Brad DeLong's Semi-Daily Journal (2004)

Err, isn't that 10^54 zeros?

Since there are 10^27 molecules and the probability per molecule is 10^-2, so 10^54???

My pedantic wife with four degrees from MIT including one in rocket science points out that Gamow was in fact right. The number does indeed begin with 54 noughts after the decimal, in fact it can be accurately be said to begin with any number between 1 and 10^54 noughts.

Actually, by my calculations, the correct figure is 1 divided by 100^10^27 zeroes -- that is, one divided by 10 followed by TWO TIMES (10^27) zeroes, not "one divided by 10 followed by 10^27 zeroes.

Still, this means that Brad was off by only half as many orders of magnitude as Gamow. (For this, I'm tempted to forgive him for his weird speculation a few months ago that the value of pi might change in curved space.)

I too was an avid reader of Gamow, although I never got around to Mr. Tompkins. One, Two, Three... Infinity kept me busy for years, and I still enjoy looking at it. My own efforts to interest younger readers in this wonderful book have not been very successful.

As for errors of calculation involving stupendous numbers of zeros, I can only say that in this case I feel just like Gamow's Hottentot, who counted "one, two, three... many." Fifty-four zeros, 10^27 zeros, 100^10^27 zeros... we have definitely entered the territory of "many"... or, in this case of a vanishingly tiny probability, "few".

The sharp-eyed Bruce is correct...

To add an additional note of pedanticism, since the molecules are indistingushable, it's not correct to say that 1/10^54 is the probability of 27 molecules being under the table. There's a correction factor that improves the probability substantially.

But I agree that Gamow is wrong, and I confess I have whizzed by that passage without thinking on it.

The quality of mercy is not strnen

"The sharp-eyed Bruce is correct."

Well, tell it to Patrick, Brad...

Explorers announcing the discovery of the world's deepest hole:

were initially set back in their claim by a miffed Secretary of War Donald Rumsfeld, who claimed the explorer's made a major mathematical error, "I cannot remember ever seeing anyone get a calculation off by so many orders of magnitude from what we don't know that we don't know."

Rumsfeld then went on to claim the US Department of War is by far the largest and deepest hole on earth, if not the entire solar system.

However, management at Guiness Book of World Records made light of "Rummy's" obstreperous behavior with a quip, "The US War Department may indeed be the deepest *financial* black hole in all of humankind's history, but is more properly described as the largest festering morass of incompetent bungling and racist empire building, that will soon be swept away with a democratic election bristle brush and lots of truth and sunlight, than as a true geological phenomenon in and of itself."

Croatia sent Secretary Rumsfeld a consolation note referring to DoD's recent interference in Serbo-Croatian politics, "Go stuff yourself in that black hole of yours. We do it deeper!"

A betrter way to get your average 14-year-old to read something is to explicitly forbid them to do so...

What about the 10^79 number, I suspect that is a bigger error. The rest of that theory is never mentioned!

Charles is wrong about the indistinguishability of the molecules mattering.

Indistinguishability matters in *choice*, as in choosing 15 things out of 27 (in)distinguishable things. There are 27!/(15!12!) ways to choose 15 things out of 27 distinguishabble things, but only one way to choose 15 things out of 27 we're regarding as identical. But once you're choosing *all* of the objects, there's only one way to do that, whether they're distinguishable or identical. So that factor goes away.

What could affect the odds here is *independence* of the molecules' behavior. Supposing you had 5*10^26 molecules already under the table, and 5*10^26 molecules in the rest of the room: I'll let the physicists tell me whether there are molecular interactions that make it less likely than 0.01 for a random molecule to 'choose' to be under the table under those circumstances. But if there were, that would lengthen the (already astronomical) odds against having all the molecules under the table at once.

Next week's topic: angels dancing on the head of a pin. ;^)

I even remember the rebuttal by another scientist, "He thought he saw the atom,
It charge and mass combine,
He looked again and found it was,
The Cosmic sounding line,
Ah now I realise, he said,
The mass of the Universe is ten to the seventy-nine.

Glad to see you corrected your entry, Brad. (I will be contacting you shortly about the matter of due payment.)

If you want the Eleven- or Fourteen-year-old to learn about Maxwell's Demon, I highly recommend Thomas Pynchon's The Crying of Lot 49. Useful preparation for the times we live in.

When I was in grad school, a couple of friends in astrophysics tried, over beer and pizza, to make a 'back of the envelope' estimate of the albedo of some heavenly body-- and turned out to be wrong by orders of magnitude of orders of magnitude. This sort of thing happens when you're doing working in that kind of 'my numbers are bigger than yours' scientific discipline...

As an astronomer who didn't try to get into grad school, I find it deeply reassuring that those who did were *able* to get multiple order-of-magnitude errors over albedo...

"Supposing you had 5*10^26 molecules already under the table, and 5*10^26 molecules in the rest of the room: I'll let the physicists tell me whether there are molecular interactions that make it less likely than 0.01 for a random molecule to 'choose' to be under the table under those circumstances."

I'm not a physicist. But this is based on the mechanics underlying air flow and air pressure. The location of air molecules in a group cannot be treated as random, because they interact. If a bunch of them suddenly found their way under the table this would create a high pressure zone under the table and a low pressure zone in the rest of the room. Air flows from a high pressure zone to a low pressure zone (basically because a molecule has a lower probability of hitting another and having its course changed when it is moving toward the low pressure zone).

