Twenty Puzzles and Understanding Entropy

A short history of energy and entropy:

Aristotle coined the word “energy,” Greek for “at work.”  Two millennia later, a scientific understanding of “at work” came into being in the study of thermodynamics.  The word “energy” got buffeted about until its scientific meaning today has, in many ways, the opposite of its colloquial meaning.

easily perceived
occult (hidden from view)
not conserved 
general, worldly
capacity for doing work
more like negative entropy
can be unavailable for doing work 
related to mass by logical equivalence

That thing we call energy in day-to-day usage is not the energy  of science.
Not even close!
National Science Teachers Association

And consider “therapeutic touch” (TT):
“…there is a sea of quantum energy between us and extending from the body” and which “moves faster than the speed of light.”

TT practitioners “can palpably sense an energy field that extends some 10 cm beyond the surface of the skin.  Treatment consists of manually smoothing the field.”

This “energy” has no relationship to modern science beyond a little plagiarism of language.  It retains the sense of importance we sense in Aristotle’s meaning, but even Aristotle’s primitive understanding got lost.

Science is one result when human beings engage "perceptions" that are capable of "seeing" in ways known only in the past few centuries to any effective degree.  "Quantum," "field," "energy," and the surprising place of speed of light in modern physics, are insights of these new perceptions.  Those uses of those words in the purple-bathed prose belong to the pre-science insights, conveying meaning unlikely to tax the understanding of contemporaries of Aristotle.  Their modern meaning, however, requires a little hard work, a bit of persistent puzzle solving.

Aristotle’s “at work” comes from life’s experiences.  Pushing, lifting, running, etc, activities that make you sweaty, tired, hungry, and eventually sleepy.  We “run out” of something that has to be replenished.  We run out of energy.

We replace it with rest, food, sleep—and in our modern age, with fuel for our engines.  We replenish “lost” or “spent” energy.

Thermodynamics asked how to wring the most out of fuel for our engines.  The answers were generally seen in terms of  “energy,” but the real insights were about probabilities. That thing science calls “energy” is necessary for “capacity to do work,” but something else is really the critical thing.  That something else is straight out of the gambling casino. The gambler calls it...

                "the odds."

We know, we feel, the importance of "energy," it's an experience central to life.

But it's a mystery...

The Odds (probablility)

What we must replenish is predictability—surrounded by, buried in, and smothered with, randomness.  The predictability is of energy transfers from atom to atom and from particle to particle: this is all at the microscopic level of happenings where energy is distributed in more or less random ways, distributed as, perhaps, kinetic energy of motions of atoms, or spinning energy of molecules, or energy of electrons getting promoted to "higher energy states."  Etc.

And so, Erwin Schrödinger said, in his essay What is Life?:

This mysterious thing that is so important to life ("energy") is really a fugitive from the gambling casino.
How does the living organism avoid decay?  The obvious answer is: By eating, drinking, breathing, and (in the case of plants) assimilating.  The Greek word (metaballein) means change or exchange.  Exchange of what?  Originally the underlying idea is no doubt, exchange of material.  (E.g. the German word for metabolism is Stoffwechsel.)  That the exchange of material should be the essential thing is absurd.  Any atom of nitrogen, oxygen, sulfur, etc, is as good as any other of its kind; what could be gained by exchanging them?  For a while in the past our curiosity was silenced by being told that we feed upon energy.  In some very advanced country (I don't remember whether it was Germany or the U.S.A. or both) you could find menu cards in restaurants indicating, in addition to the price, the energy content of every dish.  Needless to say, this is just as absurd.  For an adult organism the energy content is as stationary as the material content.  Since, surely, any calorie is worth as much as any other calorie, one cannot see how a mere exchange could help.

What then is that precious something contained in our food which keeps us from death?  That is easily answered.  Every process, event, happening—call it what you will: in a word, everything that is going on in Nature means an increase in entropy of the part of the world where it is going on.  Thus a living organism continually increases its entropy—or as you might say, produces positive entropy—and thus tends to approach the dangerous state of maximum entropy, which is death.  It can only keep aloof from it, i.e. alive, by continually drawing from its environment negative entropy—which is something very positive as we shall immediately see.  What an organism feeds upon is negative entropy.  Or, to put it less paradoxically, the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help but produce while alive.



James Watson says that Schrödinger's What is Life? inspired him to start his quest that led to his Nobel Prize.  Perhaps others might start a similar quest that leads them to "seeing" the deep absurdities in the purpled prose above.  (It would be hard to invent sillier phrases masquerading as spawn of modern science.)  The deep truthsthe subtle insightsthat Schrödinger directs our gaze toward reveal much about how we, as intelligent living organisms, relate to the world around us.

So drop your lottery tickets, and discover what the statistical world is really trying to tell us...


Puzzles #6 and #19 give a sense of  predictability buried in randomness.

Sadi Carnot established the second law of thermodynamics in about 1850.  The first law came later, about 1865.  The second law says entropy cannot spontaneously decrease: that means that the predictability of things (like transfer of energy or of momentum) can only get worse without outside influence.  The first law says that the quantity of a very abstract entity, the energy of science, is always the same no matter what happens and as long as everything affected is taken into account.

