Q: Why doesn’t life and evolution violate the second law of thermodynamics? Don’t living things reverse entropy?

Posted on March 24, 2013 by The Physicist

Physicist: In very short: nope.

The second law of thermodynamics is sometimes (too succinctly) stated as “disorder increases over time”.  That statement seems to hold true, what with mountains wearing down, machines breaking, and the inevitable, crushing march of time.  But living things seem to be an exception.  Plants can turn dirt (disordered) into more plants (order), and on a larger scale life has evolved from individual cells (fairly ordered) to big complicated critters (very ordered).

However, there are a couple things missing from the statement “disorder increases over time”, such as a solid definition of “disorder” (it’s entropy) and the often-dropped stipulation that the second law of thermodynamics only applies to closed systems.

Creatures, both in the context of growing and reproducing, and in the context of evolution are definitely not closed systems.  Doing all of that certainly involves an increase in order, but at the expense of a much greater increase in disorder elsewhere.  Specifically, we eat food which, with all of its carbohydrates and proteins, is fairly ordered, and produce lots of heat, sweat, and… whatnot.  Food, and air, and whatnot are what make living things “open systems”.

Whatnot.

Whatnot.

If a creature could take, say, a kilogram of non-living, highly disordered material and turn it into a kilogram of highly ordered creature, then that would certainly be a big violation of the second law of thermodynamics.  However, people (for example) consume along the lines of about 30 to 50 tons of food during the course of a lifetime.  Some of that goes into building a fine and foxy body, but most of it goes into powering that body and fighting degradation (blood and skin and really everything wears out and needs to be replaced).  So, about 0.15% (give or take) of that food matter is used to build a body, and 99.85% is used for power and to fight the entropy drop involved in body construction and temporarily holding back the horrifying ravages of time.

When compared to the entropy involved with turning food into the many, many bodies that make up a species, evolution is barely an afterthought.  In fact, the entropy (as used/defined in thermodynamics) of most animals (by weight) is all about the same.  A person and a mountain lion have about the same entropy as each other, simply because we weigh about the same.

The big exception is photosynthesizing plants.  They really can turn a kilogram of inert, high-disorder dirt, air, and water into a kilogram of low-disorder plant matter.  But, again, they’re working with a bigger system than just the “plant/dirt/air/water system”.

There's a huge drop in entropy between the incoming sunlight

There’s a huge increase in entropy between the incoming sunlight and the outgoing heat that’s radiated away from the Earth.

Sunlight is a bunch of high-energy photons coming from one direction, which involves relatively little entropy.  A little later that energy is re-radiated from the Earth as heat, which is the same amount energy spread over substantially more photons and involves a lot more entropy (relatively).  This huge increase in entropy, between the incoming sunlight and the outgoing heat, is the “entropy sink” that makes all life on Earth possible (with just a handful of exceptions).  In particular, green plants take a tiny amount of the sunlight that hits the Earth and turns some of the energy into sugars and other useful plant-ey material.  It all eventually turns into heat and radiates away, but instead of doing it all at once it does it through a few links in the food chain.

You can think of this huge sunlight-to-re-radiated-heat increase in entropy like water going over a waterfall, and life as being like a hydro-electric dam.  It all ends up at the bottom of the falls, but sometimes it can do some interesting stuff (life and other useful mechanical work) on the way.

Posted in -- By the Physicist, Biology, Entropy/Information, Evolution | 90 Comments

Q: Does quantum mechanics really say that there’s some probability that objects will suddenly start moving or that things can suddenly “shift” to the other side of the universe?

Posted on March 17, 2013 by The Physicist

Physicist: In a word; nope.

The Heisenberg Uncertainty Principle is a statement about how “certain” some combinations of quantities can be.  The most commonly referenced is the “position and velocity” version of the Uncertainty Principle, that says that the more exact the position of a thing (any thing) the less certain its velocity, and vice versa.  It’s basically because of the Uncertainty Principle that you’ll hear about how quantum mechanics predicts that “particles have some small chance of jumping across the universe (position uncertainty)”, or “there’s some possibility that all the atoms in a book will suddenly start moving and it’ll jump off the shelf (velocity uncertainty)”.

