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Sign problem in QCD

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The revised version of our paper 0808.2987 is up on arXiv now. Special thanks to Kim Splittorff, Mark Alford, Bob Sugar, Phillippe de Forcrand and many others for comments. See earlier discussion.

On the sign problem in dense QCD

S. Hsu and D. Reeb

We investigate the Euclidean path integral formulation of QCD at finite baryon density. We show that the partition function Z can be written as the difference between two sums Z+ and Z-, each of which defines a partition function with positive weights. If the ratio Z-/Z+ is nonzero in the infinite volume limit the sign problem is said to be severe. This occurs only if, and generically always if, the associated free energy densities F+ and F- are equal in this limit. In an earlier version of this paper we conjectured that F- is bigger than F+ in some regions of the QCD phase diagram, leading to domination by Z+. However, we present evidence here that the sign problem may be severe at almost all points in the phase diagram, except in special cases like exactly zero chemical potential (ordinary QCD), which requires a particular order of limits, or at exactly zero temperature and small chemical potential. Finally, we describe a Monte Carlo technique to simulate finite-density QCD in regions where Z-/Z+ is small.

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October 2, 2008 at 6:41 pm

Posted in physics, qcd

Survivor: theoretical physics

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Some very interesting data here on jobs in particle theory, cosmology, string theory and gravity over the last 15 years in the US (1994 — present).

Based on these numbers and the quality of the talent pool I would guess theoretical physics is the most competitive field in academia, by a large margin. (Your luck will be much, much better in computer science, engineering, biology, …)

The average number of years between completing the PhD and first faculty job is between 5-6. That would make the typical new assistant professor about 33, and almost 40 by the time they receive tenure.

Here are the top schools for producing professors in these fields:

1. Princeton 23 (string theory rules! or ruled… or something)
2. Harvard 18
3. Berkeley 16

This is over 15 years, so that means even at the top three schools only 1 or at most 2 PhDs from a given year typically gets a job in the US. The US is by far the most competitive market. If you follow the link you will see that the list of PhD institutions of US faculty members is truly international, including Tokyo, Berlin, Moscow, etc. (Note I think the jobs data also includes positions at Canadian research universities.)

The field is very much dominated by the top departments; the next most successful include MIT, Stanford, Caltech, Chicago, etc.

Here are some well-known schools that only produced 1 professor of theoretical physics over 15 years: UCLA, UC Davis, U Illinois, U Virginia, U Arizona, Boston University, U Penn, Northwestern, Moscow State University (top university in USSR), Insitute for Nuclear Research (INR) Moscow

Here are some well-known schools that only produced 2 professors over 15 years: Ohio State, U Minnesota, Michigan State, U Colorado, Brown

Here are some well-known schools that only produced 3 professors over 15 years: Columbia, CERN, Johns Hopkins, U Maryland, Yale, Pisa SNS (Scoula Normale Superiore; the most elite university in Italy), Novisibirsk (giant physics lab in USSR)

You can see that by the time we reach 3 professors produced over 15 years we are talking about very, very good physics departments. Even many of the schools in the 1 and 2 category are extremely good. These schools have all hired multiple professors over 15 years, but the people hired tend to have been produced by the very top departments. The flow is from the top down.

This dataset describes a very big talent pool — I would guess that a top 50 department (in the world) produces 3-5 PhDs a year in theoretical physics. If most of them only place a student every 5 years or so, that means the majority of their students end up doing something else!

How many professors do you think are / were straight with their PhD students about the odds of survival?

I only knew one professor at Berkeley who had kept records and knew the odds. One day in the theory lounge at LBNL Mahiko Suzuki (PhD, University of Tokyo) told me and some other shocked grad students and postdocs that about 1 in 4 theory PhDs from Berkeley would get permanent positions. His estimate was remarkably accurate.

How many professors do you think had / have a serious discussion with their students about alternative career paths?

How many have even a vague understanding of what the vast majority of their former students do in finance, silicon valley, …?

Related posts: A tale of two geeks , Out on the tail

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September 24, 2008 at 4:17 pm

Palin, RNC, Romney: the view from Italy

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I didn’t get to see any of the coverage of the RNC since I’m here in Italy.

Regarding Palin, there is a natural instability in democracy towards anti-elitism. Many voters are attracted to a leader like themselves (a hockey / soccer mom with dysfunctional family and modest IQ), forgetting that they themselves would make a terrible president or vice-president. I do think the Republican base will like / likes Palin, and there is a chance she will appeal to lot of swing voters.

