a dimensionless number can result from J/K. See:
http://en.wikipedia.org/wiki/Dimensionless_quantity
1 / (1.381 x 10-23) / ln (2) is a dimensionless number since numbers are dimensionless.
I just multiply this dimensionless number by the entropy expressed in J/K to get number of Shannon bits.
If bits are dimensionless, then by way of inference J/K must also be dimensionless, otherwise I would not get this conversion:
Joule/Kelvin = 1 / (1.381 x 10-23) / ln (2) Shannon Bits =
1.045 x 1023 Shannon Bits
This is simply rescaling, and I don’t think mere rescaling can convert a dimensional number into a dimensionless one.
Anyway, if you Allan, Neil disagree, I at least showed how I decided to claim thermodynamic entropy is an instance of Shannon entropy.
My issue with 2LOT IDists is that thermodynamic microstates are not the same as head/tails microstates (or an other microstate of interest to ID proponents). The number of microstates can be expressed in bits, but that does not mean we are talking the same microstates.
The UDers are falling prey to a simple fallacy:
1. ID arguments are probabilistic arguments.
2. The 2LoT is ultimately probabilistic.
3. Therefore, the 2LoT applies to ID arguments.
Which of course is itself an invalid argument.
Some hedge a bit and say that it’s “the principle behind the Second Law”, rather than the Second Law itself, that is problematic for evolution and OOL. It’s as if they think the Second Law is just a restatement of some underlying mathematical law regarding probabilities, rather than a physical law in its own right.
CJYman’s mistake is a bit more subtle. He recognizes that the 2LoT is a physical law applying to energy flows, but he thinks that if he can define an entropy that is related to energy flow, but not identical to thermodynamic entropy, that the Second Law will apply to it.
Basketball World is my attempt to disabuse him of that notion.
Sal:
No, because the base units don’t cancel.
In base units, J/K is (kg m^2)/(s^2 K). The only way to get cancellation is to redefine K in terms of energy.
K is only labeled a base unit in SI and is called a base unit only in practice but not strictly in theory. It can be seen to be a derived unit.
[EDIT: corrected because of subsequent comment by Keiths, see below]
K being a unit for temperature is a measure average energy per degree of freedom, and “degrees of freedom” is dimensionless while energy is dimensioned (J = kg m^2/s^2).
Degrees of freedom is
Kb ln W
which is entropy.
Thus temperature is
Average Energy / (Kb ln W) .
Thus if average energy is expressed in Joules
K = J / (kb ln W)
J/ K = J / (J/ Kb ln W) = Kb ln W
Dividing J by K results in a dimensionless number, it may be merely scaled.
Scaling a dimensionless number results in dimensionless number. If you insist Kb ln W is a dimensional number, then that conflicts with the fact I just rescaled it into a dimensionless number of bits.
EDIT:
Mistakenly wrote orgininally
“K is only labeled a base unit in SI and is called “dimensionless” only in practice but not strictly in theory.”
Keiths pointed out my egregious error
Sal,
K is dimensionless neither in theory nor in practice. The Kelvin is a unit, Sal. Units by definition are not dimensionless.
As I said:
You are doing exactly that: redefining K in terms of energy.
Sal,
The source of your error is that you are mistakenly treating your conversion factor as dimensionless. It isn’t.
Let’s do some dimensional analysis.
You are trying to find a constant C for which
S_shannon = CS
…where S is the thermodynamic entropy.
Therefore,
C = S_shannon/S
S_shannon is dimensionless, and S has units of J/K. Therefore C has units of K/J.
I meant to say K is not strictly a base unit. My mistake.
If one regards “number of microstates” as a dimension, then
S = ln W
is dimensioned.
On the other hand, the Wikipedia article on Boltman’s constant regards
S_shannon = ln W
as dimensionless. I think one can go either way, but I was using the Wikipedia claim
S_shannon = ln W
is dimensionless.
If S_shannon is dimensionless, then multiplying by Kb
S_clausius = S_shannon * Kb
should not suddenly make it dimensional.
Length is a dimension expressed in meters. We can express length in millimeters. Just because we rescale doesn’t mean we add a dimension.
If one wants to make the logarithmic count of microstates a dimension, that’s fine, but that’s different than the way wiki treats the logarithmic count of microstates.
I view the fundamental (base) dimensions of phyisics
1. length
2. time
3. mass
4. charge
I view temperature as a derived unit, unless one wants to make number of microstates a dimension, and that’s also fine with me. In that sense, if the count of microstates isn’t dimensionless, then “dimensionless entropy” really isn’t dimensionless afterall.
