Prof. Avi Schiller's departure

A post will be dedicated to his memory,

this will take some time but I hope in a couple of days
something nice will be written.

This is by no means a eulogy, but I feel it is fitting to dedicate something to this great man.

It is customary to dub people "great" for their accomplishments and otherwise professional stature.

Avi Schiller was a great man, simply because he was a "mench". One of the few. 

He would also say: A boor cannot be sin-fearing, an ignoramus cannot be pious, a bashful one cannot learn, a short-tempered person cannot teach, nor does anyone who does much business grow wise. In a place where there are no men, strive to be a man.

                                                                       

                                                                                        -Old Hillel, Ethics of the Fathers, 2,5.

Vikings, Relativity and Couch potatoes

Abstract: It's not my fault I'm lazy,
Physics made me this way

dedicated to my sister-in-law and  her fresh hubby's mawwiege

Mazal Tov!!!

Recently I attended my sister-in-law's wedding...

After a full month of being forgotten, taken as granted and ultimately being treated as some ungainly babysitter to my own girl, it has arrived...

The event dreaded and anticipated, probably by the bride and groom, but also by yours truly -

THE WEDDING

Awesome and gruesome, with a multitude of family-related potential calamities, just waiting to happen, hanging on the off chance of someone sitting in the wrong seat or uttering the wrong sentence, incurring a death sentence, with the obvious exception of the crazy and otherwise fully-certifiable-bunkers pseudo-relative  (She can say whatever the hell she wants, nobody's home thus no-one really listens or cares what she says).

The food was good, the company better, and the music divine. well, not really... the first two statements are correct whereas the third - not so much....

The DJ was poor, probably not in the fiscal sense after the presumed hefty sum he was paid, but poor taste in music, poor know-how of his trade, poor execution, and ultimately, the only reason his music was anywhere near palatable, was this was their wedding and we were so damn happy for them, that everything else vanished, not  unlike the 10th order Taylor term for a quadratic potential function. 

At some point the music was pounding so hard, I literally saw hordes of vikings working at a huge smithy, pounding away with all their might to mold some unfathomably hard metal into some equally mysterious shape.

Now, if you follow my posts you know full well what happens to me when faced with potential demise - physics comes into play!

I immediately went on to contemplate vikings and physics, and stumbled upon Dichroism, which is basically when light polarized in different directions go through a certain special material, only one polarization survives (the otherwise polarized rays are reduced to a point of vanishing altogether).

Now since sunlight is non-polarized, this normally doesn't mean squat, but when there's heavy cloud cover, the light rays disperse over the clouds so you get a general bright haze, but no obvious direction for the sun.

Enter dichroism: by using dichroic glass the light that passes through the stone is brightest in the direction of the sun, because right in front of the sun (through the clouds) the light is differently polarized.

The trick is you have to find the right direction to hold the dichroic stone in, else you'll basically get nothing.

Anyway, this dichroism trick is how vikings were able to sail across wide stretches of open sea, since they didn't have land in sight for coastal navigation, and either they didn't know their celestial-nav basics, or the skies were constantly cloud-covered.

BOOM!!

One of the loud pounding beats threw me back into consciousnesses, in time to  realize how horribly out of shape I was...
You see, I was actually dancing (well, more like contorting) on the dance floor, sweating and panting, and  the all too familiar "why am I so lazy?" question kept burning in my mind...

Which again set a chain of events in my mind culminating with physics:

It's not my fault!!! Even physics is lazy!!!

And I will prove this immediately:

First off, remember old Newt's law: 

"Every body persists in its state of being at rest or of moving uniformly straightforward, except insofar as it is compelled to change its state by force impressed"

Translated: every body is basically a fat blob. you really have to exert force to change its current state.

I will now delve into a bit of Lagrangian physics to prove a point:

<Warning: Physics ahead>
This is a simplification of deriving Lagrangian Mechanics but here goes:
Consider Newton's 2nd Law:
\[F=ma\]
And if you're a physicist you know this is a simplification of:
\[\frac{dP}{dt}=F\]
So, now, applying D'Alambert pricnciple we can constrict ourselves to "external" forces, and taking F to always mean a conservative force (i.e. arising from some scalar potential function) we can write:
\[F=-\nabla \left(U\right)\] where U is just the potential function otherwise known by the name "Potential Energy". If we constrict ourselves to one dimension we can see:
\[F=-\frac{\partial U}{\partial x}\] 

Looking at the left hand side of the above equation where we treat \(P=mv\) let's explore the relations between the kinetic energy term - \(\frac{mv^2}{2}\) and the above term.
It's fairly easy to see that:
 \[ mv=\frac{\partial}{\partial v}\left(\frac{mv^2}{2}\right) \]

Let's call the kinetic energy term T and the potential energy term U, so we now get:
\[\frac{d}{dt}\left(\frac{\partial T}{\partial v}\right)=\frac{\partial (-U)}{\partial x}\]

Understanding kinetic energy to be non-dependant on location, and potential energy to be solely location dependant we can add expressions to both sides that vanish in derivation thus defining:
\[\mathcal{L}=T-U\]
to get the Euler-Lagrange equations
\[\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial \dot{q}}\right)=\frac{\partial\mathcal{L}}{\partial q}\] 
Where here I've also switched to some generalized coordinates \(q,\dot{q}\).