In other words, you're all arguing over how to add up the probability of every individual molecule being in a room as the combination of 10^27 independent events each of 0.01 probability. But the events are not independent - the probability of one molecule being in a given volume is inversely proportional to the number of other molecules which are also in that volume.

Therefore the probability of all the air in the room finding its way into 1% of the volume is waaaa(a^943)aay lower than (1/100) ^ (10^27).

The best method I ever heard of to get that age group to read (esp. if they're male) is to introduce them to Bob Heinlein's "juvenile" SF novels. There's nothing condescending or good-enough-for-preteens about those books. Heinlein himself said that he wrote the juvenile novels just like the ones aimed at grown-up, but just "left out the sex."

Well, yeah, Ian -- but all of us (including Gamow) ignored mutual interactions between the molecules for the sake of relative simplicity. As you say, the figure we've finally come up with is an absolute upper limit.

As for CaseyL, I agree with (I believe) James Blish, who said, "The Golden Age of Science Fiction is 13." In my case, however, the man with the keys to the kingdom was Arthur C. Clarke.

It must be Clinton's fault.

Here's a quote that might be apropos:
To poke fun at his own imprecision [George Gamow] once
published a paper that contained an enormous and deliberate
error, then submitted an erratum, prepared in advance,
advising that one of its equations was off by a factor
of 10^24...which, he reassured readers of the journal,
"does not affect the result."
--Timothy Ferris, _The_Whole_Shebang_

When treating gases statistically, it's reasonable (and usual) to treat the molecules as noninteracting to a first approximation. (This approximation leads to ideal gas behavior.) The pressure of the gas arises from collisions with the walls of the container (here, the room), while the position and velocity of an individual molecule (of N2 or O2) are determined by its previous history of encounters with the walls of the container or the other molecules. This way the answer, as previously mentioned, is (10^(-2))^(10^(27)) = 10^(-2x10^(27)).

RT states that "Charles is wrong about the indistinguishability of the molecules mattering. Indistinguishability matters in *choice*, as in choosing 15 things out of 27 (in)distinguishable things. There are 27!/(15!12!) ways to choose 15 things out of 27 distinguishabble things, but only one way to choose 15 things out of 27 we're regarding as identical. But once you're choosing *all* of the objects, there's only one way to do that, whether they're distinguishable or identical. So that factor goes away."

RT is incorrect in claiming that my description was incorrect, since I only discussed the case when 27 objects are chosen out of a roomful and am very well aware of what happens when one chooses n out of n.

However, his statement is otherwise correct.

Ian Montgomerie on the other hand, is incorrect in stating that all the molecules cannot go under the table because of "pressure." This is confusing thermodynamics with statistical mechanics. "Pressure" is the macroscopic expression of the behavior of the ensemble of molecular interactions.

Did I seriously see someone suggest that you should subject an eleven year old to Pynchon? Egad. Isn't there some sort of federal agency you guys support that would prevent that sort of abuse?

Good story, Sue!

Three cheers for the memory of George Gamow, who understood even better than Einstein (but not as well as Feinman) that science is hilarious.

My favorite is the Alpher-Bethe-Gamow paper.

People born in or near Odessa around the turn of the century:

George Gamow
Isaac Babel
Sviatoslav Richter
Anna Akhmatova
Benno Moiseiwitsch
David Oistrakh
Nathan Milstein

"You must know everything...."

Where is Odessa now?

NM

"Ian Montgomerie on the other hand, is incorrect in stating that all the molecules cannot go under the table because of "pressure." This is confusing thermodynamics with statistical mechanics. "Pressure" is the macroscopic expression of the behavior of the ensemble of molecular interactions."

I try to parse that statement and can't come up with anything but nonsense. The "ensemble of molecular interactions" (where the particles can end up) is precisely what the calculation is about. Air molecules are unlikely to move toward an area of higher density (higher pressure), since while moving in that direction they're more likely to hit another and be redirected.

No. You are taking a derived concept (pressure) and trying to apply it to explain the behavior of the fundamental objects (moving particles).

"Pressure" is a quantity is derived from the statistical behavior of the particles. It does not necessarily depend on collisions. For example, the ideal gas law (PV=nkT) is derived from the assumption that particles that have zero size and do not collide or interact at at all. We can certainly correct for these assumptions, but these corrections don't matter except under unusual conditions, such as the particle concentration being *very* high. For the gas you are breathing right now, PV=nkT works just fine. The point is, gas pressure exists independently from particle-particle interaction. Collisions are a factor, but they are not the fundamental factor.

Thermodynamic quantities such as temperature and pressure are quite convenient for performing calculations on real-life stuff under real-life timescales. But rest assured, it is possible for all the gas in a room "go against the flow" and pile up in one side of the room. It's just very, very, very unlikely to happen.

Given that this blog has provided plenty of evidence that the world we live in is less Orwellian than one governed by Tyrone Slothrop's Proverbs for Paranoids (see below), it is never too soon to begin reading Pynchon.

Proverbs for Paranoids:

1: You may never get to touch the Master, but you can tickle his creatures.
2: The innocence of the creatures is in inverse proportion to the immorality of the Master.
3: If they can get you asking the wrong questions, they don't have to worry about answers.
4: You hide, they seek.
5: Paranoids are not paranoids because they're paranoid, but because they keep putting themselves, fucking idiots, deliberately into paranoid situations.