311. Video of the day: LHC and search for the Higgs
Save This Post as PDFA rather old promo video about LHC, which, I think, again may be considered interesting in light of recent news from CDF/D0 regarding constraints on the Higgs mass (no Higgs with a mass between 160 and 170 GeV, which leaves the only possibility that it is rather light albeit heavier than 114 GeV). Video features Lisa Randall 
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310. Turbulence. Statistical approach 1
Let me get back again to one of my most favourite topics in physics, that is, to developed turbulence. Last time (oh my, mid February) I have tried to explain what I consider the most important (and probably hard-to-solve) open problems in physics of turbulence. Now let me list quickly several (not too promising ) approaches to those problems we were able to develop during last hundred years or so.
It would be most natural to start by discussing statistical approach to developed turbulence.
On the Fig: 2D turbulence highlighted by laser beam. Image by I. Afanassiev (U. of Newfoundland)
1. Reynolds averaging. Correlation functions of Eulerian velocities
The first person who started thinking about turbulence as stochastic phenomenon was probably Reynolds (back in 1880s he studied transition from laminar, regular, motion of water in a pipe to turbulent motion). Since the turbulent flow is essentially stochastic, he said, one needs to deal with averaged quantities.
What do we mean by averaging? It actually depends on a particular problem we are dealing with or a question we want to answer. For example, one can average over fast pulsations of the flow or over small linear scales.
Anyway, Reynolds has managed to write down a set of equations
,
where 
is averaged (in the sense above) velocity of the flow, 
(
is viscosity) and 
is so called Reynolds tensor (here 
is velocity of the flow before averaging). The latter satisfies the equations
.
Reynolds averaging is not self-consistent, since the number of variables is larger than the number of equations. For example, if you want to find the equation for the Reynolds tensor 
, you may try to multiply eq. for by 
and average subsequently, but this will just introduce higher order tensor 
, so you’ll need to find a equation for that one etc. etc. In overall, the procedure reminds the one we perform deriving BBGKY hierarchy of equations in kinetics, with only difference that it is absolutely unclear where to truncate hierarchy in the case of developed turbulence. Still, Reynolds equations turn out to be useful when we want to study, say, averaged characteristics of the flow in pipes.
2. Transport properties of turbulent flow. Correlation functions of Lagrangian velocities
It seems that one of the most important properties of turbulence is the ability of turbulent flow to transfer matter density, energy and momentum fast compared to a laminar flow.
Initially, to describe transport properties of the turbulent flow, people used simplified models based on analogies with slow laminar flows. For example, Boussinesq has related Reynolds tensor with averaged velocity of the flow in the case when fluctuations of velocity are present only along a direction 
perpendicular to the vector itself. Roughly, in this case
.
The quantity 
is called turbulent viscosity (here 
is characteristic linear scale of the flow). Usually, 
by orders of magnitude in turbulent flows.
Talking about transport properties of turbulent flows, I cannot help mentioning work by Taylor, who has studied transport of Lagrangian markers in turbulent flow and found that
,
where 
is a Lagrangian marker (this fancy term just means that you pick a liquid particle, which is located at the position 
at 
and follow its motion with time) and the diffusion coefficient 
is given by
,
i.e., by the correlation function of Lagrange velocity. This expression basically shows how, similar to description in terms of correlation functions of Eulerian velocities introduced by Reynolds, one can describe physics of turbulence in terms of correlation functions of Lagrangian velocities. Of course, this description is also incomplete and we again get a hierarchy of equations for correlation functions, which is unclear how to close in the case of developed turbulence.
That is it for now, folks… Next time I am going to return to the discussion of Kolmogorov spectra (probably, the most beautiful result found by applying statistical approach) and cover briefly the issue of intermittency.