And in fact, if you apply Schrödinger’s equation directly (which essentially describes how quantum wave functions change with time), it does seem as through there should be no problems with things suddenly jumping around.  If you apply it directly you find that if you have a particle confined to a particular region, then any amount of time later there’s some chance (no big) that it can be anywhere else, which is pretty exciting.  Unfortunately, Schrödinger’s equation is an approximation in very much the same way that Newton’s equations of motion are approximations of the (correct) relativistic equations of motion.

The wave function of a particle, which describes its position, tends to spread out like this

Soon after a particle’s position has been measured to be near zero, the wave function of a particle (which describes the probability of it being found at that position) tends to spread out like this, getting wider and wider as time goes on.  According to Schrödinger’s equation the tails on both sides approach, but never quite reach, zero.

Schrödinger’s equation was a massive break through and provided a lot of insight into a lot of problems.  But despite that, it doesn’t work perfectly.  In general, if you have a theory and it doesn’t line up perfectly with special relativity, then you only have part of a theory.  The fact that Schrödinger’s equation is “non-relativistic”, as evidenced by the fact that it predicts that sometimes particles will blink from place to place faster than light, made a lot of physicists extremely nervous.  It took a couple more years (1926-1928) until Dirac fixed the problem with the Dirac equation, which is more or less the same, but adheres to relativity.

The Schrödinger Equation (top) and the Dirac equation (bottom).

The Schrodinger Equation (top) and the Dirac equation (bottom).  The Dirac equation takes into account relativity.  Heck, it’s even got a “c” for light speed in there.

Newton’s equations of motion are very accurate, but only up until they disagree with relativity.  For example, they imply that there’s nothing special about light speed, and you can totally go faster.  Similarly, Schrödinger’s equation is remarkably accurate in most day-to-day, electron-shell type calculations, but makes big mistakes when relativity needs to be taken into account.

Long story short, even when considering the Uncertainty Principle, nothing can ever end up someplace else that would normally require faster than light travel.

As for books suddenly jumping off of shelves; the universe according to the laws of quantum mechanics is a seriously weird place.  But ultimately, laws are laws.  In this case, the conservation of momentum and energy.

If you take the predictions of quantum mechanics at face value (and why not?), everything that can happen does (in a very specific, many-worlds, sense).  But that “can” is pretty iron-clad.  Something that’s possible, even if it’s very unlikely, will happen in one some versions of the world*, but a book (or any other object) suddenly moving involves some extra energy suddenly being added to the universe, which is no good.

So, winning the lottery 75 times in a row, while making blind free throws for a couple weeks: sure.  Books jumping off of shelves: ridiculous.

* “World” is definitely not the right word for this, because it evokes images of other dimensions à la Sliders and leads to general confusion.  Neil Stephenson uses “narrative” which seems like as good a word as any, and hits a little closer to the mark.

Posted in -- By the Physicist, Physics, Quantum Theory, Relativity | 29 Comments

Q: Using modern technology, are we any closer to turning lead into gold than alchemists were hundreds of years ago?

Posted on March 11, 2013 by The Physicist

The original question was: With the current technology, it is possible now to transmute lead into gold, or whatever element into another? What transmutations should have tried the ancient Alchemist instead of the famous lead-gold one, in order to find an easy and useful success?

Physicist: Lead to gold: no.  But you can change some elements into others.  The yield is famously tiny, and the process is prohibitively expensive.  Before the late 19th century, no body had ever observed one element turning into another, and until the 20th century there was no equipment on Earth that had the faintest prayer of successfully changing one element into another (on purpose).

Back in the day, when chemists (alchemists) were getting good at purifying samples and making fancy chemicals, they got pretty cocky about turning stuff into other stuff.  But while you can use basic chemical reactions to turn hydrogen and oxygen into water, or flour and water into bread, there’s no combination of chemicals and reactions that even start to change one element into another.  Alchemists back in the day, being unaware of these sorts of things, got very excited about lead-to-gold stuff, philosopher’s stones, and life from nothing.  Many of them were legit scientists of the day, so we legit scientists of today have inherited a lot of their symbols and short-hand (though not their methods, by and large).