Now, defining the action \(\mathcal{A}\) as the time integral of the Lagrangian \(\mathcal{L}\), or in other words:
\[\mathcal{A}=\int\mathcal{L}dt\]

We have now successfully exchanged Newton's rules formulation by the "Principal of least action", since it follows that demanding minimal Action, actually yields Euler-Lagrange equations which are the equations of motion.

A word of caution though: 
First off this derivation is a sketch, the true process is a lot more rigorous, but I assure you, it works.

Secondly, Lagrangian Mech is a heavy canon, all you need to do is describe the problem in some sensible coordinates, find the kinetic energy and potential energy, perform some derivations...

and BAM!!! 

you get equations of motion, as well as conserved quantities and symmetries of the system like magic!!

But, since we use a couple of assumptions in deriving the Lagrangian, this actually covers LESS general physical situations. In reality there's ALWAYS friction, which is by no means a conserving force, thus adaptation to Lagrangian Mech are needed and they are not always as simple.

<Warning: Physics ahead/>

The obvious conclusion of all this mess, is that Newton laws of motion are equivalent (under some basic assumptions) to a first principle that states the following:

"Every physical system aspires to minimize the action taken"

for instance, water will flow in the path of least resistance, and light will travel the path of least (optical) distance, and I, ladies and gentlemen will aspire to do practically NOTHING if I can possibly get away with it.

This means I will absolutely abhor every single time I really have to get up, for instance to go to the fridge and get some grub, I'd much rather my lovely wife make lunch for me and serve it while I leisurely sit back and enjoy the latest mind-numbing episode ofsome stupid comedy show.

By the way, this is EXACTLY why light bends in the presence of gravitation. Not to watch stupid comedy shows, the other thing - being lazy.

See, gravitation can be thought of as a property of space(-time), rather than some force or field that occupies said space. by comrade Einstein's equivalency principle you can not tell whether you are being pulled by gravity, or being accelerated in the opposite direction.

And by that virtue, even light itself (having no mass is irrelevant here) bends to accommodate for this equivalency.

One might even say (and be wrong) that light is the submissive partner in the energy-gravitation relationship... But as I said, this is not correct, since light is a form of energy, and gravitation is a product of energy concentrates. So really, it's a marriage of energy and gravity, or in oriental philosophy terms, marriage of heaven and earth, or ying and yang, or horse and carriage (err... that's from a different philosophy methinks).

To conclude, marriage is such a blessing in one hell of a disguise, since left to our own devices we will do absolutely nothing until some external force propels us to action. When in a relatively successful marriage, the same is true, only the external force is called a wife, and hopefully, most of the times she kicks you in the gonads, it's for your own good.

This also enables two people to become a system of coupled equations, thus redefining the least action to actually produce some interesting dynamics, at least until a third little equation comes into play to dominate the hell out of their lives... if you're a parent, you know what I mean, right?

My own sweet third coupled equation 

Alternatively, if your marriage sucks, well, this explains why people stay in dysfunctional relationships for too long - it simply is to much ACTION to get out of one, better (by physical standards, not mine), to become mentally dislodged and otherwise apathetic to a fault.

But we know it's the former rather than latter. we really do.

Anyway we love you guys very much.

Mazal Tov!! 

Inflations, Oscillations, and Weight loss...

Both Cosmic and otherwise

To those who had the (mis)fortune of having me as a part of their lives,  weight gain, and weight loss are all too familiar. if not by first experience, they have  probably witnessed my own physique change, all across the range spanning from wirey-thin to brown dwarf ( a so called Failed-Star, or UltraCool Dwarf, which is a stellar body of roughly 13-85 Jupiter masses), and hopefully, one day, back again.
Since my travels through mass-space have been oscillatory in nature (for the most part), it is with pain-laced mirth I give you the following analysis and analogy of myself, and the Cosmos.

I know, I know, this seems a bit megalomanic on my part, but I assure you... I am. 

Well really, since my current field of study is Cosmology, and specifically inflation models, I found it might be funny to recount my thoughts and findings and relate it to the human condition, or rather, the fatso condition...

To any and all cosmologists out there, who might be reading this right now, be gentle, I am a mere neophyte to this field, and so I appeal to your sense of humor, and sympathy (enter violins please).

In order to first pique your interest, please listen to this number by Monty Python:

Which, by the way have some major faults in cosmological terms, for instance, the universe (acording to leading theories) actually does NOT expand at the speed of light. our EVENT-HORIZON (or simply horizon) does. It's meaningless to ascribe velocity to the expansion of the universe, since very close to us, the universe recedes from us in small velocities and far from us the universe seems to recede with greater velocity, as reflected by Hubble's law - velocity is proprtional to distance from us... \(\left(v=H_{0}D\right)\) .