Further reading
1. L. Landau, E. Lifshitz, Fluid mechanics
2. Monin, Yaglom, Statistical fluid mechanics (both Vol. 1 and 2)
3. Frost (Ed.). Handbook of turbulence. Fundamentals and applications
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309. Video of the day: Quantum revolution
from BBC “Visions of the future”. Kaku is trying to argue that humankind is at a turning point to a period in which we will cease to be passive observers of Nature. Yessss, we all will be like Dr. Manhattan
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308. Dark matter via many copies of the Standard Model
The authors of this post are Alex Vikman and Iggy Sawicki. Alex is an old friend of mine (actually, a long time ago we studied English together in the same group at MIPT and even shared the same table ). After receiving his PhD from Munich U. (his advisor was Slava Mukhanov), Alex joined NYU as a postdoc. There he mostly works with Gia Dvali. Iggy is also postdoc at NYU. On the his path of scientific career he had a chance to work with Sean Carroll and Wayne Hu. Dmitry.
We would like to thank Dmitry for giving us the opportunity to blog about our latest paper (arXiv:0903.0660), “Dark Matter via Many Copies of the Standard Model” that we have written together with Gia Dvali.
In this paper, we propose that large number, 
, of copies of the standard model (“SM”) can be naturally responsible for dark matter (“DM”) . The key idea is that all the baryonic matter in the universe is a member of the same SM copy, while DM is made of protons and antiprotons which are members of all the other copies.
Image by FreakingNews.com
In particular the same number of weakly coupled copies of SM can address both the hierarchy problem and explain the existence and the observed abundance of dark matter.
As is usual, we produce the particle content at the end of inflation: we endow each copy of the SM with its own inflaton. One of the inflatons drives the last stages of inflation and, when it decays, it does so mainly to its own SM copy. The density of matter in that copy is large and it quickly thermalises. The thermal history in this sector proceeds in the usual manner with most of the baryons and antibaryons annihilating, with the remainder forming today’s baryonic matter.
However, this inflaton will also have a suppressed coupling to the other SM copies, allowing them to become populated in much smaller numbers. Given a large enough , the other sectors are not dense enough to ever thermalise, annihilation is always frozen out and they provide for an effectively collisionless fluid today that behaves as is required of dark matter. The prediction is that dark matter would be made of protons and antiprotons in other copies of the standard model.
In our model, the ratio of DM and baryon abundances is predicted in terms of measured quantities: the amplitude of the power spectrum of inflationary perturbations, 
, and the photon-to-baryon-number ratio, 
,
(1)
Our idea is motivated by the recent proofs that quantum field theories which contain a large number of species, N , have a new energy scale 
at which gravity in the theory becomes strongly-coupled. It is 
rather than 
that serves as the real cut-off of the theory. This allows us to address the hierarchy problem: in the SM, the mass of the Higgs is naturally expected to be at the cut off, i.e. around the Planck mass, since no symmetries exist to protect it from radiative corrections. However, in a theory with 
, the cut-off is actually at 
and the required weak-scale mass is naturally obtained without fine tuning. Since gravity becomes strong at much lower energies, these large- N models will have an interesting signature at the LHC. Despite this motivation, we leave N as a free parameter is our discussion.
One of the implications of the existence of this cut off is the fact that the Hubble parameter is also bound, 
, which prevents the usual chaotic-inflationary mechanism from producing a sufficiently large amplitude for cosmological perturbations. It is known that having a large number of inflatons does not change the amplitude of perturbations in first order in slow roll approximation. Thus, for theories with 
, we need an alternative mechanism for the generation of inflationary perturbations.
A simple solution is provided by modulated reheating. One introduces the modulator 
, a new scalar field, light during inflation. The modulator is common to all the species and is present in all the leading-order couplings between the inflaton and matter. Its vacuum expectation value (“VEV”) controls the decay rate of the inflaton and, therefore, the position of the reheating surface. Since it will obtain perturbations during inflation, the reheating surface will also be perturbed, resulting in density perturbations after reheating.
In particular, we assume that reheating be dominated by the following interactions
where 
is the inflaton and 
are quarks and 
and 
are dimensionless couplings. In our paper, we show that the amplitude of perturbations after reheating ends is related to the VEV of as
where 
is the Hubble parameter during inflation and therefore the VEV of the modulator is fixed by the observed amplitude. We will simplify our discussion henceforth by assuming that inflation occur around the cut-off of the theory, i.e. 
.
Another universal implication of large- theories is the suppression of couplings between different species required by the self-consistency of the theory. For example, if there exists a cross-species coupling 
, then we can construct annihilation diagrams for species 
into species 
mediated by loops of all the other species. Since the species index on each loop is free, such amplitudes will involve summations over all the species, and therefore will be relatively enhanced compared to one-species theories. In particular, diagrams with 
consecutive loops are possible, and will scale as 
. This implies that 
.
