“But Manning grew disillusioned in late 2009 in the wake of an attack that caused Iraqi casualties without costing American lives”

Why do I call this statement vile?  Why did reading this turn my stomach?  Why did this disturb me so much that I spent the last few hours of my attempt to sleep last night tossing and turning?

The use of “disillusioned” describes an event (or events) shocking severe enough that Manning risked his career, his liberty, and perhaps even his life to bring it to light.  Specifically, he had seen the video now known as “Collateral Murder”, and the subsequent coverup of the incident.  This coverup was made eventually made available by wikileaks.  This is a video record of the one incident where civilians were slaughtered because somebody perceived that they were armed with and firing AK47’s and RPGs.

The spin imposed by Potter in this article is mind boggling.  He apparently desired a way to show the whistleblowing of Manning in a negative light.  A statement like “Iraqi casualties without costing American lives” does just that.  It is anonymous enough that somebody who didn’t know what he was talking about might imagine that there had been some sort of glorious battle where the American military showed valour and skill, and managed to beat the enemy without any injury to themselves.

The problem is this.  The US Iraq invasion force has been sent in with loaded weapons, they’ve been indoctrinated to imagine they are fighting an enemy, so they see and find and create enemies that do not exist.  This is inevitable.  This is the crime of war.

There was a man carrying a camera.  Later pentagon inquiry found that there was an RPG with these men, but I’d trust that claim no further than I could spit.

A camera could be a dangerous weapon in this day and age where media manufactures the stories that support their preconceived ideas and agendas.  It could be used to show that war is nothing more than viscous profiteering.  There has been little use of cameras for this purpose in North American mainstream media.

Motivation

In [1] I saw the Euler-summation formula casually used in a few places, allowing an approximation of a sum with derivatives at the origin. This rather powerful relationship was used in passing, and seemed like it was worth some exploration.

Bernoulli polynomials and numbers

Before tackling Euler summation, we first need to understand some properties of Bernoulli polynomials [], and Bernoulli numbers [2]. The properties of interest required for the derivation of the Euler summation formula appear to follow fairly easily with the following choice for the definition of the Bernoulli polynomials and Bernoulli numbers

It is conventional to fix . Eq. 1.0.1.1 provides an iterative method to calculate all the higher Bernoulli numbers. Without calculating the Bernoulli numbers explicitly, we can relate these to the values of the polynomials at the origin

Now, let’s calculate the first few of these, to verify that we’ve got the conventions right. Starting with we have

or . Next with

or . Thus the first few Bernoulli polynomials are

The Bernoulli polynomials have a simple relation to their derivative. Proceeding directly, taking derivatives we have

or

There’s a number of difference relations that the polynomials satisfy. The one that we need is

To prepare for demonstrating this difference in general, let’s perform this calculation for the specific cases of and to remove some of the index abstraction from the mix. For we have

For (a value of that is representative) we have

Evaluating this in general, we see that the term with the highest order Bernoulli number is immediately killed, and we’ll have just one highest order monomial out of the mix. We expect all the remaining monomial terms to be killed term by term. That general difference is, for is

This last sum up to has the form of eq. 1.0.1.1, so is killed off. This proves eq. 1.0.8 as desired.

From this difference result we find for

and for

we find that either of the end points in the interval provide us (up to a sign) with the Bernoulli numbers

Integrating eq. 1.0.7 after an substitution, and comparing to the difference equation, we have

or

Evaluating this at shows that our polynomials are odd functions around the center of the interval, or

We also obtain Bernoulli’s sum of powers result

or

We don’t need this result for the Euler summation formula, but it’s cool!

To arrive at some of these results I’ve followed, in part, portions of the approach outlined in []. That treatment however, starts by deriving some difference calculus results and uses associated generating functions for a more abstract difference equation related to the Bernoulli polynomials. In this summary of relationships above, I’ve attempted to avoid any requirement to first study the difference equation formalism (although that is also cool too, and not actually that difficult).

Euler-MacLauren summation

Following wikipedia [4], we utilize the simple boundary conditions for the Bernoulli polynomials in the interval. We can exploit these using integration by parts if we do a periodic extension of these polynomials in that interval.

Writing for the largest integer less than or equal to , our periodical extension of the interval Bernoulli polynomial is

From eq. 1.0.2 and eq. 1.0.14, our end points are

Utilizing eq. 1.0.7 we can integrate by parts in a specific unit interval

Summing gives us

or

Continuing the integration by parts we have

or

References

\bibitem[Behnke et al.(1974)Behnke, Gerike, and Gould]behnke1974fundamentalsV3Heinrich Behnke, Helmuth Gerike, and Sydney Henry Gould. Fundamentals of mathematics, volume 3. MIT Press, 1974.