Thanks for the editorial suggestions. Though I think I have reason to claim I was right, I’ll withdraw the claim J/K is dimensionless.
Energy per degree of freedom, in the case of plasmas, that is exactly what is done.
See:
http://en.wikipedia.org/wiki/Electron_temperature
Sal,
I don’t regard the number of microstates as a dimension, and I’m unaware of anyone who does. Who are you thinking of?
Also, S = ln W is true for Shannon entropy, which is dimensionless, but not for thermodynamic entropy, which has units of J/K.
That’s because it is dimensionless. They got it right.
Of course it should. Kb has dimensions!
Kb = 1.3806488 × 10^-23 m2 kg s^-2 K^-1
Nobody does. That’s just your confusion talking.
Again, no one is suggesting that the number of microstates should be considered a dimension.
They’re corrections, not editorial suggestions.
No, because you can’t cancel a unit unless it appears in both the numerator and denominator. J/K in base units is (kg m^2)/(s^2 K), so the units don’t cancel.
Good, because it’s incorrect.
keiths:
Sal:
No. They are not redefining the Kelvin. They are simply expressing the electron temperature in units of energy, and then explaining how you can infer the electron temperature in K from the energy value:
Let it go, Sal. You made a mistake, and an ID critic corrected you. You should be used to that by now! 🙂
Hey sal, I left a suggestion for you on this page:
http://www.antievolution.org/cgi-bin/ikonboard/ikonboard.cgi?s=5529f5014ead2cfa;act=ST;f=14;t=7640;st=3030
Nope, not now Keiths, you’ve given it away. You’re not liking the fact you just lost this parry, hehe.
Something to realize:
http://www-mdp.eng.cam.ac.uk/web/library/enginfo/textbooks_dvd_only/DAN/units/units.html
and
https://books.google.com/books?id=oaQyO4m_kBIC&pg=PA357&lpg=PA357&dq=conversion+factors+of+physics+dimensionless&source=bl&ots=ZFVfYQS9wT&sig=GtgDbP7eqRIpU8_EjswEmgqVw2k&hl=en&sa=X&ei=7u0pVeuQDs_egwSpuoS4Cw&ved=0CDUQ6AEwBA#v=onepage&q=conversion%20factors%20of%20physics%20dimensionless&f=false
From:
http://en.wikipedia.org/wiki/Electronvolt
So if I measure electron temperature in Kelvin, am I measuring a different dimension when I measure electron temperature in terms of electron Volts? If electron temperature is measured in Kelvin is the same dimension as electron temperature measured in electron Volts, then kb is used just a conversion factor in that case.
So is electron temperature a dimensioned quantity? Yes, it can be expressed in plasma physics in electron volts because in plasma physics, electron temperature is related to the average energy per electron. Which can be converted to Joules and 1 Joule = J = kg m^2/ s^2
And for electron temperature, K can be defined as:
K = eV / kb / 11604.505
For kinetic temperature it is somewhat analogous for the average energy of each particle:
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html#c1
T = 2 /(3 kb) KEavg
in units
K = 2/(3kb) Joules
where
T = temperature
kb = Boltzman’s constant
KEavg = average kinetic energy
Incidentally, this relates to Thermal Energy
http://en.wikipedia.org/wiki/Thermal_energy
But for temperature in general, the relation is
K = J / S = J / ( kb ln W)
where S is entropy (degrees of freedom) and dimensionless (according to you), which means then S is just a conversion factor scaling Joules to Kelvin.
So you’re claiming electron temperature expressed in Kelvin is a different dimension than electron temperature expressed in electron Volts? If so, that is silly.
And recall
1eV = 1.602176565·10^-19 J
If that is not your position, the dimension for electron temperature can be described either in electron volts or Kelvin, where Kb is just a dimensionless conversion factor much like meters to millimeters is a conversion factor for length.
You’re mistake is you can’t admit you’re wrong, and you know I won’t let you get away with it if I’m right. 🙂 Saving face for you is more important than setting the record straight.
The problem with you Keiths is I will say, “I made a mistake” whereas far be it from you to be corrected.
You’ll say silly things like 500 fair coins heads is perfectly consistent with physics, and never back down even after getting toasted repeatedly for it, even after Jeff Shallit couldn’t stand the ridiculous position being defended at TSZ.
See:
🙂
Sal,
Suit yourself. Just remember that it was your choice to dig the hole deeper. No one forced you to — or even asked you to.
