A short remark on the significance of butterflies and their respective effects.

Abstarct:

"You keep using that word. I do not think it means what you think it means. "

Inigo Montoya

"The butterfly effect" is called that, not because of some cockamamie story about a butterfly farting somewhere to create the next hurricane Catrina..

I was looking at a friend's FB wall today when I realized they made a reference to the so called "butterfly effect", I want to set some things straight regarding that effect.

Usually when people use the term "butterfly effect" they use it without actual knowledge about what it really stands for, and where and when it was coined.

So, a little history lesson is probably called for here:

Of strange attractors, and non linear effects:

Some of you may have at some point or the other, had a chance to have a fatal brush with physics studies, be it in high-school, university or college or you simply had a pesky friend who was into that kind of kinky stuff and insisted on explaining, oh I don't know, general relativity to you while insisting you most certainly possess the means to understand and assimilate everything he "taught" you right there and then. well that's me.

Warning: physics rant here:
<Physics rant>
Well it just so happens, that MOST physics areas we encounter, even as PHD students or actual researchers, are areas of LINEAR PHYSICS. for instance quantum physics is strongly embedded within the framework of linear algebra, with operators that are linear by definition, another field which is strongly linear is electrodynamics, which for the most part (but not always) relies heavily on linear algebra tools ,methods and operators to provide refutable results.
</Physics rant>.

The world isn't linear, or in the words of my namesake, "it ain't necessarily so".
Looking for the linear approximation of some dynamic, is not entirely different than looking for the proverbial coin under the street-light.

It so happened that in a cold day in the late 1950s an egghead, or otherwise called a scientist, sat in his lab, punched numbers in his then top-of-the-line computer, and got meters upon meters of output sheets, containing numbers that would represent a meteorological system.

By a fluke, the researcher went out for coffee, while the computer heaved and clicked and buzzed and chirped and spewed out sheets of numbers.

When he came back, the computer was silent. no more chirping, just plain cold data. but the experiment was not over!!! since in those days the internal memory of a computer was very limited it became necessary to punch in the last output you got,and so proceed with the number spewing.

But alas, the variables were saved in the computer to a precision of six decimal places, yet the output was given in only 3! So by default the researcher rounded off the last output and introduced it into the computer.

What he got was very strange, he projected a certain result but got a result differing greatly from expectation.
So like every respectable researcher he had tried to reproduce the experiment... getting results that differ greatly from both expected values, AND previous results!!!

See, if it were me in my first year at the academy I would have just drawn the damn graph I knew was expected, shrugged my shoulders, and go to sleep.

But not our epic protagonist! He set out to delve deeper into what had happened and stumbled upon a great discovery!!!

The protagonist was called Edward (Norton) Lorentz and he found out, that in some systems, an infinitesimal change in starting conditions, translate very quickly into a gross divergence in results.

Not the same Edward Norton

To illustrate  here is a picture of a Strange attractor:

Strange attractor

Weird Al in: Strange repellent

Just to explain what's going on here, I'll disregard the right hand picture as the freak of nature that it is, and concentrate on the left hand picture.

This is a graph generated by this applet *, the applet simulates a Lorentz' system which is a meteorological model.
If you want to actually see it in action, just click anywhere on the white field, twice at the same place.

You will have generated 2 plots, or more if you have Parkinson's disease, that started at the exact same spot - or did they?
Actually no they didn't. see the computer saves the mouse pointer in a "float variable", thus it saves it to a degree of precision that we can't see, and I'm pretty sure the applet rounds off a couple of places after the decimal - thus recreating Lorentz' experiment actually...

So if you wait a while, you will see a divergence in paths taken by the two differently colored plots.

This graph is called "Lorentz' Butterfly" and THIS is why they call the effect "the butterfly effect", not because of some cockamamie story about a butterfly farting somewhere to create the next hurricane Catrina...


So this concludes the short history lesson, and here I go into proving the world is evil.

Well, no it's not, but I do have to make an attempt to show it is...

OK, so a lot of people use the so called "butterfly effect" to justify a line of reasoning that says something like this:

If you do something good now, for a certain person, it will generate more good deeds like the ripples on a pond, and like the butterfly effect, this will cause a major force for good in the world... or something like that.

Now let me be absolutely clear about this:
I DO subscribe to the notion of doing at least one random act of loving kindness a day, I think it's good for you, and if only I would remember this every day I would probably be a better human being right now.

