Meanwhile, back at the office

Closing time.  Anne and I stroll from Al’s coffee shop back to the Acme Building.  It’s a clear night with at least 4,500 stars, but Anne’s looking at the velvet black between them.

“What you said, Sy, about the Universe not obeying Conservation of Energy — tell me more about that.”

“Aaa-hmmm … OK.  You’ve heard about the Universe expanding, right?”

“Ye-es, but I don’t know why that happens.”

“Neither do the scientists, but there’s pretty firm evidence that it’s happening, if only at the longest scales.  Stars within galaxies get closer together as they radiate away their gravitational energy.  But the galaxies themselves are getting further apart, as far out as we can measure.”

“What’s that got to do with Conservation of Energy?”

“Well, galaxies have mass so they should be drawn together by gravity the way that gravity pulls stars together inside galaxies.  But that’s not what’s happening.  Something’s actively pushing galaxies or galaxy clusters away from each other.  Giving the something a name like ‘dark energy‘ is just an accounting gimmick to pretend the First Law is still in effect at very large distances — we don’t know the energy source for the pushing, or even if there is one.  There’s a separate set of observations we attribute to a ‘dark energy‘ that may or may not have the same underlying cause.  That’s what I was talking about.”Fading white satin

We’re at the Acme Building.  I flash my badge to get us past Security and into the elevator.  As I reach out to press the ’12’ button she puts her hand on my arm.  “Sy, I want to see if I understand this entropy-elephant thing.  You said entropy started as an accounting gimmick, to help engineers keep track of fuel energy escaping into the surroundings.  Energy absorbed at one temperature they called the environment’s heat capacity.  Total energy absorbed over a range of temperatures, divided by the difference in temperature, they called change in entropy.”

The elevator lets us out on my floor and we walk to door 1217.  “You’ve got it right so far, Anne.  Then what?”

“Then the chemists realized that you can predict how lots of systems will work from only knowing a certain set of properties for the beginning and end states.  Pressure, volume, chemical composition, whatever, but also entropy.  But except for simple gases they couldn’t predict heat capacity or entropy, only measure it.”

My key lets us in.  She leans back against the door frame.  “That’s where your physicists come in, Sy.  They learned that heat in a substance is actually the kinetic energy of its molecules.  Gas molecules can move around, but that motion’s constrained in liquids and even more constrained in solids.  Going from solid to liquid and from liquid to gas absorbs heat energy in breaking those constraints.  That absorbed heat appears as increased entropy.”

She’s lounging against my filing cabinet.  “The other way that substances absorb heat is for parts of molecules to rotate and vibrate relative to other parts.  But there are levels.  Some vibrations excite easier than others, and many rotations are even easier.  In a cold material only some motions are active.  Rising temperature puts more kinds of motion into play.  Heat energy spreads across more and more sub-molecular absorbers.”

She’s perched on the edge of my desk.  “Here’s where entropy as possibility-counting shows up.  More heat, more possibilities, more entropy.  Now we can do arithmetic and prediction instead of measuring.  Anything you can count possibilities for you can think about defining an entropy for, like information bits or black holes or socks.  But it’ll be a different entropy, with its own rules and its own range of validity.  … And…”Riding the Elephant

She’s looming directly over me.  Her dark eyes are huge.

“And…?”

When we first met, Sy, you asked what you could do for me.  You’ve helped me see that when I travel across time and probability I’m riding the Entropy Elephant.  I’d like to show my appreciation.  Can you think of a possibility?”

A dark night, in a city that knows how to keep its secrets.  On the 12th floor of the Acme Building, one man still tries to answer the Universe’s persistent questions — Sy Moire, Physics Eye.

~~ Rich Olcott

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At The Turn of A Card

Not much going on today.  I’m dealing myself a hand of solitaire when I hear a familiar fizzing sound.  “Hello, Anne.  Good to see you again.”

She’s freshened up that white satin outfit and is looking very good.  “Hello, Sy.  Busy?”

“Not so’s you’d notice it.  What can I do for you?”

“Can’t a girl just drop in when she wants to visit?  Playing with real cards, I see.  That’s good, but your tens and treys are frozen.”white satin and cards

“That’s the way the odds break sometimes.  The elephant‘s in the room.”

Entropy again?  What’s it look like this time?”

“These cards and surprise.  How surprised would you be if I were to draw a queen from the stock pile?”

“No queens showing, so some surprised but not very surprised.”

“You know me, I’m a physicist, we put numbers to things.  So put numbers to the situation.”

<sigh>  “OK, there are 52 cards in the deck and you’ve got … 28 cards in that triangle, so there are 24 left in the stock.  Four of them have to be queens.  Four out of 24 is one out of 6.”

“Or 17%.  And the odds for the queen of hearts?”

“I’m here so it’s 100% until I leave.  Oh, I know, you’re talking about the cards.  One in 24 or 4%.  So I’d be four times as surprised at seeing the heart queen as I would at seeing any of them.  Pooh.”

“Now how about the odds of drawing all four queens?”

“One in 24, times one in 23, times one in 22, times one in 21.  Whatever, it’s a very small number and I’d be very surprised.”