By the way, this also implies that Galaxies which are far enough from us, recede at velocities greater than the speed of light (yes it IS possible), and thus they "drop off" our horizon, since they fade away faster than their light travels to us.

This also foretells a dark and lonely eventual demise, for our universe.

At any rate,  I would like at this juncture, to acquaint the reader with Friedmann's equation, which I will first write down and derive LATER (in two ways by the way...)
\[\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
Where \(R\) is the radius of an arbitrary sphere in space. this equation describes the evolution of \(R\) due to gravitational forces alone.

In layman terms:

The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.

For those who have some interest of keeping their sanity, you are invited to skip these derivations.
For those of brave soul and not so sound mind, please take special interest in the general relativity case...
(Important conclusions, in green)

<derivation - Newton style>
from Newt's 2nd law: \[F=ma\\
\Rightarrow F=m\ddot{R}\]
Gravitational force due to enclosed mass, on the perimeter of a sphere:
\[F=-\frac{GMm}{R^2}\]
Equating these yields:
\[\ddot{R}=-\frac{GM}{R^2} \Rightarrow \dot{R}\ddot{R}=-\frac{GM\dot{R}}{R^2}\]
Integrating, we get:
\[\dot{R}^2=\frac{2GM}{R}-\kappa\]
So, now, for matter in a sphere with a radius of  \(R\),   \(M=\frac{4\pi R^3 \rho}{3}\), thus:
\[\dot{R}^2=\frac{8\pi G \rho R^2}{3}-\kappa \Rightarrow \left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
taking \(R=R_0\cdot a(t)\) we get:
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G \rho}{3}-\frac{\tilde{\kappa}}{a^2}\]
</derivation - Newton style>
It's fairly easy to see that, according to Newton, the term \(\frac{\kappa}{R^2}\), is simply a term that is a result of integration, i.e. an integration constant. And so according to this derivation \(\kappa \) is simply given by initial conditions, and given no other incentive, we can gauge it away.

Let's see what old Einei have to say about this...
<derivation - Einstein style>
I'll try to be brief yet informative:
The metric for a spherical symmetric curved space is given by:
\[ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-\kappa r^2}+r^2d\Omega\right]\]
Or in matrix form:
\[g_{\mu\nu}=\left(\begin{array}{cccc}
-1&&&\\
&\frac{a^2}{1-\kappa r^2}&&\\
&&r^2&\\
&&&r^2\sin^2(\theta)
\end{array}\right)\]
From here it's relatively easy to find the Christoffel connections:
\[\Gamma^{0}_{11}=\frac{a\dot{a}}{1-\kappa r^2}\;;\;\Gamma^{0}_{22}=a\dot{a}r^2\\
\Gamma^{0}_{33}=a\dot{a}r^2\sin^2(\theta)\;;\;\Gamma^{i}_{0j}=\delta_{ij}\frac{\dot{a}}{a}\\
\Gamma^{1}_{11}=\frac{\kappa r}{1-\kappa r^2}\;;\;\Gamma^{1}_{22}=-r(1-\kappa r^2)\\
\Gamma^{1}_{33}-r(1-\kappa r^2)\sin^2(\theta)\;;\; \Gamma^{2}_{12}=\Gamma^{3}_{13}=\frac{1}{r}\\
\Gamma^{2}_{33}=-\sin(\theta)\cos(\theta)\;;\; \Gamma^{3}_{23}=\cot(\theta)\]
all other symbols vanish.

The Ricci scalar is then given by:
\[R=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]\]
The Einstein equation:
\[G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}\]
or, equivalently :
\[R_{\mu\nu}=8\pi G \left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\]
Taking the 00 term we get:
\[3\left[\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]=8\pi G\rho\Rightarrow \left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G\rho}{3}-\frac{\kappa}{a^2}\]
And, if we use the other notation also we get:
\[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}
(\rho+3p)\]


</derivation - Einstein style>
For now, we'll leave the second equation be, and look at the first.
We'll notice a couple of things first - Although it SEEMS as though the equations (Newt's, and Einei's) are the same, really the real Friedmann equation (derived from General Relativity) is the more general case. that is due to the former derivation relying on dynamics of MATTER only, and the latter does not, though, a careful massage of the former with reletavistic ideas, might give the correct equation for radiation, and other cosmic stuff...
The second thing I wish to emphasize is the existence of \(\kappa\) and it's meaning: In the Newtonian analysis, it was simply an integration constant, but in the Einsteinian analysis this is an intrinsic factor to the very fabric of the universe, this is the CURVATURE signature of the metric, that governs whether the cosmos is positively, negatively or null curved (i.e. flat) like in this picture:

Curvature types: positive, negative, and null.