The second law came from the observation that you can have an enormous amount of energy (science meaning)—something like, say, the thermal energy in the ocean—and still not be able to put it to work.  If all that thermal energy is at the same temperature, it is not available for doing work.  At the particle level, energy transfers are random and unpredictable.  [the windmill; an exemplar]

If things are at different temperatures, energy can flow in a somewhat predictable manner (higher to lower temperature) and that predictability is what makes some of the thermal energy available for doing work.  (This available energy is called free energy because it is free to do work.)


Gamblers become gulls of the casinos and lotteries because they don't really understand statistics.

be a swift not a gull

Before the late 19th century, statistics was a mystery to just about everybody.  Thermodynamics unlocked a lot of the mysteries.

"Life is two locked boxes, each containing the other's key."

Piet Hein
Entropy is a measure of probability that something is in some state.
That “state” is usually something of interest to an engineer designing a “heat engine”: a steam engine or Diesel engine or gasoline engine, engines that derive mechanical energy from flow of heat.  The “something” is the steam or burned fuel.  Thermal energy is random energy, like the randomness of the device that chooses the numbers that win the state lottery.  On the other hand, mechanical  energy is organized energy, like the kinetic energy of all the individual atoms of iron in a sledge hammer.

If you know the motion of one atom in a moving sledge hammer, you can state with high probability the motions of all the others.  The hammer does work on what it hits and that work is a transfer of mechanical energy.  The sledge hammer is like a lottery in which the number selecting machine is completely rigged and all the money goes exactly where the operator wants.

If  lottery numbers were picked by a set of “loaded” dice, and you knew what that loading is, you could make the lottery work for you.  Statistically speaking, that is.  Getting mechanical energy out of a steam engine is more like getting money out of a lottery that uses loaded dice, not a rigged selecting machine. 

If you know the motion of one molecule of steam in the cylinder of a steam engine, you have very little idea of the motions of the other atoms.  Their motions are highly random, and you can describe them only in terms of statistics.  You can get a little mechanical energy out of the steam engine by letting the steam expand against the piston, but you can't begin to get all of that kinetic energy tied up in the random motions of the steam molecules.  You must go take a course in engineering thermodynamics to learn how to maximize what you do get.  (For example, when everything related to your heat engine is at the same temperature, getting work out of the engine is like getting money out of a fairly run lottery: it happens only by the most remote of odds.)

Thermodynamics will teach you the statistical nature of heat, temperature, and entropy.


lower entropy

rigged wheel
a little
temp gradient
loaded dice
uniform temp
fair coin toss
higher entropy


LOOK HERE!  Entropy is simple...if we avoid the briar patches!

Statistics is about predicting outcomes.

Heat is energy that is transferred from one place to another place at a different temperature, and it's energy that gets transferred solely because of that temperature difference.  Heat is energy transferred by statistical exchanges of energy between particles. Temperature is a parameter in a statistical distribution of energy—and the greater the temperature difference, the greater the amount of heat transferred.  Entropy was originally a measure of that part of the total thermal energy which is rendered unavailable for doing work because those exchanges are so random.  (Free energy is total energy minus the energy which is unavailable for doing work.)  Later, Ludwig Boltzmann demonstrated that the entropy of something in some state is simply the logarithm of the probability of it being in that state: S = k ln W.  That profoundly insightful equation is the epitaph on his tombstone.
Information is a key concept here, too.  Information selects from alternatives.  A mailing address selects which mailbox the envelope must end up in.  The more mailboxes, the more numbers in the address.  That’s why the cost of the hardware in Puzzle #5 increases (roughly) as the logarithm of the number.  (One costs $.20; ten costs $.40; one hundred costs $.60; One thousand costs $.80; etc.)

More generally, living organisms use information to help select from alternatives.  We then take action, and are successful to the extent that we anticipate the outcome of the action.  “Order” is a kind of predictability.

At the level of molecules in our cells, it's the predictability of chemical reactions, the predictability of osmotic diffusion through membranes, the predictability of heat transfer...these are the issues of entropy that Schrödinger spoke to us of.

These are statistical matters, and that makes them matters of modern "insight," insight just like that which goes pervasively and persistently unseen in the gambling casino.  (And then there's that "sea of quantum energy" that "moves faster than the speed of light"...    Woweeeee!!!)

Information is about selecting from alternatives...

so that we can predict the outcomes of actions we take.

January 26, 2000
The next step in this look at entropy is to examine dot patterns.  Dot patterns can be highly ordered, completely random, and everything in between.  We have several patterns of the "in betweens" that give a picture (literally) of information content.  It also gives a picture of how ordering lets us predict some things about that part of the world mapped by the dot patterns.  It's the kind of  prediction that we see reflected in the stock market, the gambling casino, and the molecular events in and about steam engines and living cells.  Entropy is a measure of "the odds" at the molecular and particle level of happenings.

It'll take a while to prepare our old materials for the Web.  In the meanwhile please tell us how these materials might better help you.
March 11, 2000