NEQNET: Non-equilibrium Phenomena

Cosmology, turbulence, markets, non-equilibrium QFT and much more. No nonsense, just science

On gun politics and culture in US

In Russia, any talk about personal weapons/gun policy gets immediately reduced to the question of how actually effective are guns for personal self-defence on the street. On the other hand, in US general attractor seems to be discussion of the statement that personal weapons is your defence against tyranny, i.e., “armed man=free man” etc. etc.

Personally, I think that the statement is badly flawed (and, ironically, the very reason why people from US cannot immediately figure this out is that US is a free country). After Civil War of 1920s in USSR population had lots and lots of guns in their hands: legal and illegal, gratuity and stolen, etc. Did all those weapons make people free from tyranny of 1930s-…? Not really.

Geoff here have mentioned Solzhenitsyn – I think, it would be instructive to recall one interesting example. Solzhenitsyn once wrote that it would be great if all interested persons started laying ambushes against NKVD officers during Great Purge in Leningrad. That’s a creative thought, however, it is known that Solzhenitsyn himself was arrested on the front line. He was an active officer, he had an authorized gun(s) in his hand, his soldiers would follow his order, and finally he was well aware what would arrest mean for people like him. Yet…

Objective reality is that personal weapons cannot be really used against authorities What stays on the first place is Organization, large and powerful, cemented by discipline. Single revolted shooters are easily overridden. Why? Well, because Organization’s is always longer than yours! That is, you have a gun – they have machine guns, you have a machine gun – they have rocket launchers, you have a sniper rifle, they have tanks.

This is the very basis of contemporary America. Gangs have lots and lots of arms, but police and feds have much more.

Note that the opposite situation is not impossible -take for example Columbia

Via Boris Ivanov.

Average life expectancy or more on data visualization

In continuation of my series of posts about data visualization (Dynamical maps and Gapminder), let me show you today another cool resource: interactive map of the world StatPlanet.

Here is for example a map showing average age in different countries (data from 2006):

Russia seems to be rather mature country on this map – average age is about 37 years, even higher than in US. Nice? Not quite. Here is another map – a trickier one. What’s shown here is the total number of 15-year-old people who are expected to die before achieving their 60s (this number is calculated using current known mortality rates).

The world record belongs to Lesoto (72.2%), while the rate for Russia is 30% and, surprisingly, 29.6% for Sudan. That is, in this respect the situation in Russia is worse than in Sudan.

The data are collected by WHO (World Health Organization) and published on their site, in a somewhat tricky format though – Adult mortality rate (probability of dying between 15 to 60 years per 1000 population) both sexes.

Now it would be nice to compare the first map with the second and, say, determine percentage of population to die out before 2050 (the higher is average age of the population and the higher is the mortality rate, the larger is this percentage), but unfortunately StatPlanet does not allow this.

Via Sergey Schegloff.

Quantum tunneling in flux compactifications

Delia Schwartz-Perlov is a postdoc at Tufts U. working with Alex Vilenkin. Her interests include quantum field theory, string theory, general relativity and cosmology. Dmitry.

I am very happy to find myself writing a blog about a recent paper written by Jose Juan Blanco-Pillado, Alex Vilenkin and myself, and titled “Quantum tunneling in flux compactifications“. In this paper we studied bubble nucleation rates in a 6-dimensional Einstein-Maxwell theory. The two extra dimensions are compactified into a 2-sphere, and their radius is stabilized by a magnetic flux through that sphere. We picked this toy model because it is simple enough to allow a quantitative analysis, yet it also includes some of the essential features of string theory compactifications (a related paper by Sean Carroll, Matthew Johnson and Lisa Randall was posted on the same day as ours!).

But why do we care about bubble nucleations, extra dimensional theories and compactification? To help set the mood, let’s come down to earth and think about a vanilla galaxy. Mmm, nice. How about the Hubble deep field? Mmmm – very nice! We can see thousands of galaxies effortlessly suspended in space billions of light years away. The universe we see is undeniably beautiful and incomprehensibly vast. But still, we can’t help but notice that our laws of physics are telling us that there may be more to the story – infinitely more!