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

[2] Wikipedia. Bernoulli number — wikipedia, the free encyclopedia, 2013\natexlab{a}. URL
http://en.wikipedia.org/w/index.php?title=Bernoulli_number&oldid=556109551
. [Online; accessed 28-May-2013].

[3] Wikipedia. Bernoulli polynomials — wikipedia, the free encyclopedia, 2013\natexlab{b}. URL
http://en.wikipedia.org/w/index.php?title=Bernoulli_polynomials&oldid=548729909
. [Online; accessed 28-May-2013].

[4] Wikipedia. Euler-maclaurin formula — wikipedia, the free encyclopedia, 2013\natexlab{c}. URL
http://en.wikipedia.org/w/index.php?title=Euler%E2%80%93Maclaurin_formula&oldid=552061467
. [Online; accessed 28-May-2013].

Somebody told me how to disable this brain dead “feature” on facebook, and I’m sharing it here.   You need to use File -> preferences -> sametime -> notifications, but once you are there what shows up is “Location awareness” :

You have to individually click on all the other options (like One-on-one) to actually disable the model and flashing nastiness.  For example:

Once this is done, then sametime windows hide in the background where they should be, until you actually get around to looking at them, if you ever choose to.

Question: Bose gas specific heat above condensation temperature ([1] section 7.1.37)

Equation 7.1.33 provides a relation for specific heat

Fill in the details showing how this can be used to find

Answer

With

we have for constant

From the series expansion

we have

Taken together we have

or

We are now ready to evaluate the derivative and find the specific heat

This is the desired result.

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

In section 6.2 [1] we have the following notation for the sums in the grand partition function . Note that I’ve switched notations from as used in class to as used on our final exam. The text uses a script Q like but with the loop much more disconnected and hard to interpret.

This was shorthand notation for the canonical ensemble, subject to constraints on and

I found this notation pretty confusing, since the normal conventions about what is a dummy index in the various summations do not hold.

The claim of the text (and in class) is that we could write out the grand canonical partition function as

Let’s verify this for a Fermi-Dirac distribution by dispensing with the notational tricks and writing out the original specification of the grand canonical partition function in long form, and compare that to the first few terms of the expansion of eq. 1.0.5.

Let’s consider a specific value of , namely all those values of that apply to . Note that we have only for a Fermi-Dirac sysstem, so this means we can have values of like

Our grand canonical partition function, when written out explicitly, will have the form

Okay, that’s simple enough and really what the primed notation is getting at. Now let’s verify that after simplification this matches up with eq. 1.0.5. Expanding this out a bit we have

This completes the verification of the result as expected. It is definitely a brute force way of doing so, but easy to understand and I found for myself that it removed some of the notation that obfuscated what is really a simple statement.

Once we are comfortable with this Fermi-Dirac expression of the grand canonical partition function, we can then write it in the product form that leads to the sum that we want after taking logs

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

I’d seen this book recently in the Markham public library. It’s been republished with additions, but I didn’t feel the new author added much value.

It’s interesting to see that this project also makes the tex sources available. Because of that I can include the awesome prologue and first chapter from this text in this post. Check it out. Doesn’t it whet your appetite for more calculus?

Prologue

Considering how many fools can calculate, it is
surprising that it should be thought either a difficult
or a tedious task for any other fool to learn how to
master the same tricks.

Some calculus-tricks are quite easy. Some are
enormously difficult. The fools who write the textbooks
of advanced mathematics—and they are mostly
clever fools—seldom take the trouble to show you how
easy the easy calculations are. On the contrary, they
seem to desire to impress you with their tremendous
cleverness by going about it in the most difficult way.

Being myself a remarkably stupid fellow, I have
had to unteach myself the difficulties, and now beg
to present to my fellow fools the parts that are not
hard. Master these thoroughly, and the rest will
follow. What one fool can do, another can.

To deliver you from the Preliminary Terrors

The preliminary terror, which chokes off most fifth-form
boys from even attempting to learn how to
calculate, can be abolished once for all by simply stating
what is the meaning—in common-sense terms—of the
two principal symbols that are used in calculating.

These dreadful symbols are:

(1) which merely means “a little bit of.”

Thus means a little bit of ; or means a
little bit of . Ordinary mathematicians think it
more polite to say “an element of,” instead of “a little
bit of.” Just as you please. But you will find that
these little bits (or elements) may be considered to be
indefinitely small.

(2) which is merely a long , and may be called
(if you like) “the sum of.”

Thus means the sum of all the little bits
of ; or means the sum of all the little bits
of . Ordinary mathematicians call this symbol “the

integral of.” Now any fool can see that if is
considered as made up of a lot of little bits, each of
which is called , if you add them all up together
you get the sum of all the ‘s, (which is the same
thing as the whole of ). The word “integral” simply
means “the whole.” If you think of the duration
of time for one hour, you may (if you like) think of
it as cut up into little bits called seconds. The
whole of the little bits added up together make
one hour.

