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<   No. 3449   2015-12-20   >

Comic #3449

1 Mordekai: We haven't even burnt down a tavern!
2 Lambert: We're not burning down a tavern in my home village!
3 Kyros: Someone want to burn down a tavern?
3 Dwalin: Soonds like foon!
3 Draak: Draak like!
4 Alvissa: <sigh>
4 Lambert: I begin to see why you sigh so much...

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A tavern is an establishment which serves alcoholic drinks and food. The distinction from an inn seems to be that traditionally an inn also provides lodging for travellers. (The Wikipedia article also has a [citation needed] statement that originally a tavern served wine while an inn served beer and ale. So that could change at some point.)

But either way, over the years the words "tavern" and "inn" have come to be more or less interchangeable, together with the word "pub", depending on your dialect of English. I myself don't associate any particularly strong distinctions between the meanings of the three words.

And going off on a complete tangent...

We like to think of the laws of physics and motion as deterministic, in the sense of the following example:
If you know the speed and trajectory of a billiard ball when it strikes another, stationary, billiard ball, then after the collision you can calculate the speed and trajectories of both balls.
You might need to know some other stuff like coefficients of elasticity or whatever, but the idea seems plain. And if you rerun the experiment with the same setup each time, the outcome will be identical. Absolutely the same. If you saw a collision of billiard balls on a table, and then exactly the same collision again, you'd expect the balls to go in the same directions after the collision as the first time - it would be surprising if they didn't.

Billiards room at Martindale Hall, Mintaro
A billiard table.

So where does randomness come into our world? If I roll a die, I can't tell in advance what number will come up. And if it lands on 6, and I roll it again in the same way, it might well land on some other number. Why?

The basic reason is that many processes, like rolling a die, are chaotic. Technically, this means that a small change in the initial conditions leads to a large change in the outcomes.[1] If we did know, exactly, the initial position, trajectory, and rotation of the die as it leaves your hand, and the relevant coefficients of elasticity and friction and so on, plus the details down to a microscopic level of the shape of the die and the surface it lands on, and also the speed and direction of any wind currents in the air caused by your hand moving or the breeze blowing in the window, then in theory we could calculate how it would bounce and spin and what number it would end up showing. But of course we don't have this level of knowledge.

Furthermore, not only do we not know these factors, but each time we roll the die again, many of these variables are different, even if we attempt to roll the die in as close as possible to exactly the same way. And even a really tiny change in the initial position or speed of the die can end up magnifying when it comes into contact with the table and bounces around. If it lands on an edge with a certain rotation speed it will tumble this way, but if it lands on the same edge with a slightly different rotation speed it will tumble that way instead. Multiply all this by a few bounces and tumbles, and by the end of it the orientation of the die is well and truly unpredictable, or randomised.

As a way of dealing with such complex physical systems (and also some more abstract things such as economic systems), we have developed a mathematical framework for describing and analysing apparently random behaviour.

The first step is to define random variables. These are quantities which can take on any of a certain range of values, with the specific value being determined at random. For example, the result of rolling a standard 6-sided die is (for our intents and purposes here) a random variable which can take on the values 1, 2, 3, 4, 5, or 6. Random variables don't need to be discrete like die rolls. A board game spinner which you flick with your finger generates a random variable, being the angle at which the spinner ends up pointing (relative to some reference mark), which varies continuously from 0° to 360°. Random variables don't need to be evenly distributed either. If you roll three dice and take their sum, the result is a random variable that varies from 3 to 18, with the totals 10 and 11 being most likely and more extreme numbers progressively less likely.

283/365: Natural 20
A 20-sided die.

Next, we define a stochastic process to be a process which is governed by one or more random variables. For example, a collection of die rolls made one after the other. But things get more interesting when we consider the behaviour of some other object which is governed by the random variables. For example, imagine tossing a coin and then taking a step forwards if it's heads and a step backwards if it's tails, then repeating this procedure many times. The process of you taking steps is a stochastic process, governed by the random variable of the coin flip. You can imagine that after a long time and many steps, you could be a long way from where you started, or you could be quite close still. (This particular stochastic process is of a type known as a random walk.)

