On perpetual motion machines (and where to find them).

The invention of a perpetual motion machine will revolutionize the world, and Dr. Harvey Trussbloom has done it!  Well, perhaps he didn’t invent it, but he found it.  Or, no, not quite found it, but he knows where it is.  Roughly.  Although perhaps it isn’t entirely accurate to refer to it as a “perpetual motion machine.”  Because it will stop, eventually.  But if you’re looking for a machine that behaves as though it were a perpetual motion machine for a length of time X, continuing to do work without loosing any internal energy, where X can be as large a number as you desire, well, Dr. Trussbloom knows where that machine is!  Roughly.  And that machine will revolutionize the world just as resoundingly as would a true perpetual motion machine.  Although, yes, it’s true, Dr. Trussbloom did not quite pin down the location of this ingenious device.  But if you tell him your number X, any number, pick a number, he has written an equation that will tell you the radius of a sphere centered around your current location in which a device that operates for time X will almost assuredly (>99.9% probability) be.  For this press release, we started small.  We chose one hour, a single glorious hour of work extracted from the universe for free.  And as it happens, such a device is only ________ kilometers away.


Perpetual_Motion_by_Norman_RockwellI launched myself out of bed in the middle of the night to type the preceding passage.  It seemed like a good story at the time; sounds ridiculous despite the basic content of the story being realistic, which is something I like.  After hacking it out, I trundled back to bed feeling rather pleased.

When I read it again in the morning, though, I realized that the story probably fails for a general audience.  Because there are a few scientific concepts underlying it that (hopefully) the average person doesn’t spend much time obsessing about.

The first important idea for the story comes from the Second Law of Thermodynamics (often written with capital letters, I guess to show off how important thermodynamics is — and maybe people involved with the field feel like they have something to prove, because chemistry majors everywhere seem to dread thermodynamics as their most boring required class.  For undergraduates, that is.  At the grad level, statistical mechanics probably wins the “most boring” ribbon; coincidentally, stat mech is the course where you learn a more rigorous formulation of thermodynamics).

There are many ways to phrase the second law, but the most popular

(yes, I realize this is perhaps equivalent to the dudes at the nerd table in a high school cafeteria debating amongst themselves who is most popular, but we all fight for what accolades are within our grasp)

is probably Max Planck’s formulation, which states (roughly) that entropy is always increasing.

Okay, here’s thermodynamics in brief: there are two factors that decide what’s going to happen.  One factor is how easy a thing is; easy things happen more often (this is “enthalpy”).  The other factor is how abundant an outcome is; abundant things also happen more often (this is “entropy”).  When these factors are aligned, it’s simple to guess what’s going to happen; when they’re opposed, we have to think.

Sipping_BirdImagine there’s an assignment in high school where the teacher brings in a big sack of hundreds of books and dumps them on the floor and says, “Each student has to pick a book, read it, and write a report.”  If there are, oh, thirty or so students, and the books all appear to be roughly the same length, they might each just pick a book at random.  If the teacher had a hundred copies of To Kill a Mockingbird and only one copy of each of a few other books, it’s probably reasonable to guess that the outcome will be many reports on To Kill a Mockingbird.  But then, if there are hundreds of copies of Ulysses and only a few copies of To Kill a Mockingbird, we have to consider how hard the students are willing to search through the pile in order to find a book they might enjoy reading.

With something like an ideal gas in a box, the situation is dictated almost entirely by entropy.  It’s just as easy for each molecule of a gas to move upward as opposed to any other direction.  But there’s only one possible permutation that has every molecule moving upward simultaneously, and there are many, many, many arrangements that has them moving every which way, unproductively.

The Second Law of Thermodynamics states that, for a bunch of randomly-oriented gas in a box, the molecules won’t all suddenly find themselves moving in the same direction. All moving upwards, say, which means they could press on the top lid together and do thermodynamic work.  The second law says you don’t get work out of a system for free.