When you see an expression that begins with this
terrifying symbol, you will henceforth know that it
is put there merely to give you instructions that you
are now to perform the operation (if you can) of
totalling up all the little bits that are indicated by
the symbols that follow.

That’s all.

References

[1] Silvanus P Thompson. Calculus made easy. Macmillian, 1914. URL
http://www.gutenberg.org/files/33283/33283-pdf.pdf
.

Here’s a bash at one of the exam questions, where I get the time to think things through properly. I think I did something like this on the exam itself, but may have also made some arithmetic errors.

Question: Ultra relativisitic spin zero condensation temperature (2013 final exam pr 2)

Consider a Bose gas with particles having no spin and obeying an ultra relativisitic dispersion . Unlike photons or phonons, these particles are {\bf conserved}, and hence we must determine the chemical potential which fixes their density. Working in three dimensions, show whether or not these particles will exhibit Bose condensation, and find if it is nonzero.

Answer

For the number of particles in the gas, as with photons, we still have

As in the discussion of low velocity particles in [1] section 7.1, the ground state term has been split out, before making any continuum approximation of the sum over the energetic states.

Writing

where the number of particles in the ground state is chemical potential and temperature dependent

We proceed with the continuum approximation for the number of particles in the energetic states

So we have

Note that , a fixed number. The key feature of Bose condensation remains. There is a finite limit to the number of particles that can be in the energetic state at a given temperature and volume. Any remaining particles are forced into the ground state.

In general the number of particles in the ground state is

and we will necessarily have particles in this state if

That temperature threshold is the Bose condensation temperature

With , , we have for the ground state average number density

This is plotted in fig. 1.1.

Fig 1.1: Ratio of ground state number density to total number density

From the figure it appears that the notion of any sort of absolute condensation temperature is an approximation. We can start having particles go into the ground state at higher temperatures than , but once the chemical potential starts approaching zero, that temperature for which we start having particles in the ground state approaches . The key takeout idea appears to be, once the temperature does drop below , we necessarily start having a non-zero ground state population, and as the temperature drops more and more, the ratio of the number of particles in the ground state relative to the total approaches unity (all particles are forced into the ground state).

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

Central limit theorem

If and , and , then in the limit

Binomial distribution

where was something like number of Heads minus number of Tails.

Generating function

Given the Fourier transform of a probability distribution we have

Handy mathematics

Heavyside theta

volume in mD

area of ellipse

Radius of gyration of a 3D polymer

With radius , we have

Velocity random walk

Find

Random walk

1D Random walk

leads to

The diffusion constant relation to the probability current is referred to as Fick’s law

with which we can cast the probability diffusion identity into a continuity equation form

In 3D (with the Maxwell distribution frictional term), this takes the form

Maxwell distribution

Add a frictional term to the velocity space diffusion current

For steady state the continity equation leads to

We also find

and identify

Hamilton’s equations

SHO

Quantum energy eigenvalues

Liouville’s theorem

Regardless of whether we have a steady state system, if we sit on a region of phase space volume, the probability density in that neighbourhood will be constant.

Ergodic

A system for which all accessible phase space is swept out by the trajectories. This and Liouville’s threorm allows us to assume that we can treat any given small phase space volume as if it is equally probable to the same time evolved phase space region, and switch to ensemble averaging instead of time averaging.

Thermodynamics

Example (work on gas): . Adiabatic: . Cyclic: .

Microstates

quantum

Ideal gas

Quantum free particle in a box

Spin

magnetization

moment per particle

spin matrices

spin addition

Canonical ensemble

classical

quantum

Grand Canonical ensemble

Fermions

(so for large temperatures)

Bosons

For , .

BEC

Density of states

Low velocities

relativistic

Question: Low temperature Fermi gas chemical potential

[1] section 8.1 equation (33) provides an implicit function for

or

In class, we assumed that was quadratic in as a mechanism to invert this non-linear equation. Without making this quadratic assumption find the lowest order, non-constant approximation for .

Answer

To determine an approximate inversion, let’s start by multiplying eq. 1.0.2 by to non-dimensionalize things

or

If we are looking for an approximation in the neighborhood of , then the LHS factor is approximately one, whereas the fractional difference term is large (with a corresponding requirement for to be small. We must then have

or

This gives us the desired result

References

[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.

April 21, 2013 Fermi function expansion for thermodynamic quantities

April 20, 2013 Relativistic Fermi Gas

April 10, 2013 Non integral binomial coefficient

April 10, 2013 energy distribution around mean energy

April 09, 2013 Velocity volume element to momentum volume element

April 04, 2013 Phonon modes

April 03, 2013 BEC and phonons

April 03, 2013 Max entropy, fugacity, and Fermi gas

April 02, 2013 Bosons

April 02, 2013 Relativisitic density of states

March 28, 2013 Bosons

plus everything detailed in the description of my previous update and before.