<derivation - Gangnam style>
And this is just for fun:

</derivation - Gangnam style>

Admission of Guilt, errr... Ignorance

But wait!! What does all of this have to do with inflation?
Well, in fact, suppose space is sufficiently dillute, the matter density is sufficiently close to zero, and if it's sufficiently cold, radiation density drops to zero (almost), with an (almost) flat curvature, we are then left with some quantity we'll call \(\rho_0\), ("Rho naught"), and we get a dynamic equation, with an inflationary/deflationary solution:
\[\left(\frac{\dot{a}}{a}\right)=\pm\sqrt{\frac{8\pi G\rho_0}{3}} \Rightarrow a=A\exp\left(\sqrt{\frac{8\pi G\rho_0}{3}}t\right)+B\exp\left(-\sqrt{\frac{8\pi G\rho_0}{3}}t\right)\]
The second part drops off rapidely, and so we are left with an inflationarry solution.

"What is this witchcraft?!?!" you ask - where does this \(\rho_0\) come form?
well, suppose you're a fat guy, and you go into a fasting mode, note that you are actually GAINING weight (at least in the short run)... and by the time you've lost the battle against hunger, and went on a burger binge, guess what? you've now underwent inflation.

Well, actually this has nothing to do with \(\rho_0\), here's the REAL explanation for this:

WE DON'T KNOW!!!!

Oh my god, I can't believe I said that!!! this is the absolute NO NO for physicists!!!
To actually state that I don't know something? to recognize that everything we *THINK* we know is simply an approximate modeling of the awsome and complex reality we live in???!?

I should stop. NOW!!! I hear them knocking already... don't let them take me! NOOOOOOOOOOOOOOOOO!!!!

Ok, done with that gag, are we?

Moving on, there are several possible explanations for this, most are problematic to say the least, but hey, this is what it means to be in the front lines of science. you either get promoted or get diced....

At any rate, Inflation is a widely accepted theory of the adolescent universe, whatever mechanism manifests it.

What about oscillations?
In a nutshell - at the onset of the early universe, after inflation, some major ocillations occured in matter density, affected by ordinary (barionic) matter as well as dark matter, these oscillations are known as BARIONIC ACOUSTIC OSCILLATIONS (or modes) , and they could be seen quite nicely in analysis of the cosmic background radiation.

Moreover, when the universe was matter dominated, the same dynamics might have happened on a cosmic scale - Suppose matter is the dominant part of the universe, the dominant force then is gravitation, thus the universe itself, aspires to CONTRACT, offset only by radiation pressure, there MIGHT have been an epoch of slight contraction on the universe's part.

Sadly I didn't find a sufficiently fascinating animation to show, of the acoustic oscillations, but maybe some other time...

In conclusion, much like myself, the universe was "born", and began inflating.
Undergoing some oscillations, and at a certain point (just about.... now!) the universe moved from matter driven dynamics into  moderate inflationary epoch.

In layman terms:
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.

In the words of our mutual friend:

Entropy and Einstein's turnover time

Abstract
A failed attempt at explaining Entropy,
and one zombie, coming right up...

And so it was, on a nice evening, much like this one, that we had all sat around the table, and a question popped up...

The question was along the lines of "what is the Entropic principle?", and it was asked by my brother, a brilliant man, and a science-fiction aficionado, who unfortunately for the physicist community, never had the chance to dabble with physics, and so they have to find a poor substitute in the image of your humble servant here...

At first I asked if he meant the Anthropic principle, but he just wanted to understand Entropy.

Thus I found myself trying to explain Entropy, and the 2nd law of thermodynamics to the uninitiated, in layman terms, and, after a fashion, follow the Einstein grandmother rule - "You do not really understand something unless you can explain it to your grandmother." (A.Einstein).

Incidentally , you could probably calculate the period time \(T\) of Einstein's turning over in his grave, by someone misquoting him, or otherwise deifying him, and justifying a falsehood or plain ol' stupidity by attributing something to old Einei that he never would have meant in a million light years.  

We'll start with some observational data - I have around 400 people in my human network, and on average I get an Einstein quote which falls under the aforementioned category, maybe once every two weeks.

Now, suppose only a third of the world's population leads a somewhat western lifestyle (either connected to facebook, google+, twitter etc. or alternatively reads the paper and or listens to the radio at least once a day), we have about 2.3 Billion people.

let's be harsh and assume each of the 400 people sub-networks are non-connected between them and so we neglect back-propagation we can put a lower limit of
\[\frac{2.3\cdot 10^{9}}{400}=5.75\cdot 10^{6}\,\text{instances in 2 weeks}\]
Divided by the number of seconds in a two-week period we get:
\[\frac{5.75\cdot 10^{6}}{14\cdot 24\cdot 60\cdot 60}\approx 4.75\,\text{times per second}\]

And so the period time of Einstein's turning in his grave would be \(T\approx 0.21 \,sec\).

So basically even the lower limit states that Einstein, by now, is a zombified Olympic athlete, even considering the initial rigor mortis....

By now, he would have a solid six-pack.

Anyway, I digress, I was going to explain Entropy and then a random rant stole my attention... sorry for that.

In a nutshell, Entropy is a measure of disorder, and I will explain.

<Failed attempt at an explanation : but is still worth a read>

Imagine a group of four coins, each with two sides - heads, and tails - right? (we'll have non of that Two-Face shenanigans here!)
Now suppose every coin is perfectly balanced so there's a fifty-fifty chance of getting heads or tails for each coin flip.