WaterScrewPerpetualMotionBut in my story, Dr. Trussbloom found a machine that does get work for free.  Because it isn’t actually disallowed.  It’s just very, very, very unlikely.

To begin, imagine two molecules, randomly moving.  If we pixelate their world such that each can move in only six directions (up down, forward back, left right), then the chance they’ll be moving in the same direction is 17%; the first molecule has to go in a direction, the second has a one in six chance of moving the same direction.  For three molecules, the chance they’ll be moving in the same direction is 3%.  For N molecules, (1/6)^(N-1).

Of course, our world, if it is course-grained at all, has more than six directions to move in at any time.  So instead you’re looking at something like (1/an almost-if-not-actually-infinitely-large-number)^(very large power).  Which is small.  So small that the second law rounds it off to zero.

And if space is actually continuous, in which case there would be an infinite number of directions that any molecule could move in, the probability of a mere two molecules moving in exactly the same direction is in fact zero.  But near-alignment will allow work to be extracted almost as well as perfect alignment, so we could imagine a cone of directions centered around the movement vector for our first molecule that we would want the movement of the second, and third, and fourth, and so on, to fall within.  Then the probability won’t be zero, even if it’s very small.

That’s half the background that I think would’ve been necessary for a general audience to think the story was funny.  The other half is simpler, conceptually — not about the nitty-gritty of extracting work from our perpetual motion machine, but about the likelihood of finding the machine somewhere.  This, you could read about in Max Tegmark’s Our Mathematical Universe (which, sure, I’ve offered some criticism of in the past, and which does devolve into non-science toward the end, but the first two-thirds of the book really is a high-quality, very accessible description of some cool ideas from physics).

The basic idea is that, if space is unbounded, there’s a high probability that an arrangement of particles equivalent to your immediate surroundings will arise again somewhere in the far-off distance.  There should be a room virtually identical to the room you’re sitting in, and then many rooms that are quite close but with some strange deviations — the flowerpot is to the right of your computer, not the left, or the flowerpot is full of luminescent fungi instead of ornamental moss, or the flowerpot is a hammer dull red with encrusted gristle — and then many more that are increasingly bizarre as compared to your current environs.

The basic idea being quite similar to the above explanation of entropy; there is only one arrangement of particles that will match an environment exactly, but many more arrangements that seem similar but somehow strange.

“Schroedingers cat film” by Christian Schirm – Own work. Licensed under CC0 via Wikimedia Commons.

So, we imagine a room with our prototype perpetual motion machine sitting inside.  Activate the thing, and, voila!  It gives us no work for free.  It is not, in fact, a perpetual motion machine.


But then we translate laterally through the universe, looking for near-identical copies of that room.  In most of those copies, the machine gives us… no free work.  But if we look at enough copies — and we should be able to look at infinitely many, if the universe is infinitely large — we will eventually find one where, as unlikely as it may be, the motions of randomly moving particles align, heat flows from a cold object to a warm one, work is done for free, our perpetual motion machine continues operating for whatever your chosen length of time.

And, uh, I didn’t bother trying to estimate the distance for the story above.  But it is very large.  Large enough that I’m not really sure what adjective to use; often when I’m trying to describe a large distance I’ll write something like “astronomically far away,” but it’d be silly to use the word “astronomical” for this.  Because astronomy can’t really address distances larger than 14 billion light years.  We are trapped within a sphere where the radius is defined by the product of the speed of light and the age of matter in our corner of the universe.  Anything outside that sphere, we can’t see, we can’t hope to interact with (as far as we know).

That perpetual motion machine?  It is well outside our sphere.  Many orders of magnitude farther away.

K did suggest that there was a reading of the story that might seem (slightly) humorous from a non-scientist’s perspective, though.  The idea that scientists often get quite excited about advancements in their narrow fields, despite their findings having little to no impact on the rest of the world.  Like, hey, a perpetual motion machine!  But it’s… where?  Then, why?  Why is this what you’re studying?

Hell, you could even extend that last question to my own work.  Why is this what you’re writing?