“Well, here’s where we get another look at the elephant.  There’s a definition of entropy that links directly to those percentages AND can handle extremely small ones.  What do you know about logarithms?”

“A little.  I read your   last   series  of  posts.”

“Wonderful, that simplifies things.  Let’s start with strange dissociation thought up by Claude Shannon to whom we owe the entire field of information theory.  His crucial insight was that he had to distinguish between information and meaning.”

“How can they be different?  If I say ‘green’ that means, well, green.”

“It’s all about context.  If you’re telling me what color something is, saying ‘green’ is telling me that the thing isn’t white or red or any of the other umm, nine colors I know the names of.  But if you’re telling me someone is really inexperienced then I know not to trust them with a complicated task that has to be done right the first time.  From Shannon’s point of view, the information is the signal ‘green,’ and the meaning is set by the context.”

“You’re going somewhere with this, I suppose?”

“Mm-hm.  In Shannon’s theory, the more surprising the message is, the more information it contains.  Remember when you told me that in one of your alternate realities you’d seen me wearing a green shirt?  That was a surprise and it told me you’d visited an unusual reality, because I rarely wear green.  If you’d told me the shirt was black or grey, that would have been much less surprising and much less informative.  Shannon’s trick was in putting numbers to that.”

“You’re just dragging this out, aren’t you?”

“No-no, only two more steps to the elephant.  First step is that Shannon defined a particular signal’s information content to be proportional to the negative of the logarithm of its probability.  Suppose I’m maybe 1% likely to wear green but equally likely to wear any of the other 11 colors.  Each of those colors has a 9% probability.  log10(1%) is -2.00, information content is 2.00, but -log10(9%) is only 1.04.  By Shannon’s definition when you said ‘green’ in this context, you gave me nearly double the information as any of the other color names.”

“Why’d you use base-10 logarithms?”

“Convenience.  It’s easy to figure log10(1%).  Information scientists tend to use base-2, physicists go for base-e.  Final step — Shannon took the information content of each possible signal, multiplied it by the probability of that signal, added those products together and called it the signal system’s information entropy. For our colors it’d be 2.0+(11×1.04)=13.44.  Regardez, voici l’éléphant!”

“Ooo, French!”

“Aimeriez-vous un croissant et un café?  My treat at Al’s.

~~ Rich Olcott

Two Sharp Dice

<further along our story arc>  “Want a refill?”

“No, I’ve had enough.  But I could go for some dessert.”

“Nothing here in the office, care for some gelato?”

We take the elevator down to Eddie’s on 2.  Things are slow.  Jeremy’s doing homework behind the gelato display.  Eddie’s at the checkout counter, rolling some dice.  He gives the eye to her white satin.  “You’ll fit right in when the theater crowd gets here, Miss.  Don’t know about you, Sy.”White satin and dice

“Fitting in’s not my thing, Eddie.  This is my client, Anne.  What’s with the bones?”

“Weirdest thing, Sy.  I’m getting set up for the game after closing (don’t tell nobody, OK?) but these dice gotta be bad somehow.  I roll just one, I get every number, but when I roll the two together I get nothin’ but snake-eyes and boxcars.”

I shoot Anne a look.  She shrugs.  I sing out, “Hey, Jeremy, my usual chocolate-hazelnut combo.  For the lady … I’d say vanilla and mint.”

She shoots me a look.  “How’d you know?”

I shrug.  “Lucky guess.  It’s a good evening for the elephant.”

“Hey, no livestock in here, Sy, the Health Department would throw a fit!”

“It’s an abstract elephant, Eddie.  Anne and I’ve been discussing entropy.  Which is an elephant because it’s got so many aspects no-one can agree on what it is.”

“So it’s got to do with luck?”

“With counting possibilities.  Suppose you know something happened, but there’s lots of ways it could have happened.  You don’t know which one it was.  Entropy is a way to measure what’s left to know.”

“Like what?”

“Those dice are an easy example.  You throw the pair, they land in any of 36 different ways, but you don’t know which until you look, right?”

Dice odds

“Yeah, sure.  So?”

“So your uncertainty number is 36.  Suppose they show 7.  There’s still half-a-dozen ways that can happen — first die shows 6, second shows 1, or maybe the first die has the 1 and the second has the 6, and so on.  You don’t know which way it happened.  Your uncertainty number’s gone down from 36 to 6.”

“Wait, but I do know something going in.  It’s a lot more likely they’ll show a 7 than snake-eyes.”

“Good point, but you’re talking probability, the ratio of uncertainty numbers.  Half-a-dozen ways to show a 7, divided by 36 ways total, means that 7 comes up seventeen throws out of a hundred.  Three times out of a hundred you’ll get snake-eyes.  Same odds for boxcars.”

“C’mon, Sy, in my neighborhood little babies know those odds.”

“But do the babies know how odds combine?  If you care about one event OR another you add the odds, like 6 times out of a hundred you get snake-eyes OR boxcars.  But if you’re looking at one event AND another one the odds multiply.  How often did you roll those dice just now?”

“Couple of dozen, I guess.”