So, now, what are the chances of getting all 4 heads, when you flip 4 coins?
if you do the experiment enough times, you get an average of 1/16 chance.
the state of all heads, or equivocally all tails is the most "ordered" result, why?
because it is the most homogenous result (and we humans like homogeneity, symmetry and by the same token order).
Now, what's the most plausible result?
that's easy - the result where two coins are tails up, and two coins are heads up (regardless of their locations), happens ideally \(\frac{3}{8}\) of the times you flip (almost half of the times you flip the coins, you'll get this result).

That is the least "ordered" result, since we don't care about locations, and the coins show the most diversity in results.


Now, I won't go through the whole derivation, but if you'll try the same logic with 6 coins and then 8 coins you'll get a breakdown of \[\frac{1}{64},\frac{6}{64},\frac{15}{64},\frac{20}{64},\frac{15}{64},\frac{6}{64},\frac{1}{64}\,\text{for six coins}\]
and a breakdown of \[\frac{1}{256},\frac{8}{256},\frac{28}{256},\frac{56}{256},\frac{70}{256},\frac{56}{256},\frac{28}{256},\frac{8}{256},\frac{1}{256}\,\text{for eight coins}\]
And so on and so forth, the reason I'm sticking to even numbers is because it LOOKS more clear that way, but really, it makes no difference. You could go on and on until kingdom come, and you'll find the middle, most unorganized result will be the most common.

It turns out that this kind of dynamic is best approximated by a gaussian function called the \(g\) function (or the multiplicity function) and I'll spare you the details in favor of a graph:

Probable results graph

So, what you see here, is basically an overlay of 4 graphs that show the relative probability of results as they stray from the middle "disorganized" and probable result.
What is interesting, is the bigger the experiment is (i.e. instead of 8 coins, let's say a 100 or 1000 coins) the sharper the peak is, meaning it's narrower, and higher in respect to other possible results. that means by the way that the most probable result is highly probable, and the others highly improbable.  Now imagine an experiment of \(10^{23}\) coins, every result other then the most probable and it's immediate neighbors is SO improbable, it virtually is IMPOSSIBLE (in the sense that it would take a ludicrously impossible amount of experiments to perform to actually get a significant chance to get such a result).

A word of caution though - this is probability we're talking about, so in theory a highly organized result MIGHT happen, in actuality - yeah, not so much...

By the way, there are roughly no more than \(6\cdot 10^{14}\) coins in circulation today in THE WORLD, meaning even if you took all the coins in the world today you couldn't perform such an experiment, even once!

Incidentally the ridiculously high number of participants in a single experiment, makes all the difference between "hard sciences" even if they are statistically oriented, and "soft sciences".

Even if we take all the people in the world, and get them to participate in one of our experiments, the result will produce some correlation that may, or may not apply to a single participant.

In physics, while the same is true, you could say a statistic result applies and be absolutely correct on a macro level (with deviations so small as to be insignificant for most purposes), and be correct almost every time on a micro level as well!.

So anyway, Entropy is defined as the logarithm of the multiplicity function.

The reason for taking the logarithm is for the sake of defining a cumulative quantity, as opposed to multiplicative.

< /Failed attempt at an explanation : but is still worth a read> 

So anyway, obviously I failed at this attempt but let's try it in a simpler manner:

Entropy is a quantity that signifies how probable a result is.
by a fluke of chance, which isn't a fluke at all, more of a deep connection really, the most probable result is also the most diverse one, or differently put, the most disorganized.

Thus, Entropy becomes a measure of disorder of a system.

Entropy is a cumulative property in the sense, that when two non-interacting experiments are done the combined entropy is the sum.
However, when systems are allowed to interact, the combined entropy is typically larger than the sum of individual entropy.

it is by that sense, that entropy tends to increase over time (and interactions).

<example of entropy increase>

Suppose, we have two systems, each of 4 coins.

The most probable state is given by 2 heads, and 2 tails for a single experiment right?
as was explained in the above failed attempt, the chance for that happening is \(\frac{3}{8}\).

Now, what is the chance of each of the experiment to get the most probable state independently? you guessed it - the product of the two independent probabilities i.e. \(\frac{9}{64}\) right?

OK, but now, let's put all the coins in a single experiment, an flip all of them, the chance of hitting 4/4 division of heads/tails is given by \(\frac{70}{256}=\frac{35}{128}\) which is almost double the size of the product of individual probabilities.

So what happened here, really?
in essence, the combined system has more places to choose from, meaning more diversity of scenarios that lead to the same end result, thus the combined system is more "disordered" thus bigger probability that leads to bigger Entropy.

</example of entropy increase>

OK, that wraps it up for this time.
Obviously I don't understand Entropy enough, since I feel I have failed at explaining it,
but I will try again ("if at first you don't succeed" etc...).

Oh by the way, for all us \(\LaTeX\) geeks out there, isn't it cool that every time we use a fractional, we immediately get a reference to Battlestar Galactica?

Anyway, till next time...