We can apply this sort of argument to the values of our couplings and : for , for example, consider a diagram for annihilation of quarks to 
mediated by a virtual (with an additional two external inflaton legs). We can integrate the modulator out and obtain an effective cross-species six-point coupling proportional to 
. We can then apply the loop argumentation presented above to find that the coupling is constrained to be 
. A similar argument can be made for , except here we can take diagrams, each mediated by a different species of virtual inflaton in the presence of a non-zero VEV of . This constrains the coupling to be 
We assume that these couplings take their most natural values, the largest for which the theory remains consistent, i.e. ones saturating the above bounds. This then allows us to predict the ratio of densities of matter in the each of the dark species, 
, to that in our species, 
, immediately after reheating,
As is evident from the above, the total density in the dark sector summer over all the species is independent of .
The final element of the calculation is provided by the fact that, in our sector, the plasma thermalises and mostly annihilates, with only a fraction 
of the baryons surviving. In the paper, we show that given 
the annihilation in the other sectors is always frozen out and no depletion of density occurs. This leads to our final result that the ratio of the dark matter to baryonic abundance is given by equation (1) given that the number of species lies between 
.
In summary, we have shown that, in models with a large number of species, we can naturally reproduce the observed dark-matter abundance. The parameters of such models are constrained by the requirement for perturbative unitarity: the effective coupling between species must be suppressed in such a way that amplitudes with loops of species do not diverge. Saturating these constraints, coupled with a modulated-reheating mechanism for generating cosmological perturbations during inflation allows the inflaton to decay to our copy of the standard model (which we interpret as baryonic matter) and equally to the other copies. Since interaction between species are required to be suppressed, the resulting fluid is collisionless, replicating the properties of cold dark matter.
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307. Scientist’s gadgets: Tablet PC and handwriting formulae recognition
As usual, on Sunday I’ll try to feed you with as less technicalities as possible but still make the post useful for you… Today the subject of our discussion will be related to Tablet PCs and functionality they offer to a scientist (in particular, to a theoretical physicist).
As far as I remember, Steven Jobs once said that Tablet PC (and especially Windows XP Tablet PC Edition) is a joke and nobody really wants their computers to support handwriting recognition (“Who wants the stylus? Ehrrh” Sorry, Steven, I do ). At that time he was probably right, especially regarding Windows XP. But nowadays, I think, Tablet PCs matured quite a bit, handwriting recognition support is built directly into Windows Vista – and this support is quite good as I can assure you based on my personal experience. Actually, I find that Tablet PC did become an ultimate scientist’s gadget. But before explaining why do I think so, let me first give you my personal answer to the question
0. Which one?
Although I am very well aware how much you love Apple, guys , I am afraid currently no Apple product supports handwriting recognition and related functionality (although rumours are circulating that Apple might eventually come out with Tablet PC). So, yes, you’ll have to kiss Bill Gates’ glasses to enjoy handwriting recognition support.
I am writing this post on Acer TravelMate C210 bought for me almost two years ago by the the Helsinki Institute of Physics (hey, guys, that’s another good reason to apply here, especially to Kari Enqvist’s group, if you are currently looking for a job – HIP is really well supported).
What can I say? Acer is terrific and reliable, it worked without a hitch for 2 years and hardware of C210 supported everything I wanted. Formfactor of C210 is especially nice which you’ll feel immediately as soon as you take this fellow to a flight in economy class with you.
Nevertheless… Tablet PC of the first choice for me was Fujitsu Siemens P1510, simply because it was so much smaller than Acer and the battery life was much better (8 hours, as far as I remember). Unfortunately, a single copy of any FS tablet pc was sold out in Helsinki at the time we were ready to make a purchase (I am not kidding), and hard decision had to be made. If I would have an option to buy a new Tablet PC today, Fujitsu Siemens would probably again be my first choice – a newer model of course, say, P1620.
So, what are Tablet PCs good for and why do I think the one’s purchase can be worth considering for a theoretical physicist?
