Quantum Bayesian Networks

October 20, 2012

My TIT Infatuation

Filed under: Uncategorized — rrtucci @ 11:08 pm

Wow, check out these Great Tits!

And listen to these
great tit songs.

I’m currently reading a lot about a subject which I like to call TIT (Thermal Information Theory). TIT is closely related to but not quite the same as SIT (Shannon Information Theory). Whereas pure SIT says nothing about energies, TIT does.

Two of my favorite authors on the subject of TITs are Sagawa and Ueda. One reason I like their papers is because they use a language that is very similar to the one quantum SIT people use. Sagawa and Ueda have recently written two lovely papers that review their previous TIT work:

• Information Thermodynamics: Maxwell’s Demon in Nonequilibrium Dynamics, by Takahiro Sagawa, Masahito Ueda, arXiv:1111.5769

• Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction, by
Takahiro Sagawa, arXiv:1202.0983

I’ve been fascinated with thermodynamics for a very long time. Funny thing is, I didn’t understand it too well when I took my first course on it. But I kept coming back to it, and, nowadays, after much perseverance, I believe I have finally become quite good at it. I think there were several reasons why I found thermodynamics so difficult and mystifying initially.

1. It took me a while to see what is essential and what is just icing on the cake—what are the main rules and what is just an application or a straightforward consequence of the main rules.
2. It took me a while to find any books about the subject that agreed with my style of thinking. I tend to prefer books that give you all the rules at the beginning of each chapter and then say, okay, now let’s look at some applications. I found Feynman’s book “Statistical Mechanics ” to be like that. On the other hand, I tend to dislike books that mix the rules and applications into one gray goo.

For example, in my early college years I was recommended the book by Reif. I tried learning some thermo from Reif’s book and ended up thoroughly confused. So I gave up on that book and looked for others that were better for me. I found some that I really like, such as

• “Thermodynamics”, by Joachim E. Lay. A stunningly beautiful masterpiece. This book is very complete theory-wise, but being a book for mechanical engineers, it also includes lots of beautiful applications.
• “Statistical Mechanics” by Feynman.
• Thermodynamics, by H.B.Callen

There are many other good ones. The thermo books by mechanical engineers and chemists discuss some cool applications that are native to their fields.

Let me end this post by quoting the first page of Feynman’s book “Statistical Mechanics”. Long, long ago, the first time I read this passage, it was an epiphany moment for me. I thought. Wow! So that’s all there is to thermo? I still find this passage to be a stunningly beautiful and highly effective way to begin a book on statistical mechanics:

CHAPTER 1

INTRODUCTION TO STATISTICAL MECHANICS

1.1 THE PARTITION FUNCTION

The key principle of statistical mechanics is as follows:

If a system in equilibrium can be in one of $N$ states, then the probability of the system having energy $E_n$ is $(1/Q) e^{-\frac{E_n}{kT}}$, where

$Q = \sum_{n=1}^{N}e^{-\frac{E_n}{kT}},$

$k=$ Boltzmann’s constant, $T=$ temperature. $Q$ is called the partition function.

If we take $|i\rangle$ as a state with energy $E_i$, and $A$ as a quantum mechanical operator for a physical observable, then the expected value of the observable is

$\langle A \rangle = \frac{1}{Q} \sum_i \langle i|A|i \rangle e^{-\frac{E_i}{kT}}$.

This fundamental law is the summit of statistical mechanics, and the entire subject is either the slide-down from this summit, as the principle is applied to various cases, or the climb-up to where the fundamental law is derived and the concepts of thermal equilibrium and temperature $T$ clarified. We will begin by embarking on the climb.