P.S. some quick ideas to utilize Einei's incredible turnover time of 0.21 seconds :
1. Attach him to a turbine and generate electricity.
2. Display him as the 8th world wonder - the quickest man on the planet (faster than Usain Bolt).
3. Use him as the engine for a horse-ride carousel for my kid.
4. Use him to refute the 2nd law of Thermodynamics as he is both dead (Entropy supposed to get bigger), but in the greatest shape of his life (which requires work and decrease in local Entropy)...

Just sayin'...




 

Short Derivation of Archie's Law

 "captain's log: supplemental"

It's been quite a while since I wrote the previous post about Archie, his wife, a crown etc.
While I was writing that, I did not want to derive Archimedes' law on my own, so I looked around on the net, to maybe find someone who already posted that, sadly, I couldn't.

Also, this morning my wife and I had an argument about this law, and guess what, she was right, I was wrong... Isn't it weird these are the arguments my wife and I have?!? I mean, seriously, how nerdy can you get? 

So for the interest of completeness (which is actually a mathematical axiom, but let's leave that be for now), I give you a short derivation of Archie's law:

Deriving the law (I AM THE LAW)

Let us consider an infinitesimal volume element of a material with density \(\rho\) , and incident area and height \(A,h\) respectively.
So now let's suppose the material is lighter than the medium surrounding it we get a force equation that looks like this:
\[\Sigma F=A\cdot P_{up} -A\cdot P_{down}-Mg\]
Where the force equivalent (sum of all forces) is positive.

Now, let's wrap the forces that are the result of pressure and name it the Buoyancy force thus:
\[F_{Buoyancy}=A\cdot\left(P_{up}-P_{down}\right)\]
and with a little massage we get:
\[A\cdot\left(P_{up}-P_{down}\right)=A\cdot h \left(\frac{P_{up}-P_{down}}{h}\right)=-V\nabla P\]

So, now, let's take a small detour to understand who \(\nabla P\) is shall we?

suppose we are dealing with an element that is filled with the same material as the medium it's in, in that case we get a mechanical equilibrium and \(\Sigma F\) is simply zero. and so we get:
\[Mg=F_{Buoyancy}=-V\nabla P\]
and breaking up the \(Mg\) element we get:
\[V\rho_{medium}\cdot g=-V\nabla P\Rightarrow \nabla P = -\rho_{medium}\cdot g\]
and so we get quite simply:
\[F_{Buoyancy}=V\cdot \rho_{medium}\cdot g\]

Proper usage

What's useful with this representation, is that we get the force outright, thus we can make force calculations directly.

An important thing to understand is that the buoyancy force is directed not UPWARDS (which is a common misconception) but against the direction of the pressure gradient . which is quite different. that means for example that in a spinning tube of air, the buoyancy force will be mostly inwards as the pressure gradient is directed outwards (in cylindrical coordinates).

That's it.

this time it was really short.

Crowns,Boats, Subs, and other things that float in water...

Abstarct:

A short story about a friend,

and things that go "ping" in the dark.

It has been some time since last I wrote something, but life, the multiverse and everything, got in the way.

By the way, if the answer to life universe and everything is 42, then for the multiverse it should be what? a Vector? a Tensor? and what are the entries? in which linear basis?
I guess I could get cute and say that its a \(42^n\) Tensor, but that's just my regular idiotic nonsense at play...

So yes, in layman terms I was simply very otherwise occupied, but today, as my wife is away, and after almost a week of sleep deprivation and other torture methods, I found the will and the way to actually start writing stuff.

Anyway the idea for this post came from a rather dubious experience, and I might have a follow-up post on the physics of water-closets, otherwise known as restrooms, bathrooms or the unsavory "latrines".

What's the connection?

And Seeing as approximately a quarter of my life was spent on boats or in the context thereof, I deem it fit to dedicate a post to the physics of these floating wonders.

What makes things float? 

Most of us at one point or the other have had a chance to hear that quite famous cry "EUREKA" but I daresay less of us actually know the origin of this cry. so here goes...

Some 2500 years ago in the golden age of the great kingdom of Crete, there lived a king and his queen, or, more probably a queen and her lackey of a king.

Now that queen was little capricious, not unlike the queen of hearts from Lewis Carroll's Alice's adventures in wonderland, and at some point in time decided that the crown allotted to her, was not fancy enough. Thus, she ordered a new crown to be made, a crown of solid gold.

 The word spread, and a huge congregation of goldsmiths, jewelers, and their apprentices, flocked to the isle of Crete, in hopes of gaining the queen's grace and be chosen to forge the queen's new crown. 

But as is often the case with tyrannical rulers, this queen was a very suspicious being, I could theorize she had a bad experience with  goldsmiths, or might be she was simply a bitch.

In any event, she was absolutely terrified by the prospect of being swindled and as a result she wanted to make sure the crown was actually made of solid gold. 

But here's the pickle... suppose she WASN'T swindled and she melted the crown to check if it's solid gold, she's now left with a pot of molten gold, while still having to pay the goldsmith.

Thus a method had to be devised to check the crown without damaging it.

