## Some perils regarding interpretations of entropy

Posted in Uncategorized on May 22, 2012 by quantummoxie

I have been involved in an interesting discussion on Google+ with a few people regarding the proper way to interpret entropy,

$S=-k\sum_{i}p_{i}\textrm{log}p_{i}$.

The Bayesian interpretation takes S to be a state of ignorance such that when we update our probabilities, the entropy subsequently updates.  The question is, is the updating of the probabilities themselves subjective or objective?

Here’s a very simple example.  Consider a chamber divided into two sub-chambers of equal volume.  This setup can be observed from two sides, with no communication occurring between the two sides as shown in the figure.

Suppose that Observer One thinks that Gas A = Gas B.  On the other hand, suppose Observer Two thinks that Gas A ≠ Gas B.  In short, Observer One believes that Gases A and B are indistinguishable.  Observer Two believes the opposite.  Both these states of knowledge can be reflected in terms of the probabilities for the associated microstates of the gases and thus in the entropies.

Now let us suppose that the partition is removed and the two gases are allowed to mix.  Since Observer One thinks that Gas A = Gas B, he will find that the entropy of mixing will be zero.  Since Observer Two thinks that Gas A ≠ Gas B, she will find the entropy of mixing to be non-zero.  It is important to note here, that both observers have correctly carried out their experiments with the equipment and knowledge that they have at their disposal.  That is not under debate.  What is under debate is how do we interpret these results?

Let us take entropy to be a measure of the state of knowledge, then.  Let us assume that the number of molecules of Gas A is the same as the number of molecules of Gas B, i.e. N(A) = N(B).  Since this is a case of free expansion, there is also no change in internal energy, U, of either Gas A or Gas B.  The thermodynamic identities, then, reduce to

$TdS=PdV$

where the PdV term can be interpreted as mechanical work.  Suppose both Observer One and Observer Two have an identical device attached to the chamber on their respective sides that measures mechanical work.  Call these measurements M(A) and M(B).  According to the above thermodynamic identity, Observer One expects to find that M(A) = 0 while Observer Two expects to find that M(B) ≠ 0.

Now let us adopt a principle of consistency for classical physics:

Experiments on classical systems must yield consistent results.

This is a slight variation on the Principle of Relativity.  If this principle is correct, then it must be that M(A) = M(B).  That is, either the devices both measure non-zero mechanical work (in which case there’s something wrong with Observer One’s assumptions) or they both measure zero mechanical work (in which case there’s something wrong with Observer Two’s assumptions).  In other words, classically, the measurements of mechanical work represent a state of reality independent of the Observer’s knowledge of the system.  As such, the thermodynamic identity can be interpreted to imply that

[knowledge] = [reality].

Now, this can be interpreted in two ways. The usual Bayesian interpretation (or what passes for it), at least when applied to classical situations, would say that reality should inform our knowledge, i.e. with new information, we can update our knowledge (in other words, the equal sign is really directional, in a way).  On the other hand, it is possible to interpret this as implying that by changing our knowledge of the system we can change reality!  While this may be true for certain quantum systems, it nevertheless implies that, for instance, prior to our discovery of the existence of Antarctica, it didn’t exist which is absurd!

I prefer to interpret entropy as a measure of the number of possible configurations of a system.  This removes the ambiguity in the classical case and it still works in the quantum case.  In quantum situations we assume there is some subjectivity to the measurements (in fact there is in some classical cases as well).  But at the quantum level, we become part of the system.  As such, there are degrees of freedom (configurations) that the act of measurement can introduce.  These degrees of freedom are minor in classical systems, but not so in quantum systems.  Nevertheless, they are always still there.

So, yes, to some extent, we do “make” reality, i.e. it is a “participatory universe”, as Wheeler has suggested, just not quite in the way that everyone assumes.  Or, another way of looking at it is to say that, if we apply Ockham’s razor to all the possible interpretations, this is the simplest and most consistent.  Are there still issues with it?  Sure.  No theory is perfect.  But until someone comes up with something more consistent, I’m sticking with this one.

## Physics meets baseball on the mound: the physicist-pitchers

Posted in Uncategorized on May 19, 2012 by quantummoxie

Over the course of the six years of this blog, my most popular post continues to be – year after year – this post about the physics of baseball.  I love baseball.  It’s a great, very nuanced game.  Over the past few years I have also been lucky enough to have two baseball players as physics majors.  Both graduated today, both were/are pitchers (Bret Bartlett “retired” last year), and both took my quantum mechanics class this semester (and did very well – both have good, inquisitive minds though, much to my dismay, neither is heading straight to graduate school).

Today was unique, however.  Neil Hesek, our one still active physicist-pitcher, could not attend the regular graduation – along with 6 of his fellow seniors – because the Saint Anselm baseball team was, for only the second time in history, in the NCAA Division II playoffs and had to play a game smack in the middle of graduation.  Neil wasn’t pitching today, though he was the winning pitcher last night as the Hawks beat cross-town rival Southern New Hampshire University.  I watched him pitch a gem against another cross-town rival – and nationally ranked – Franklin Pierce University a few weeks ago (he pitches a mean splitfinger fastball).

Sadly, they were eliminated from the tournament today at the hands of LeMoyne (home of fellow FQXier and Quantum Times editorial board member David Craig).  Nevertheless, they went further into the post-season than any previous Hawks team.  Both Neil and Bret were a pleasure to have in class over the years (I taught them a number of courses and Neil was also my advisee) and I wish them luck in their future endeavors (and I hope at least one of them decides to give grad school a try down the road!).

As a final note, we had three other terrific graduates today as well from our department: Michael Sheridan, Tim Moreau, and Chris Benoit.  It was one of our strongest classes in awhile.  Good luck to all!

## The downside of bibliometrics: let’s reinvent the wheel – and patent it!

Posted in Uncategorized on May 16, 2012 by quantummoxie

Bibliometrics such as the h-index and g-index are often used to judge the quality of a researcher’s work.  They are often used in the tenure review process, to help evaluate grant applications, and to promote entire academic departments (side note: I really like the folks at Albany so I mean them no disrespect – they really are a quality department).  But here’s where the danger comes in.  As absolutely hysterical as it sounds, some doctor at St. Luke’s in New York has reinvented calculus and named it after herself.  What’s worse, is that this article, according to Google scholar, has received 161 citations!  That certainly bodes well for Dr. Tai’s h-index.

Incidentally, pre-med students at my own institution are not required to take calculus (which is why we, in physics, are required to offer a separate, non-calculus-based introductory physics course).  It doesn’t make me confident in the medical community (not that I was to begin with).

Hey, while we’re at it, let’s reinvent the wheel – and patent it!