I don’t know who said it first, but Palin has that naughty librarian look from 80’s heavy metal videos! If you are not familiar with the term MILF, you might look it up — only because it’s being used in a lot of discussion 🙂

Is Romney the favorite for 2012? I’ve heard a lot of good things about him, but his RNC speech is pretty thoroughly middlebrow. Not that I disagree with every point, and certainly he had to tailor it to his audience, but I detect no signs of a large brain (unless you normalize to the MBA population).

I know few Americans care, but people in Europe think Palin is a joke. Another thing I’ve heard is that they don’t believe Obama can overcome all the (perhaps hidden) racism to win the election. We will see!

Exciting action photo from Trento:

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September 4, 2008 at 10:08 am

Posted in obama, physics, politics

ECT fun

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26 hours of travel yesterday (EUG-PDX-FRA-VCE + train) but so far I feel pretty good, even after seminars from 9:30 AM until 6 PM 🙂 Trento is a beautiful mountain town — I will try to post more photos.

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September 1, 2008 at 4:21 pm

Posted in conferences, physics, trento

A warning from von Neumann

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I can’t resist reproducing this quote from John von Neumann, which I think applies well to certain branches of particle theory today. Thank goodness the LHC is coming on line soon…

As a mathematical discipline travels far from its empirical source… it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. …In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration.

From the opening material of the book John von Neumann and Modern Economics. If you can get a copy of this book, I highly recommend the chapter by Paul Samuelson.

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August 28, 2008 at 9:24 pm

ECT in Trento

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I’m off soon to the following meeting at ECT: European Center for Theoretical Studies in Nuclear Physics, which is located in the mountain town of Trento, in the Italian Alps. The picture above was taken just north of Trento. I’m excited to see the dolomiti!

I’m flying in and out of Venice — any tips on what to do there or in Trento would be appreciated 🙂

Meeting: The statistical model of hadron formation and the nature of the QCD hadronization process

program , poster , slides of my talk.

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August 28, 2008 at 4:38 pm

Posted in dolomites, physics, qcd, trento

Dense nuclear matter: intuition fails!

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I usually don’t get into detailed physics exposition on this blog, but I thought I would make an exception with regard to the paper 0808.2987 which I recently wrote with my student David Reeb. (See earlier blog post here.)

In the paper we conjectured that there might be regions in the QCD phase diagram where the sign problem does not prevent monte carlo evaluation of the Euclidean functional integral. We rewrite the partition function as

Z = Z+ – Z-

where Z+ and Z- are sums with positive weights, and each define independent statistical ensembles. Defining Z+ = exp( – V F+ ), and similarly with Z-, so that F+ and F- are the (piecewise) analytic free energies of the two ensembles, we conjectured that

F+ < F-

is the generic situation. Note Z > 0 so F+ > F- is not possible, but they can be exactly equal: F+ = F- , which is where the sign problem is most severe (see below). Since the F’s are analytic except at phase boundaries, we reasoned that if they are equal in a region they must be equal everywhere within that phase region. At mu = 0 we know Z- = 0, so we assumed that there would be a region of small mu where F+ < F- and that this region would extend into the mu-T plane.

It turns out this last assumption is probably wrong! We were unaware of results which strongly suggest that even at arbitrarily small (but positive) mu and small T, Z+ does not dominate Z. That is, in the thermodynamic limit F+ = F- exactly even at small nonzero mu. The order of limits matters: taking V to infinity for fixed nonzero mu (no matter how small) leads to large phase fluctuations. The only way to avoid it is to take mu to zero before taking V to infinity. (See 0709.2218 by Splittorff and Verbaaschot for more details. Note their results rely on chiral perturbation theory, so don’t apply to the whole plane.)

It is quite strange to me that zero density QCD can only be reached in this way. The case we are most familiar with turns out to be the oddball.

To make a long story short, our conjecture is probably incorrect: what we thought would be “exceptional” regions in the phase diagram are the typical ones, and vice versa — at least as far as anyone knows.

Note to experts: we used the term “sign problem” a bit differently than apparently it is used in the lattice community. We refer to dense QCD as having a sign problem even though we don’t know for sure (i.e., for all mu and T) that Z is exponentially small in V due to cancellations (i.e., a “severe sign problem”). Our usage probably translates to “potential sign problem” — the functional measure isn’t positive, so potentially such cancellations can occur, although we do not know if in fact they do. We got a lot of emails from people who thought we were claiming to have a method for dealing with severe sign problems, but in fact we were claiming something else entirely: that there should be regions in the phase diagram where the sign problem is not severe.

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August 27, 2008 at 2:35 pm