Enter stage left: Archimedes.

Archie, our friend happened to be in the vicinity and having a reputation of the genius he was, was charged with the daunting task of finding that method.

Well, Archie thought hard, maybe losing some weight (why can't I?) and some hair (why am I?) at the prospect of failing the queen and subsequently failing to breath, and at the end of 3 excruciating days, his wife decided she would have none of it anymore, "You stink!" shrilled the shrew "Go have a bath or it's the couch tonight, for you!".

Archimedes, being the thoughtful husband that he was, climbed into a warm bath, and noticed, that when he submerged more of his body, the water level rose and spilled over the sides of the tub. 

"EUREKA"

he shouted then, followed by his wife's "Shut up already you git! you'll wake the baby and then you'll have to deal with it!!" 

Anyway, I will leave you wondering as for how the story ends, did old Archie indeed wake the baby, how long was spent in the dog-house, and whether or not a goldsmith found his premature demise.

Now for the physics:

Suppose an object with a volume V is partially submerged in water - the elevation force is due to pressure differences, the partially exposed part of the object experiences just the atmospheric pressure, but the underside experiences the upward pressure from the water, so let's see what that pressure is:
Let's consider a column of water and a thin strip \(\Delta Z\) thick
\[\Sigma F=0 \Rightarrow s\cdot\left(P(z+\Delta z)-P(z)\right)-s\cdot\Delta z \cdot \rho_{water} g \\
\text{or in other words} \frac{\partial P}{\partial z}=\rho_{water}\cdot g\Rightarrow P(z)=\rho_{water}\cdot g\cdot z+ P_{atm}\]

Where z is the depth of water.

So, we have to consider \[mg=s\cdot\rho_{water}\cdot g\cdot z \Rightarrow h\cdot\rho_{object}=\rho_{water}\cdot z \]

In other words, the depth of immersion is given by the height and relative density of the object and the fluid.

Now, there's an easy way to see that this dynamic is correct, simply take a piece of wood , and see that it submerges deeper when you hold it length up, than when it's laying flat on the water.

So that takes care of boats, we just have to make sure the average density of the boat is lesser than the density of the water, mind you we're talking the average density of the space occupied by the boat meaning also the air inside the boat, unless you start to take on water, and then guess what? your downward bound.

That also might provide a hint why we sometime encounter "unflushables"...

That also might provide an insight as to how submarines stay submerged at a constant depth:
When the sub is at "bubble up" state, basically the density of the sub is lower than the surrounding water thus the sub tends to float up. to hasten the process the sub might or might not apply it's propeller or other means of propulsion.

When the sub is at "bubble down" state, the average density of the sub is higher than that of the surrounding water making the sub "heavier" and thus sinks down, again applying propulsion or not is at the captain's discretion.

By the way, subs mostly have compressed air tanks, which they discharge into  ballast sections, to change the average density of the vessel, and then use compressors to re-compress said air to the tanks, evacuate the ballast air ballast sections to increase the average density (water then flood the ballast sections).

So it seems fairly simple right? WRONG, we actually took the water's density to be constant where it really isn't, cold water is denser than hot water, and deep water is a tad denser than shallow, so what's the deal? 

Well the physics for this is fairly complicated in terms of the math involved, but the IDEA is fairly simple, water density is a product of the mutual forces between water molecules, that are essentially electric in nature, and so external pressure is somewhat involved in this, but even more so, temperature.

So with pressure \(\rho_{water}\) rises linearly at first but pretty quick stabilizes to a constant.

Like so:

Courtesy Windows To the Universe (NESTA)

With temperature the change is much more pronounced, but still pretty much the same applies, A linear rise in density when temperature drops, and then exponential decay to a constant.
I suspect somewhere in the middle there's actually a point where it all turns to ice...

That remind me of the cool Thermometer where there are different glass bells with different nifty colored liquids in a glass water tube, and when the water in the tube is in thermal equilibrium with the area, some bells float up, some sink down, and the one left in the middle shows the right temperature on it.... pretty cool if you ask me...

Told you water density changes with temperature!

So anyway making the calculations needed to predict the density at a certain depth and temperature is a pretty nasty undertaking thus usually subs employ feedback loop mechanisms to apply the right density. either that or they do it by hand and eye i.e. "bubble up"\"bubble down" mechanism.

By the way, remember the couple of pictures in the beginning? well there you go:

Baby Ruth is swimming pool - not quite what you think...

I could go on and on about this, about weighing ships, (as opposed to sheep), and using partially submerged sonar buoys, Thermocline, and using different water densities to mislead enemy vessels as to your true location etc. etc. But I'm pretty sure if you read Clancy's "The hunt for Red October" you'd learn all this and have great time doing so...

a short post about Wind Power and shams

ABSTRACT: 

Wind power, commercial jets, and politics.

And almost no humor this time...

A couple of weeks ago, maybe 3~4 days after posting the little thingy about Solar energy, two things happened.