Newton loved himself some alchemy.

Newton loved himself some alchemy.  Sure he did calculus and science, but he also did pioneering work into finding the holy grail and even a variety of crazy pursuits.

Fancy chemicals and molecules are different because they use different combinations of elements, but elements and isotopes are different from each other because they have different numbers of protons and neutrons in their nuclei.

There are basically three ways to change the number of protons and neutrons in the nucleus of an atom.  Fusion, radioactive decay, and neutron bombardment.

Fusion is tricky.

Fusion is tricky.

There are some issues with practical fusion.  To date we’ve managed to fuse deuterium (hydrogen) into helium, which is the easiest fusion there is, even then just barely, and only to useful effect in the middle of very big bombs.  To use fusion to make gold (which is the way gold is created in nature) you need a super nova, which would probably be expensive.  Also impossible.

A lot of atoms have unstable nuclei that will occasionally “pop” and turn into another element or isotope.  So, technically, being patient is one way to turn a sample of some material into another.

Start with some Uranium 235, then wait a few billion years, and you've got lead.

Start with some Uranium 235, then wait for several hundred million years, and you’ve got mostly lead.

Unfortunately, the material in question has to be radioactive beforehand.  Radioactive atoms “decay” until the number of protons and neutrons are in balance (not equal, just balanced in a particular way), and for heavy elements that balance is almost always reached with lead or thallium.

The last way to change one isotope to another, and the only technique we can really use and control, is “neutron bombardment“.  Neutron bombardment isn’t the best option, so much as it’s the only option.  The idea is that since neutrons are electrically neutral (hence the name) they can enter and join the nucleus of an atom without being repelled by the positively charged nucleus (this repulsion is why this technique doesn’t work with protons, and why fusion in general is so difficult).

Bombardment is how plutonium is manufactured from uranium.  Bombarding a sample with neutrons sometimes makes the atoms in question decay into higher elements, and almost always makes them more radioactive (so this is a “bombard then wait” sort of thing).  In some cases it makes them so spectacularly radioactive that they immediately fly apart, and if they also produce a spray of neutrons, then you’ve got yourself the makings of a bomb or a power plant.

Here’s a map of all of the known isotopes and their preferred means of decay (many isotopes have several ways they can decay).  The full chart, in detail, can be found here.  It’s a very big picture.

All of the isotopes, with the number of neutrons increasing as you go up, and the number of protons increasing to the right.

All of the isotopes, with the number of neutrons increasing as you go to the right, and the number of protons (which is the “atomic number” or “element number”) increasing as you go up.  The black squares are the stable isotopes, and this region is called the “valley of stability”.

The different colors indicate different decay paths.  For example, pink is β+ decay, which turns a proton into a neutron, and an extra anti-electron, which in this case is the “radiation” we detect flying out.  So, on the chart the pink isotopes decay down and to the right, by losing one proton and gaining one neutron.

How to "play the game". At best, we can cause a tiny fraction of a sample of an isotope to move one to the right (gain one neutron).

How to “play the game”. At best, we can cause a tiny fraction of a sample of an isotope to move one to the right (gain one neutron).

By looking at the chart you can figure out what elements can reasonably be made from others using neutron bombardment.  For example, you might look at this little part of the chart (picture above) and think that you should be able to create gold by bombarding platinum 196 (which is directly below gold 197, the only stable gold isotope).  This would add a neutron, which changes some of the sample into platinum 197, which would then execute a β– decay, moving up and to the left, and turn into gold.  As it happens, this is exactly how you create gold from platinum.  That β– decay has a half-life of about 20 hours, so once you irradiate your platinum you only have to wait a few days before extracting the trace amounts of gold from your sample.
There’s also an isotope of mercury, mercury 196, that can be turned into gold (it’s above and two to the left from gold 197).Lead, on the other hand, is in a bad position to form gold.  Using neutron bombardment you can move to the right on the chart, but if you follow the decay path from every heavy isotope of lead, they all lead back to either lead or bismuth.

All we can do in add neutrons (white arrows), but that just takes puts us on "decay paths" that lead back to lead or bismuth.