I understand that you’re unhappy with the definition of the kelvin. Too bad — the scientific community is not going to change the definition just to soothe your bruised ego.
Here’s the definition:
It’s a base unit:
…and it’s defined in terms of the triple point of water, not in terms of energy.
Them’s the breaks, Sal.
Look at the list of dimensionless constants:
http://en.wikipedia.org/wiki/Dimensionless_quantity#Dimensionless_physical_constants
This may seem strange since speed of light may obviously appear to have dimensions of meters/second, but under relativity it can be expressed simply as c = 1.
So under naturalized units, Einstein’s famous equations is:
E = m c^2 = m
Suggesting energy and mass are equivalent, c is dimensionless. The notion that c = 186,282 miles/second has dimensions is an artifact of representation, not that c really has dimensions under Einsteinian relativity.
Similar considerations for Boltzmann’s constant. It is dimensionless in the physical sense, but J/K is an artifact of representation.
See also:
http://en.wikipedia.org/wiki/Natural_units
Sal:
It’s true. Every head/tail sequence of length 500 is consistent with the laws of physics. Flip a fair coin 500 times and you will get a sequence, Sal. This isn’t rocket science.
Those old defeats really chap your ass, don’t they?
Sal:
keiths:
Sal:
I did, and Boltzmann’s constant is not on the list. It isn’t a dimensionless constant, after all.
Good grief, Sal.
Sal:
Right. It’s exactly like I said:
The joule is already defined in terms of kg, m, and s. The kelvin is already defined in terms of the triple point of water, and no one is going to redefine it tonight so that you can save face.
Your statement is incorrect:
You didn’t read correctly:
kb = 1 under natural units, not man made units, hence kb is dimensionless were it not for historical man-made reasons. Hence kb expressed in J/K is just a conversion factor, and as pointed out conversion factors are dimensionless.
Good grief get a clue.
Sal:
Right, which is exactly what I said near the beginning of our exchange:
However, J and K are not natural units. They are SI units, and K is a base unit in that system..
So my statement is correct:
And yours is incorrect:
Keiths’ reasoning.
Entropy S stated this way has no dimension (according to Keiths):
S = ln W
Entropy S stated this way must have a dimension since kb (according to keiths has real dimensions):
S = kb ln W
Hence according to Keiths one entropy has no dimensions (S = ln W) but the other does (S = kb ln W). Silliness.
This implies then if I had an electron temperature scale where I measured electron temperature by using
S = ln W
that according to Keiths it is fundamentally a different dimension than the electron temperature using
S = kb ln W.
Silliness, Keiths, silliness. The two expressions of entropy better result in measuring along the same dimension if we are measuring electron temperature, and that implies kb is just a dimensionless conversion factor. They yield different numbers along the same dimension because in one case kb = 1, and in the other kb=J/K.
If kb = 1 naturally, this implies:
1.3806488(13)×10^−23 J/K = 1
which means
J/K = 1/ 1.3806488(13)×10^−23
which shows J/K is just a dimensionless conversion factor.
Keep digging, Sal.
I’ll explain your latest errors tomorrow, if someone else doesn’t beat me to it.
Good night.
Keiths has problems with conception, so I will help him.
A thermodynamic system of 500 pure copper pennies has hypothetically
2^(8.636 x 10^25) microstates.
So
W = 2^(8.636 x 10^25)
If I use natural units, kb = 1
S_natural = kb ln W = ln 2^(8.636 x 10^25)
S_shanon = kb log2 W = log2 [ 2^(8.636 x 10^25)]
if I use SI units, kb = 1.3806488(13)×10^−23 J/K , thus
S_si = kb ln 2^(8.636 x 10^25)
So according to Keiths, S_nautral has no physical dimensions but somehow S_si has acquired real (not just perceptual) physical dimensions just because we used a different value of Boltzman’s constant. Doesn’t that sound a little, um, incongruous since after all entropy is fundamentally just a logarithmic count of the microstates?
I mean gee, this would be as bad as saying “when measuring the length of the same object, one measures a different physical dimension when expressing the length in yards vs. expressing the length in meters.”
If the SI unit for entropy (J/K) measures the same thing that the natural unit for entropy measures, then one measurement can’t be physically dimensionless while the other is dimensioned. The SI unit (J/K) and the natural unit for entropy are logarithmic measures of the same number of microstates, yet Keiths swears one measurement measures actual physical dimensions while the other doesn’t.