1. I was a guest at an eminent solar scientist BBQ lunch, and we had a little chat about what I wrote.
   He said he'd send me a paper he wrote, so I could avoid making the same mistakes as everybody, and go on to make some mistakes of my own... :)
   
   I promise when I finish reading it, I'll atone for my sins, stop eating meat, cease all gasoline use and switch over to the green side. Well no. but I will try to rework my logic and share my thoughts to the extent possible, since it is article material and I don't want this blog to jump the gun on hard work, not my own.
   
2. I came across this nice video :
   
   
   
   Now, a word of caution, this video is very political, and actually, the field of environmental science is highly political in nature, which by the way brings the question whether Thomas Kuhn's largely misinterpreted  view on science is actually correct - i.e. science is as politicized as everything and social trends actually effect scientific truth.
   
   Kuhn probably never intended to say or write that, but that's a different story altogether.
   
   Anyway my only intention here is to derive the part of how much energy an electro-windmill can produce.

Since wind have several sources, I can't simply rely on Ideal gas equation plus the Euler equation (though I've done the math, it actually accounts for very little of the wind regiment in the world), so I'd have to start from observational data...

After digging a bit, I found several sources, but I'm going to be extremely optimistic and take the average wind velocity, averaged over the globe over a period of about 50 years to be about 7.5 m/s.

So, the energy is given by \(\frac{1}{2}mv^2\) but we need a way to evaluate the total mass of the moving gas that comprises the wind right?

Let's do that real quick shall we?
the volume of air passing through an incident area \(A\) in one second is basically given by \(A\cdot v\), thus the wind power that goes through a windmill with incident area \(A\) is simply:
\[P=\frac{1}{2}\rho_{air}Av^3\]
with \(\rho_{air}\) being roughly \(1.25 kg\cdot m^{-3}\), we get that the power as a function of incident area is given by:
\[P_{\text{wind averaged}}\approx 264\cdot A\,\,\, W\]

So supposedly its enough for every person to have about 9 square meters to his name in the constellation of a private windmill to account for all our earthly needs.

Enter windmill efficiency at a staggering  36% (the theoretical boundary is actually about 60% but wind drag, mechanical stress etc. account for further efficiency loss), so every person must now have about 27 square meters, which is not that much right?

Really? 2% of the energy a commercial aircraft needs to take off ?

The Energy needed for an empty Boeing 747 at 170 tons to take off and achieve cruise altitude of about 10km is about \(2\cdot 10^{10}\,\,\,W\) which means we need a windmill with a blade length of about 5km or about 1000 150m blade turbines, or 10,000 50m blade turbines.

What about just take-off speed? average takeoff speed for a 747 is about 80 m/s, and you'd have to maintain that speed for the duration of the takeoff, but let's calculate just for one second shall we?
\[E_{\text{one second}}=10^9\]
So we would need about 1km long blade to account for that plane taking off, and a 50-store blade spells about 150 meters blade which means about \(1.9\cdot 10^{7}\, W\)  and that really is about 2% of the energy needed for an empty 747 to take-off...
So you see? the guy in the video was right!!! or was he? 

Cheap rhetoric or outright sham?

well no, that argument is just throwing sand in the public's eyes because supposedly if we had enough turbines to cover the world's energy needs, a spike in demand as reflected by a taking-off event would not even show on the overall scale, with about 4 to 5 orders of magnitude difference...

The real consideration has to be this one:
in order to provide for humanity's power needs, we need about \(2\cdot 10^{11}\) square meters dedicated to power production. factor in spacing issues (a factor of 6 times the rotor blade length means about 36 times the area). thus we get about \(8\cdot 10^{12}\,m^2\) which means about 5% of the land area available on earth.

Again, this is about the size of Europe, and I didn't even go into wind regiment as a function of height consideration, or considering blackout times and backup requirements as the wind DOES tend to stop at rather inconvenient times (for instance when it's hot as hell outside and there's no wind,   and you want to turn on the AC, but guess what - no wind means no electricity means no AC!!! damn you green energy advocates!!)

I read somewhere (an incomplete) analysis that concluded it calls for covering the surface of the earth with some kind of a wind capturing apparatus a 150 meters high in order to manufacture enough energy for us to  live the way we do. I'm not that pessimistic, but let's just say wind power is no more a true messiah than solar power is...

So you see, this field is indeed highly politicized, the green energy advocates would have you believe that so called green energy is a magic pill solution, with no downsides, when really, at best it's a small part of the solution.

While the people who basically side up with the petroleum industry will try to sell you imaginary numbers, or otherwise hammer you down with irrelevant comparisons, conclusions and data.

If I might add my 2 cents - Wind power as well as Solar power, are part of the solution and it's very important for us to develop these further. a 100% efficiency is impossible, but imagine what could happen if we had 80% efficiency on Solar&Wind energy production? we could maybe minimize (not eliminate mind you!) our trace on the environment, while also lessening our demand for energy production dedicated land. together with another 2-3 sources of clean energy, we may very well create a society that is both abundant in food,water and energy and as non-obtrusive as can be.


I know, this post might have bored you to death, I was sick when I wrote it so it might be lacking in humor, wits or logic, but I do hope later posts will be better that way...