All we can do is add neutrons (white arrows), but that just takes puts us on “decay paths” that lead back to lead or bismuth.

So, using the one and only technique available to us, we definitely cannot turn lead into gold.  Not even a little bit.  Platinum and one fairly rare isotope of mercury, sure.  But not lead.

Also, both the platinum and mercury processes are substantially more expensive and dangerous than digging gold out of the ground.  Among other things you need to get your hands on a neutron source, which is generally an extremely radioactive (illegal and expensive) metal, or a multi-billion dollar accelerator used to blow apart heavy isotopes into buckets of neutrons.  There are easier ways to lose money.

Posted in -- By the Physicist, Particle Physics, Physics | 21 Comments

Q: How do you turn/change directions in space?

Posted on March 5, 2013 by The Physicist

Physicist: There’s more to this than you might think.  If you’ve seen a movie involving spaceships of any kind, then you’ve probably seen the wrong answer.  We’re used to thinking about airplanes (flying through the air) and walking (on the ground), so the basic intuition we have about how to move around (turning and starting and stopping) doesn’t apply in space.

In space it could barely matter less what shape your ship is, because

In space it could barely matter less what shape your ship is, because it doesn’t interact or push off of air (left).  Some epic sci-fi franchises choose to ignore that fact (right).

Every possible motion always conserves momentum, which just means “if you want to move, you need to push on something else”.  Airplanes can bank in order to turn because they can push on air, and we can get up and walk across the room whenever because we can push on the ground.  But in space those luxuries are missing, since there’s nothing to push on.  Moving in space is the most frustrating damn thing ever.  Think about trying to maneuver on infinitely slippery ice.  Worse than that.

Space: Worse than this.

Space: Worse than this.

If you want to turn and face a new direction in space there aren’t a lot of options available to you.  One technique is to literally throw things and getting a push from the recoil.

Manuevering thrusters firing on the back of the shuttle (left) and different thrusters just looking pretty on the front of the shuttle (right).

Maneuvering thrusters firing on the back of the shuttle (left) to pitch the nose up and different thrusters just sitting around looking pretty on the front of the shuttle (right).

If you want to move to the right you need to chuck a bunch of stuff to the left, and if you want to turn one way you have to chuck a bunch of stuff in the other.  Flying around like an airplane in space doesn’t work at all (other than slowing down, this is a fair approximation).

However, if you just want to turn in space without moving (thrusters always push you around), you can “push on yourself”.  This is how the Hubble telescope points at stuff.  If it used tiny thrusters it would run out of fuel pretty quick (that thing is always looking at stuff) and it would pollute its tiny corner of space with exhaust.  Instead Hubble uses flywheels, which run on electricity, to turn.

Turn a wheel in space, and turn yourself.

Turn a wheel in space, and turn yourself.  Hubble has small flywheels (right) to turn, and tiny flywheels (left) to zero in on an exact direction.

Scattered throughout Hubble are an arrangement of little motors attached to basically nothing.  Just by turning on those motors, and by spinning them in one direction, the entire craft turns (slowly) in the other direction.

So, if you want to move in space, you’ve got to move something else, and if you want to turn, turn something else.

Posted in -- By the Physicist, Physics | 19 Comments

Q: If a man hangs on an un-insulated wire using both his hands what will happen and why?

Posted on February 28, 2013 by The Physicist

Physicist: Just a quick note before answering this; hanging off of power lines holds a special place simultaneously in both in the Very Long List of Stupid Things to Do and the Somewhat Shorter List of Last Things to Ever Do.  Hanging off of power lines is a very effective way to get yourself killed a lot.

Assuming the dude in question was in contact with exactly one wire, and absolutely nothing else, and wasn’t even close to touching anything else, and that the wire didn’t snap, then he’d be fine.  Getting into that situation and back out again however involves getting electrocuted pretty good.  Turns out that the voltage in power lines is high enough that it can jump a fair distance (given the chance).

Electricity is a lot like water, with electrical current being a lot like water current, and voltage being a lot like water pressure.  It turns out that for slightly obscure reasons it’s a lot more efficient to transmit electricity using low current and high voltage.  So you can think of power lines as being like big pipes that are holding slowly moving water that’s under a lot of pressure, and that are looking for any chance to “spring a leak”.