BUT! (and it's a big butt) this line of reasoning sucks.
it's flawed in so many ways I can't even start counting all of them so I will just give up and point the one I was aiming at.

First off the butterfly effect is all about a small change in starting condition having a huge effect later on, what is described in the above dynamic is called a "chain-reaction".

A chain reaction is in essence a LINEAR dynamic, it has absolutely NOTHING to do with the butterfly effect (as far as I know).

Secondly and more importantly is this:

Invoking the butterfly effect assumes a non-linear system.
Assuming that if we do something good things get better in the world assumes strong linearity of the system.

These two assumptions are mutually exclusive and while the former is actually well documented, experimented and reproduced, the latter is a figment of our wishful thinking (that may be true in a convoluted way as far as I know, but is neither proven nor reproduced).

In general MOST dynamics in the world are non-linear, and we physicists make assumptions, and approximations to linear or at most quadratic cases in order to get prevailing approximate results we can later compare with experimental results.

For the most part I think most of us would agree that human behavior is non-linear as well thus, I tend to lean towards the first assumption rather than the second.

And if we take in account Murphy's law we instantly come to the grim realization that it is much more probable that if you do a good deed, something will go terribly wrong.
or differently put:
"No good deed goes unpunished" in-deed (see what I did there?).

But also, at the same time no bad deed goes unpunished, as almost EVERY change in starting conditions results in a huge divergence in results thus I propose that:
"No deed goes unpunished" -  and so we are all constantly punished... whether we do something or not - thus the world is an evil place.
QED.

But to end this on a lighter note, two things.
the first: if we all do random acts of loving kindness every day, the whole graph jumps up on the scale of good vs. evil, and so the whole dynamic is shifted up by a constant, it might be just the constant that may save us as a species.
so don't stop doing good things!!!

the second is this...
If I learned anything in my 30 odd years of life experience it is that when it comes to physics I am usually wrong the first time around and I always need further study.

oh and something else - you can argue with your wife, but ultimately, she's right and you know it.

And I mean this in the best possible way.... *GULP*

again, this was longer than expected... bummer.

I also almost died -again- yesterday, maybe I will tell you all about it in a future post...


* with permission. Copyright 1996, James P. Crutchfield. All rights reserved.

Physics of marriage and hypoxia

Abstract:

Got married, took a hike,

got high, got dehydrated and almost suffocated.

oh and a gerbil.

I lied.
I said the next post would deal with how being fat can save your life, and there are actually several ways that can happen. but that will have to wait for another instance.

A couple of years ago I got married.
Yes. I, like loads of better men than I, donned the world's smallest handcuffs, and walked the "green mile" as it were of single men, to happy cheers of "dead man marrying".

his: +5 ring vs. reason&accountability.
hers: one ring to rule them all and in the darkness bind them.

Well, actually I count myself lucky. most men get fucked on the deal, but as for myself, I got myself a ninja for a wife and a Jedi knight to boot. You should see her performing Jedi mind tricks on me or using The Voice on people - damn she's good!

Anyway after about a year of marriage we decided it's time to go for our much overdue honeymoon.
so somehow we came up with the obscene amount of money it now costs to fly to the USA, and we went to visit grandma, the family and take a hike. no really. we traveled California's national parks and hiked some...
at some point we even had oxygen shortage due to height sickness.

What had happened was, we trekked from Camp White Wolf to Ten lakes in Yosemite national park, and back, in one day.
Now Camp White Wolf is at \(\approx 2,400 m\), and the mountain overlooking Ten lakes is at \(\approx 3000 m \). so all in all it doesn't seem like a distinctive height difference right?
BUT (and it's a big butt for sure!) being overzealous and not so experienced hikers we spent the night near Briceburg which is at \(\approx 350 m\).

Now for a bit of physics:
Humans usually breath air which is comprised among other things of roughly 21% \(O_2\).
Assuming air is an ideal gas we get :
\[PV=NRT\]
Which means in a cubicle meter at sea level pressure, at \(298^\circ K\), we get \(\approx 40.9 {} moles \) of air which translates roughly to \(8.6\text{ moles of }O_2\).