1. Making notes
Have you ever had that unpleasant feeling after relocating to a place of your next postdoc position that you forgot to pick up some really important notes in your old office? Or that you lost a piece of paper containing some truly nice calculation (and somehow it’s hard for you to reproduce it)? Well, with Tablet PC you don’t need to worry about that staff anymore since you go completely digital. All your notes are made on your Tablet and stored there forever – your handwriting can be recognized, indexed and searchable.
As for note-taking software, currently, there are essentially two options on the market: Agilix GoBinder and MS Office OneNote (I am talking about ver. 2007). Sorry to say, but from my point of view Bill Gates rules and the latter way outsmarts the former, simply because it supports handwriting recognition, indexes your notes and allows full text search later. It allows many other things such as recording voice notes with later speech-to-text transformation (sic! and it does work even with my thick Russian accent), collaborating with other people in real time (that is, doing whiteboard meetings using OneNote) etc. etc.
Actually, I think that OneNote is the second most useful program in the Office Suite after PowerPoint.
If you are lucky enough, your IT department already has an Enterprise license for MS Office, that is, they will install MS Office to your Tablet for free (as it was in my case), but if you are unlucky – OneNote is included into MS Office Student Edition which costs about 90$.
Currently, all my notes (and not only science!) are getting written and saved in OneNote format.
2. Annotating papers from Arxiv
This is very convenient. To make annotating of downloaded PDF papers possible, you need either Adobe Acrobat Standard (or Professional) or FoxIt Reader. The latter is free and allows highlighting text in the PDF file – the way you do it by a marker on the hard copy of the paper (with your Tablet pen this is not just easy-peasy, this is fun). You do want to go for the former though, since Adobe Acrobat supports PDFs natively and handles them so much better than FoxIt. Acrobat does cost quite a bit, so ask your IT department guy if they have an enterprise license for Acrobat and can install it to your Tablet for free.
3. Online whiteboard session
If you want to chat with a friend or colleague and exchange some formulae during the conversation, there are three options I know of.
a) Create a notebook in MS OneNote and share it with your friend. Use phone or Skype to chat with her/him in the mean time This option can be problematic, if your IT department firewalls ports that OneNote uses to share notebooks.
b) Use MS Messenger Yes, that awful green shiny-looking crappy analogue of Skype The point is that it does natively support handwriting recognition during conversations. Personally, the best option ever – you can do real time collaborating online with collaborators overseas.
c) Use Skype Whiteboard Meeting support. I am going to write about it in details in one of the next “Scientist’s gadgets” posts.
4. Finally, the most interesting part – formulae handwriting recognition
Well, recognition of handwritten text is fine and fun, but what we really and ultimately want is recognition of handwritten formulae, don’t we? Would formulae handwriting recognition be supported in our Tablet PC (say, you can export written formula to TeX), it would really make our life and writing papers a bit easier.
That’s where Microsoft falls short. The reason, I guess, that technically formulae handwriting recognition is so much more complicated (roughly, 2 dim problem) than the recognition of handwritten text (roughly, 1 dim problem). Anyway, as it turns out, there exists a Japanese group (the leader is Masakazu Suzuki (Kyushu University)) working on handwriting formulae recognition for decades as well as OCR of scanned images of scientific papers. The group was able to come out with a real product, which is even free.
There are three products actually in the Infty suite. One is called InftyEditor (the last version is 3.06 released on the end of Dec 2008). The program supports formulae handwriting with subsequent export of your writing to LaTeX.
The second product is InftyReader (the last version is 2.7.9, dated Dec 2008 as well). This one supports recognition of scanned scientific papers and export of images into LaTeX, which is also quite useful (maybe, not directly of us, but for, say, libraries).
Finally, the third product is ChattyInfty that has a text-to-speech support of LaTeX source files (i.e., this fellow will read your LaTeX source files for you in the way which is understandable).
Among those three, InftyEditor is certainly the most interesting and useful one for me. My experience with it wasn’t smooth at all – while writing a formula, one should be really as accurate and precise as possible, since the hit/miss rate of the recognition software is terrible otherwise. InftyEditor (especially, its latest versions) does work though – in comparison, say, to redundant handwriting recognition support built into the latest version of Maple. I really hope that the Suzuki group will continue developing this extremely nice piece of software…
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