A simple process that produces power law distributions is exponential growth over a period of time t where t is sampled from an exponential distribution. A contrived example might be growing bacteria in a dish until an atomic nucleus decays. A more realistic example would be the total profits a company makes over its lifetime, where a very simple model would be to say that the company grows exponentially until it is acquired or is destroyed by some disaster. (Assuming the the chance of getting acquired/destroyed in a given month stays constant.)
Some power law distributions are weirder than others. If the doubling period of the growth is at least as large as the half life of the process, then the expected value is infinite, even though the distribution itself is still perfectly well defined. Other, slightly less extreme, power law distributions have a finite mean, but infinite variance.
Our human utility functions are bounded whether we like to admit it or not. If they were unbounded, we'd all be broke from paying Pascal muggers. Even eternal suffering has a large but finite negative utility. Consider all the Christians who have believed in eternal torment in hell with probability much higher than 10−1200 but sinned anyway. You're already much better off than they are (for one thing, nothing you do will make much of a difference to the probability of a mega-torture AI, so even if the idea keeps you up at night, it shouldn't interfere with your everyday life).
Bounded utilities come from the fact that at a large enough scale, everyone is an average utilitarian, not a total utilitarian, and this average is taken across time, as well as across individuals. Eternal suffering just means that average utility goes to nearly the lowest possible value. Which is very extremely bad, don't get me wrong. But it's not infinitely bad.
There's a fun experiment you can try at home with infinite expected suffering called "the St. Peppersberg game" , and it goes like this. You'll need a coin and a selection of spicy peppers. You start out with 1 point. Start flipping the coin. If you get heads, double your points. The first time you flip tails, the game ends, and you must eat a pepper of spicyness n on the Scoville scale , where n is your final score in the game. If you don't have a pepper with such a rating, then you must eat several peppers whose collective Scoville rating sums to n. Buy more peppers if necessary. If the coin is fair, then a simple calculation shows that the expected value of n is infinity.
Large genomes have (at least) 2 kinds of costs. The first is the energy and other resources required to copy the genome whenever your cells divide. The existence of junk DNA suggests that this cost is not a limiting factor. The other cost is that a larger genome will have more mutations per generation. So maintaining that genome across time uses up more selection pressure. Junk DNA requires no maintenance, so it provides no evidence either way. Selection pressure cost could still be the reason why we don't see more knowledge about the world being translated genetically.
A gene-level way of saying the same thing is that even a gene that provides an advantage may not survive if it takes up a lot of genome space, because it will be destroyed by the large number of mutations.
An interesting difference between the drowning child situation and the "could donate to effective charity to save children's lives" situation is that the person who happens to be walking by that pond has a non-transferable opportunity to save a child's life for $500 (or whatever the cost of the clothes are, plus some time cost, and the inconvenience of getting wet and muddy). In the case of effective charity, even if one declines to donate, other people will still have the same opportunity. In the case of the drowning child, the fact that you are the only one who can act makes jumping in to save the child somehow seem more urgent. If you don't save the child, then you'd be somehow "wasting" a valuable opportunity. For a mostly selfish person who values all lives other than their own at less than $500, the opportunity would be valuable to others but useless for themselves.
If the going rate is $1000 to save a child through effective charity, then a mostly altruistic person would be willing to pay a mostly selfish person $600 to compensate for the costs of their clothes. There would have to be some "honour" involved, since the selfish person couldn't exactly unsave the child after the fact. If they could make the deal work anyway, then the mostly selfish person would have succeeded in selling non-transferable opportunity for $100, and it would be worthwhile for them to save the drowning child.
There would be quite a long period of time between the initial formation of these micro-pore membranes and the point in time where they could form free floating cells. In that time, they would need to evolve all the machinery to exist unsupported: better membranes, RNA, maybe even DNA, and some method of generating their own proton gradient. And to evolve at all, they would need to reproduce.
The problem is that these creatures are stuck in their respective pores. In order to evolve beyond a stage where they could barely even be considered alive at all, these things would need a way of colonizing other pores. I'm not sure if Lane discusses the question at all, but it seems like a difficult problem. The process would initially have to be so simple it could happen naturally without any molecular machinery.