This is an artificial example, but some very similar things happen in nature. Consider particles suspended in a fluid (either a liquid or a gas). Such particles are continually being buffeted on all sides by the motions of the fluid molecules around them. For large objects, this buffeting consists of numbers of the order of 1023 or more molecules hitting on all sides during every small fraction of a second. Because there are so many molecule collisions, they are relatively evenly spread out: 1023 collisions on the left side, 1.000001×1023 on the right. Any imbalance is quite a small fraction of the total number of collisions. An imbalance in the total collision momentum from different directions results in a net momentum being imparted to the particle. But the momentum of 0.000001×1023 atoms is small compared to the mass of a large particle, so any transfer of velocity is very small. The result is that the effect on the large particle at a macroscopic level is nice and even, and this produces the effect of pressure - air pressure or water pressure, or what-have-you.

However, when a particle is small enough, the unevenness in the random fluctuations of the fluid molecules hitting it from all sides starts to become larger compared to the mass of the particle. The result is that a really tiny particle needs to move with appreciable velocity to carry the transferred momentum. So smaller particles tend to get buffeted around by fluid molecules more than larger ones.

You can see this if you look at tiny particles suspended in water under a microscope. This was first noticed by the Scottish botanist Robert Brown[2] in 1827, while studying pollen grains. In the small cavities of the pollen grains were tiny particles ejected by the pollen, and Brown observed them moving erratically in the water. This form of erratic motion of tiny particles is named Brownian motion. As a diligent scientist, Brown performed an experiment by observing other small particles too, including inorganic matter such as rock dust, to rule out the possibility that the motion was caused by life processes of the pollen particles. And they showed the same erratic movement.

Pollen Central
Pollen grains in a flower.

Brown gave no explanation of what might be causing this motion. Decades later, in 1880, the Danish statistician Thorvald N. Thiele proposed a mathematical model which attempted to characterise the apparently random motions, but he also offered no physical explanation for why they occurred. It wasn't until 1905 that someone proposed a reason for why these motions occurred. That someone was Albert Einstein, and he pointed out that it could be an observable effect of the motions of molecules.

To show this, Einstein had to invent a way of mathematically modelling the effects of collisions and momentum transfers from the order of 1020 particles, producing roughly this many collisions per second. This is way beyond tractability with any simplistic approach to Newton's laws of motion, and so Einstein had to calculate the accumulated effect of all these interactions with a statistical model. Such a model assumes that the physical processes involved are stochastic processes - i.e. random - even though they are in fact governed by the laws of physics. But when you're dealing with the motions of so many particles, it's a reasonable assumption that the behaviour of any given particle is essentially random, because the accumulated effect of so many physically deterministic particles approaches very closely to the accumulated effect of the same number of randomly behaving particles.

This insight - that when you have enough particles you can treat them as all being random, even though they're not, but the accumulated behaviour comes out virtually the same - leads to a vast simplification in the mathematics needed to calculate what that overall behaviour will look like. In the case of Brownian motion, you assign probabilities to all the various things a molecule can do (speeds and directions), and then you sum or integrate the probabilities over the large number of particles, to calculate average and expected large scale behaviour. By tossing other physical terms into the calculation, you can figure out the cumulative effect of the collisions on the larger (visible under a microscope) particle. If you do this, you can calculate what are called the moments of the particle's position over time.

The first moment describes the expected average direction of movement of the particle at any given time. This comes out to be zero, because of the symmetry of the situation: at any given time you're equally as likely to have more molecules hitting the left side of the particle as you are to have more hitting the right side. In macroscopic terms, this means the particle is equally likely to move either left or right (or up or down, or whatever) at any given time. The second moment, however, comes out to be non-zero. This moment describes the average distance of the particle from its starting position over time. This second moment turns out to be a function of the time elapsed, and equal to a constant multiplied by the square root of the time. Macroscopically, this means that if you pick a start time and watch the particle move in its erratic path, and repeat this experiment many times, then after time t the average distance of all the particles from where they started (ignoring the direction) will be proportional to the square root of t. In other words, they tend to drift away from their starting locations, but not at a linear speed (because of the erratic jerking motion they show).