Almost sure Keith is male, Sal, so it’s pretty much guaranteed he’d have problems conceiving.
stcordova,
Boltzmann entropy from Wikipedia:
“The value of W was originally intended to be proportional to the Wahrscheinlichkeit (the German word for probability) of a macroscopic state for some probability distribution of possible microstates—the collection of (unobservable) “ways” the (observable) thermodynamic state of a system can be realized by assigning different positions(x) and momenta(p) to the various molecules. Interpreted in this way, Boltzmann’s formula is the most general formula for the thermodynamic entropy. However, Boltzmann’s paradigm was an ideal gas of N identical particles, of which N_i are in the i-th microscopic condition (range) of position and momentum. For this case, the probability of each microstate of the system is equal, so it was equivalent for Boltzmann to calculate the number of microstates associated with a macrostate. W was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today.”
(my italics).
I think you are applying a naive view (ideal gas particle energies) to the whole world of entropy.
Consider a system with 500 molecules of hydrogen in a closed box. All have exactly the same energy and direction – all heads, if you will. This has the maximal capacity to do work. They are all heading to the same side of the box, and would drive a fan if one was available. Gradually, there will be fewer and fewer molecules at this extreme of the energy distribution, as the capacity to do work converts into actual work. Absent the fan, the entropy of the overall system does not change during equilibration, and the only ‘information’ you have is that the box has temperature T. But, you could insist, there is real ‘information’ in there. Each subdivision of the larger system is a macrostate in itself, with a distribution of microstates that can account for it, so if you pick an appropriate level, it can be informative in a Shannon-like manner – you can peek in the box, and find 500 bits of info: all heads. But the fineness of those subdivisions is somewhat arbitrary – you have a nesting of microstates, rather than 2 levels, ie a macrostate and the permutations of infinitely small microstate that can account for it. Each time you drop a level, you gain information. So if you cut the box in half, you get 2 bits, and so on. Now, I’m not sure which of these is a ‘Shannon state’. The molecules are I suppose digital, so if your microstates are sufficiently fine that each contains one (or none), then you have your equivalence (a mathematical one – it is due to the mathematics of permutation).
But now, replace the molecules with 500 hydrogen atoms. Now, you have added chemistry. Eventually you’ll get 250 molecules of hydrogen and a fair bit of energy. What happens to the ‘thermodynamic Shannon info’?
If it were dimensionless, then normalization wouldn’t do anything. The Wikipedia paragraph title is misleading.
Fifteen years ago, the largest disk drive one could purchase was around 10G. Today, 4T disk drives are commonplace. Shannon entropy is not tied to material, or this would be impossible. Thermodynamic entropy is tied to material. And if it is tied to material, it is not dimensionless.
Neil,
I don’t have any problem if someone wants to say thermodynamic entropy is dimensioned or dimensionless. But I do have a problem if someone says (as Keiths is insinuating)
If I had a conversion factor of meters/ 1000millimeters that I apply to length, I don’t think that counts as creating a new physical dimension when I multiply a length expressed in millimeters by meters/ 1000millimeters. So if I apply a converstion factor of
1.045 x 1023 Shannon Bits / (J/K)
to an entropy expressed in SI units, namely in J/K, I’m not measuring something in another physical dimension or removing a physical dimension. It is either the same dimension, or the quantity was dimensionless to begin with.
I claimed entropy was dimensionless to begin with, but I have no problem saying others will reasonably disagree. I do have a problem saying “thermodynamic entropy is a measure of a physical dimension when expressed in J/K and after it is multiplied by a factor of 1.045 x 1023 Shannon Bits / (J/K), thermodynamic entropy becomes physically dimensionless.”
As far as microstates in a computer, they are physically realizable in principle, the correspond to the variety of ways the bits can be individual set to 1 or 0. They are physical states. But I have no problem if one wants to call computer bits dimensioned or dimensionless. The literature I’ve seen is a bit vague on the matter.
Thank you for your response.
You lose possible microstates, which reduces the number of bits. An illustration is if I draw the thermodynamic boundary around 1 copper coin vs. 500 copper coins, I have fewer thermodynamic bits.
500 copper pennies
S_thermodynamic = 500 * 33.2 Jolues/Kelvin/Mol * 3.11 grams 65.546 grams/ mol = 826.68 J/K = 826.68 J/K / (1.381x 10^-23 J/K) / .693147 = 8.636 x 10^25 bits
1 copper penny
S_thermodynamic = 500 * 33.2 Jolues/Kelvin/Mol * 3.11 grams 65.546 grams/ mol = 1.6316 J/K = 1.7272 x 10^23 bits