“Let’s start with three.  Suppose you got snake-eyes AND you got snake-eyes AND you got snake-eyes.  Odds on that would be 3×3×3 out of 100×100×100 or 27 out of a million triple-throws.  Getting snake-eyes or boxcars 24 times in a row, that’s … ummm … less than one chance in a million trillion trillion sets of 24-throws.  Not likely.”

“Don’t know about the numbers, Sy, but there’s something goofy with these dice.”

Anne cuts in.  “Maybe not, Eddie.  Unusual things do happen.  Let me try.”  She gets half-a-dozen 7s in a row, each time a different way.  “Now you try,” and gives him back the dice.  Now he rolls an 8, 9, 10, 11 and 12 in order.  “They’re not loaded.  You’re just living in a low-probability world.”

“Aw, geez.”

“Anyway, Eddie, entropy is a measure of residual possibilities — alternate conditions (like those ways to 7) that give identical results.  Suppose a physicist is working on a system with a defined number of possible states.  If there’s some way to calculate their probabilities, they can be plugged into a well-known formula for calculating the system’s entropy.  The remarkable thing, Anne, is that what you calculate from the formula matches up with the heat capacity entropy.”

“Here’s your gelato, Mr Moire.   Sorry for the delay, but Jennie dropped by and we got to talking.”

Anne and I trade looks.  “That’s OK, Jeremy, I know how that works.”

~~ Rich Olcott

Keep calm and stay close to home

Again with the fizzing sound.  Her white satin still looked good.  A little travel-worn, but on her that looked even better.  Her voice still sounded like molten silver — “Hello.”White satin and drunkard walk

“Hello, Anne.  Where you been?”

“You wouldn’t believe.  I don’t believe.  I’ve got to get some control over this.”

“What’s the problem?”

“I never know where I’ll be next.  Or when.  Or even how it’ll look when I get there.  We’ve met before, haven’t we?”

“Yes, we have, and you told me your memory works in circles.  We figured out that when you ‘push,’ you relocate to a reality with a different probability.”

“But it could also be a different time.  Future, past, it’s so confusing.  Sometimes I meet myself and I don’t know whether I’m coming or going.  We never know what to say to each other.  It’s horrible way to be.”

“It sounds awful.  Here, have a tissue.  So, how can I help you?”

“You do theory stuff.  Can you physics a way to let me steer through all this?”

<fizzing sound> Another Anne appeared, next to my file cabinet on the far side of the office.  “Don’t mind me, just passing through.”  <more fizzing>  She flickered away.  My ears itched a little.

“See?  And she always knows more than I do, except when I know more than she does.”

“I’m beginning to get the picture.  Mind if I ask you a few questions?”

“Anything, if it’ll help solve this.”

“When you time-hop, do you use the same kind of ‘push’ feeling that sends you to different probabilities?”

“No-o, it’s a little different, but not much.”

“We found that you have to ‘push’ harder to get to a less-probable reality.  Is there the same kind of difference between past and future hopping?”

“Now you mention it, yes!  It’s always easier to jump to the future.  I have to struggle sometimes when I get too far ahead of myself.”

“Can you do time and probability together?”

“Hard to say.  When I hop I mostly just try to work out when I am, much less whether things are odd.”

“Give it a shot.  Try a couple of ‘nearby places’ and come back here/now.  Just use tiny ‘pushes.’ I don’t want you to get lost again.”

“Me neither.  OK, here I go.” <prolonged flickering and fizzing> “Is this the right place?  I tried a couple of hops here in your office, and <charming blush> stole some of your papers.  Here.”

“Perfect, Anne, objective evidence is always best.  Let’s see…  Yep, this report is one I finished a week ago, looks OK, and this one … I recognize the name of a client I’ve not yet hooked, but the spelling!  The letter ‘c’ isn’t there at all — ‘rekognize,’ ‘sirkle,’ ‘siense’ — that’s low probability for sure.”

“Actually, it felt like higher probability.”

“Whatever.  One more question.  I gather that most of your hops are more-or-less good ones but every once in a while you drop into a complete surprise, something you’re totally not used to.”

“Uh-huh.”

“I’ll bet the surprises happen when you’re in a jam and do a get me out of here jump.”

“Huh!  I’d not made that connection, but you’re right.”

“I think I’ve got the picture.  When you ‘push,’ you somehow displace yourself on a surface that has two dimensions — time and probability.  You move around in those two dimensions independently from how you move in 3-D space.  I take it you’re comfortable dong that but you want more control over it, right?”

“Mmm, yeah.  It’s kind of my special superpower, you know?  I don’t want to give it up entirely.”

“Good, because I wouldn’t know how to make that happen for you.  Best I can do is give you some strategy coaching, OK?”

“That’d be a big help.”Drunkard

“Stay calm.”

“That’s it?  Where’s the physics in that?”

“Ever hear of the Drunkard’s Walk?”

“I’ve seen a few.”

“Well, you’re doing one.”

“Beg pardon?”

“It’s math talk for a stepwise process where every step goes in a random direction.  Your problem is that some of the steps are way too big.  Keep the steps small and you’ll stay in familiar territory.”

<molten silver, coming closer> “Like … here?”

“Stay calm.”

~~ Rich Olcott