Interestingly, in the above example of the coin toss where you walk forwards or backwards depending on whether the coin shows heads or tails, you can show by doing some relatively basic probability calculations that the expected distance from your starting point increases as the square root of the number of coin flips. This is no coincidence, as the net effect of Brownian motion also corresponds to a random walk process. More precisely, since time is not discrete like coin tosses, Brownian motion is really the limit of a discrete random walk as the discrete element size reduces to zero. Such processes are very common in the physical world, and are known by the general name of Wiener processes, after the American mathematician Norbert Wiener.[3]

Berrem Gorge walk
A non-random walk.

Beyond just doing the maths, Einstein also put in the physical parameters of expected molecular mass (from atomic theory). The observations of Brownian motion agreed with this physical model, and so showed that the atomic model of matter was correct. That is, Brownian motion is a visible and quantifiably understandable effect of the fact that matter is made of discrete particles at scales too small to see.

The statistical modelling which allowed this conclusion is an example of a branch of mathematics known as stochastic calculus. In short, this is an area of mathematics which deals with stochastic processes which are (effectively) random on small scales and integrating them to determine larger scale effects.

Stochastic calculus is used wherever there are small events which are either random or can be treated as random, which accumulate to produce larger scale effects. As described above, this occurs in many physical interactions, such as the diffusion or mixing of gases or liquids. It also occurs in fields as diverse as:

This diversity of applications shows off the fact that what might seem to be abstruse and abstract mathematics often has uses far broader than in the initial field where it was developed. (It's like most pure science research, in that way.) It's a reminder that underlying the macroscopic reality we interact with on a daily basis, there are physical laws acting on myriad tiny particles. And in the specific case of stochastic calculus, it's a fascinating insight into the complex interactions between determinism and randomness on different levels. Things which seem random are often based on underlying deterministic actions, while conversely things which seem predictable (like the diffusion of gases) are often based on underlying randomness.

Paraphrasing the words of a popular expression: It's an alternating layer cake of determinism and randomness all the way down.

Note: This completely non-sequitur annotation was inspired by reader Iain A., who requested an essay on this topic as part of his Patreon supporter reward. If you'd like me to write an extended annotation on any topic you care to name, or if you just want to show some support for the comics and other creative work I share, please consider becoming a patron.
[1] There are a couple of other necessary conditions for a system to be considered chaotic, but these are often implicitly assumed or occur naturally in most cases where people are considering whether behaviour is chaotic or not. These conditions are topological mixing and denseness of periodic orbits. If you're interested, you can read about them on Wikipedia's chaos theory page.

In these terms, the outcome of a die roll as described by the number rolled actually violates the condition of denseness of orbits, as the outcome is restricted to one of a finite set of discrete numbers. So generating numbers from 1 to 6 by rolling a die is, technically speaking, not a chaotic system, but merely a random one. But on the other hand, if you describe a die roll by the path through space and orientation of the die at all times, then it is chaotic, as this path can vary by tiny amounts, and slight changes in initial conditions can densely fill regions of the space of potential paths.

[2] By the way, check out Robert Brown's disambiguation parentheses on Wikipedia. Robert Brown (Scottish botanist from Montrose), as opposed to Robert Brown (Scottish botanist from Caithness). Imagine if they'd both come from the same town, and been born and died in the same years.

[3] A friend of mine says "Norbert Wiener" is an epic name, and another friend says, "Yeah. He probably chose to do mathematics because he knew he'd get picked on at school regardless." Norbert Wiener is also responsible for one of the potentially most surprising Wikipedia pages of all time for most of the people who accidentally stumble across it while looking for something else: Wiener sausage.

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