Being in a high-pressure environment isn’t so bad, but being between high and low pressures is so bad.  For example, you can dive to 250 feet (with air tanks and whatnot), and despite being exposed to lots of pressure you don’t get pushed around or hurt.  But if you try to hold back the water in a fire hose, which operates at about the same pressure, you’ll get pushed around plenty (also, point of fact, you won’t stop the water at all).

A difference in pressure will push you around, and a difference in voltage will cause electricity to flow trough you.

A difference in water pressure will push you around and, similarly, a difference in voltage will push electricity through you.

The same sort of thing is true with voltage.  If you stand on the ground you’re at the same voltage as everything around you, and there’s no need for electricity to flow through you.  And, somewhat surprisingly, if you’re dangling from a power line you’re at the same voltage as the power line and there’s no need for electricity to flow through you.  But, if you get close to anything else, the voltage difference may be big enough for the electricity to “make a break for it” and flow through you and out onto whatever you’re near.  Same thing would happen if the wire snapped; you’d become part of the wire.

Unlike a lot of the “what if” question we get, this is one that’s been tested very, very extensively, both in terms of safety and danger.  If it were dangerous to touch a power line (while not touching anything else), then there wouldn’t be many birds left (or those that were left would wise up real fast).

The birds on these power lines are exposed to the same voltage as the wires, more than 100,000 volts, but since they’re not in contact with anything at a lower voltage (like the ground) no electricity is flowing through them.

But since contacting anything else (especially another wire) is dangerous this is a serious issue for birds big enough to bridge the gap between lines.  As a result you’ll often find weirdly shaped power poles, or very widely spaced wires in areas with large birds (large flying birds that is).

In some areas specially designed power poles

In some areas specially designed power poles are in use to keep larger birds from contacting more than one wire at a time.

The people who work with power lines take a lot of precautions.  Everything is insulated, so that they don’t ever touch the wires directly, and even if they do, they’re not touching anything else that can conduct.  By the way, and I can’t emphasize this enough, as safe and fun as hanging on power lines might seem, it’s both deadly and boring.

The fire hose picture is from here and was taken during a celebration (not a protest or anything).  The bird pictures are from here and here.

Posted in -- By the Physicist, Physics | 28 Comments

Learning intro number theory

Posted on February 24, 2013 by The Physicist

Physicist: We occasionally get questions about free learning resources.  Khan academy is excellent, and if you poke around you can find a smattering of free class notes and text books, but generally speaking the more more detailed/advanced the material, the more difficult it is to find/understand.

In keeping with that tradition, here’s this ↓.  It was originally written for a group of “mathlete” high school students to teach them number theory and erode their egos a bit.  It’s Socratic, so rather than just presenting stuff to be known, it’s mostly a series of leading questions.

Although number theory is a huge, horrifying field, what’s presented here is fairly accessible.  If you’re comfortable with arithmetic and have patience and time and paper, you can work through this (but keep in mind, it’s not supposed to be easy).  Most of the problems benefit from making up an example or two.

If you notice any typos or omissions, or have questions or answers, put that mess in the comments, and if you happen to be a fair hand at LaTex and feel like constructing some solutions, I’ll post the hell out of that.

 

Part 1: In which Euclid’s algorithm is considered and linear Diophantine equations are seen to be over-named.

Part 2: In which modular arithmetic is shown to be easier than regular arithmetic and primes are counted and accounted for.

Part 3: Wherein modular arithmetic is shown to be harder than was previously implied, Wilson’s theorem is briefly noted, and divisor functions are dissected.

Part 4: Whereby Euler’s theorem is revealed to high praise, and Fermat’s little theorem is proposed as an after-thought.

Part 5: That the hidden structures of modular math might better be understood, Professor Emeritus Chinese’s invaluable Chinese Remainder Theorem is revealed.

Part 6: The end of this short and arduous journey of discovery finds us face to face with RSA encryption, which is now seen as it truly is; no biggie.

Update: The Gang of Five came up with these solutions for part 1.

Posted in -- By the Physicist, Brain Teaser, Math, Number Theory | 5 Comments