Now we will consider a thin spherical layer \(\Delta r\) of gas:
\[ A(r)\cdot P(r)-P(r+\Delta r) = V\cdot\rho (r) \frac{GM_{\oplus}}{r^2} \]
\[\Rightarrow -4\pi r^2 \Delta r \frac{\partial P}{\partial r}=4\pi r^2 \Delta r \rho (r) \frac{GM_{\oplus}}{r^2} \Rightarrow -\frac{\partial P}{\partial r}= \rho (r) \frac{GM_{\oplus}}{r^2}\]

For all intents and purposes the gravitational acceleration on the earth's surface and at 3 km differs by a factor of \(1 \pm 10^{-4}\) so really there's no need to take into account the gravitational change in that small a difference in distance.
Thus we will treat the gravitational potential as \(\Phi_N=gr\), and thus the force per kg is directed downwards and is given by \(F_g=g\hat{r}\).
\[\rho = \frac{N\cdot m}{V} \Rightarrow \rho=\frac{Pm}{RT}\]
and so we quickly get:
\[\frac{dP}{P}= -\frac{gm}{RT}dr\]
Which assuming a constant temperature (which is a WRONG assumption as we will see immediately) we get:
\[ P=P_0\,e^{-\frac{gmr}{RT}} \]
Where \(m\) denotes the molar mass of air, \(R\) the ideal gas constant, and \(T\) the temperature.

Now using this formula we get that the air pressure at 3000 m above sea level is \(\approx 52595\) pa. which means the amount of moles in a cubic meter in 3000 meters of height is 21.2 moles. keeping in mind that \(O_2\) is only slightly heavier the \(N_2\) we allow ourselves the assumption that the \(N_2\sim O_2\) ratio in 3000 meters is similar to that of about ground level we get that the air up there has only ~11% oxygen in the same volume.
Just for the sake of good form - at 15-19% you get decreased ability to work strenuously.  impaired coordination may occur and there is a chance of induced early symptoms with individuals that have coronary, pulmonary, or circulatory problems. at 12-15% respiration and pulse increases; impaired coordination, perception, and judgment occurs. , at 10-12% respiration further increases in rate and depth; poor judgment and bluish lips occur. At 8-10% symptoms include mental failure, fainting, unconsciousness, an ash-colored-face, blue lips, nausea, and vomiting.

you get the picture right?

So 11% is quite close to being knee-deep in shit.

Let's have a more realistic estimate though, because we know the temperature decreases with height (at least in the atmosphere).

So basically the temperature is a physical function and we like to think of those as analytical so the first approximation of \(T\) around \(T_0\) is given generally by:
\[T(r)\approx T_0 - ar\]
The minus sign owing to our understanding of temperature decreasing with height.
and so the above differential equation becomes:
\[\frac{dP}{P}= -\frac{a\,gm\,dr}{a\,RT_0\left(1-\frac{ar}{T_0}\right)} \Rightarrow P=P_0\left(1-\frac{ar}{T_0}\right)^{\frac{gm}{aR}}\]
And so using the corrected formula with \(a\) denoting the linear-approximate rate of temperature decline in the atmosphere which is given by \(a=6.5 \cdot 10^{-3}\) we get the pressure up there was about 70970 pa which is much likelier, whereas the effective air percentage there is almost 15%.

Meaning that we we're only in an ankle-high depository.

Just as an additional reference, in normal circumstances the oxygen percentage in our exhaled breath is about 16%, and in some circumstances could go as low as 14%.
So imagine yourselves in a hot, non-ventilated room, stuffed with cheap foreign workforce, and imagine the wicked headache you'll get. Now multiply it, as we were climbing up at a quick pace thus taking even shallower and quicker breaths.

At some point we actually sat there at the top of the mountain, breaking for lunch, not realizing we are only making matters worse...

The payoff was worth it though - it's such a beautiful view up there - it's AMAZING!

Ten Lakes view, headache and dehydration included.

Anyway, on our way back we also came to the undeniable conclusion we were idiots, as we ran out of water, we had a pounding headache, and we had to virtually run all the way back to Camp White Wolf before dark, since we were really afraid to walk in the woods at night, with no light, no fire, no reception, and loads of wolves howling in the distance.

We got there in the nick of time,  the sun was already setting as we hit the marked trail back to camp, about 20 minutes later, in full dark, we got there.
Sore-foot, dead-tired and otherwise completely wretched, we decided that we are NOT making camp to sleep on cold bare ground. I drove us to a nearby motel outside the national park, and we took a shower, and slept the deepest sleep achievable by mankind...


oh, and a gerbil:

Gerbil

Next time: Lenses and idiots.