Also, primordial soups don't necessarily have to be in chemical equilibrium. If the soup is sitting in sunlight, then it could very easily be out of chemical equilibrium, and probably would be. (Soups could be out of equilibrium for a variety of other reasons too, like a consistent influx of chemicals coming out of the ground.) Sunlight comes from a surface with a temperature of 5700K. Compared to a puddle with a temperature of 300K sitting on the Earth's surface, those photons are absurdly energetic, and could easily bump up the prevalence of some molecules that would otherwise have too much free-energy to exist.
Once you have excess free energy, copying information becomes possible, and it becomes conceivable that you could have self-copying information, aka life.
If you believe that our existence is a result of a mere fluctuation of low entropy, then you should not believe that the stars are real. Why? Because the frequency with which a particular fluctuation appears decreases exponentially with the size of the fluctuation. (Basically because entropy is S=log W, where W is the number of micro states.) A fluctuation that creates both the Earth and the stars is far less likely than a fluctuation that just creates the Earth, and some photons heading towards the Earth that just happen to look like they came from stars. Even though it is unbelievably unlikely that the photons would happen to arrange themselves into that exact configuration, it's even more unbelievably unlikely that of all the fluctuations, we'd happen to live in one so large that it included stars. Of course, this view also implies a prediction about starlight: Since the starlight is just thermal noise, the most likely scenario is that it will stop looking like it came from stars, and start looking just like (very cold) blackbody radiation. So the most likely prediction, under the entropy fluctuation view, is that this very instant the sun and all the stars will go out.
That's a fairly absurd prediction, of course. The general consensus among people who've thought about this stuff is that our low entropy world cannot have been the sole result of a fluctuation, precisely because it would lead to absurd predictions. Perhaps a fluctuation triggered some kind of runaway process that produced a lot of low-entropy bits, or perhaps it was something else entirely, but at some point in the process, there needs to be a way of producing low entropy bits that doesn't become half as likely to happen for each additional bit produced.
Interestingly, as long as the new low entropy bits don't overwrite existing high entropy bits, it's not against the laws of thermodynamics for new low entropy bits to be produced. It's kind of a reversible computing version of malloc: you're allowed to call malloc and free as much as you like in reversible computing, as long as the bits in question are in a known state. So for example, you can call a version of malloc that always produces bits initialized to 0. And you can likewise free a chunk of bits, so long as you set them all to 0 first. If the laws of physics allow for an analogous physical process, then that could explain why we seem to live in a low entropy region of spacetime. Something must have "called malloc" early on, and produced the treasure trove of low entropy bits that we see around us. (It seems like in our universe, information content is limited by surface area. So a process that increased the information storage capacity of our universe would have to increase its surface area somehow. This points suggestively at the expansion of space and the cosmological constant, but we mostly still have no idea what's going on. And the surface area relation may break down once things get as large as the whole universe anyway.)
The static friction is high enough that the beads will only move if you push them, but low enough that they are easy to push. The necklace is made of 2 strands, joined at the endpoints. Put one strand on the table in the shape of y=sinx, the other goes on top of it in the shape of y=−sinx. The beads go around the crossing points of the 2 strands, as rings in the x,z plane. (You don't have to make an exact sine wave of course, anything that yields the same topological result will work, and to loop the beads around the crossings the way I've described, you''l have to thread the strands through the beads before joining them at the end.) Having the crossings pass through the beads increases the friction, since the bead is redirecting some of the tension in the strands.
In theory, putting beads in the normal way on a single strand could also work, if the diameter of the strand and the hole size of the bead were well matched. (You'd want fray-proof string for threading that not to be a huge pain, though.)
Have you read Ursula K. LeGuin's book "Changing Planes" by any chance? If I recall correctly, there's a chapter where the viewpoint character is visiting a library, and reads a legend about two rival cities with a river border, and the outcome is rather similar.
σ2 isn't actually the variance here. The variance is σ2/2. Sorry for the confusing choice of variable name.