For over two decades eternal inflation has been telling us that our entire observable universe is just a small part of an infinitely large universe (that was spawned some 13.7 billion years ago in the Big Bang), which itself is only one out of an infinite number of other universes (each the product of their own “local” big bang). Here’s the thing: once inflation starts it never ends! To illustrate this, let’s think about a simple model with a scalar field potential that has two metastable de Sitter minima separated by a barrier. Let vacuum A have a larger cosmological constant than vacuum B. If the universe starts in vacuum A, bubbles of vacuum B can nucleate and begin to expand at a speed approaching that of light within A. However, vacuum A is itself expanding, always leaving room for new bubbles to form. Furthermore, since B also has a positive vacuum energy, it can itself become a parent vacuum to type A bubbles. This simple recycling universe is an example of a “multiverse” which gets populated by the two possible vacua in the theory. This idea generalizes to theories with many different vacua – all possible types of bubbles are nucleated one within the other in an everlasting effervescent cosmic extravaganza!

More recently String theory has echoed the same sentiment, suggesting the existence of a multitude of vacua characterized by different values of the low-energy constants of Nature. String theory (currently our best candidate for a quantum theory of gravity) demands that we consider  or  dimensional spacetimes instead of the mundane  that we’re used to! The idea of extra dimensions predates string theory, going back roughly a century to Kaluza-Klein theory, which attempted to unify gravity and electromagnetism by considering a  gravity theory which precipitates electromagnetism in a reduced  perspective. At any rate, it does seem as though we live in a  4d world, so where do all the extra dimensions go? This is where the idea of compactification comes in. Physicists have been able to show that if we start with a higher dimensional world, some of the extra dimensions can be “compactified” so that we don’t “experience” them directly (although what’s going on in those compactified dimensions does influence our effective  reality).

It turns out that there are many ways to compactify extra dimensions. In string theory, the role of scalar fields is played by the moduli that characterize the sizes and other geometric aspects of these extra dimensions. String theory vacua also involve additional objects, such as fluxes and branes. There are so many ways to combine these ingredients (we’re talking numbers in the googols here!) to produce different vacua, that we land up with a “string landscape” of possible vacuum solutions. When the string landscape is combined with inflationary cosmology, the picture of an eternally inflating “multiverse”, populated by all possible types of vacua comes into sharper focus. The calculation of bubble nucleation rates is an essential part of the irresistible task of quantitatively understanding the multiverse and it’s evolution.

Plot of the effective potential, in units, as a function of the modulus field . We show the potential for 3 different values of the flux quantum .

So now that we have motivated why we study bubble nucleation rates, let’s get back to the  Einstein-Maxwell model we investigated in the paper. Our action included a  cosmological constant term and we assumed that a 2-form magnetic flux permeates the extra dimensional  sphere. For our metric ansatz we assumed a maximally symmetric  [/tex] Riemannian manifold, and compactified the extra dimensions on a 2-sphere. While the model can be studied directly in  it is easier to see what’s going on when we dimensionally reduce our model so that we have an effective  potential as shown in the figure. Each minimum in the figure corresponds to a metastable vacuum with a given quanta of the Maxwell field flux  . The set of minima with different values of  , constitute a “small” landscape.

We set out to describe “flux tunneling” from a configuration with a given value of  to a neighboring minimum with flux quantum  (upward jumps may also be allowed if the initial vacuum has positive vacuum energy ). We showed that this process of vacuum decay occurs through the nucleation of magnetically charged 2-branes, which look like expanding spherical bubbles in the large 3 spatial dimensions and are localized in the extra 2 dimensions. The vacuum inside the bubble has its extra-dimensional magnetic flux reduced by one unit compared to that of the vacuum outside. We estimated the instanton action corresponding to this flux tunneling nucleation event, and used it to calculate transition rates.

While the effective  potential has stable vacua under small perturbations in the compactification radius (modulus field  ) for any given value of the flux  , we see that it tends to zero for large values of the radius/modulus field  . This in turn means that positive-energy vacua should be able to decay by tunneling through the barrier, leading effectively to decompactification of space. This seems to be a generic situation for four dimensional effective potentials for moduli fields that represent the size of internal manifolds and that are stabilized at non-negative values of the  cosmological constant. We estimated the decay rate of the above vacua towards decompactification and compared it with the flux tunneling decay rates.

We found that for light and extremal branes (extremal branes have a simple relation between their tension and charge), flux tunneling proceeds far more rapidly than decompactification tunneling, while for superheavy branes the two tunneling rates are comparable.

There is another flux compactification sector of our  theory. The existence of this branch of the landscape is more easily understood in the dual picture, where we have a four-form field flux that one could turn on in the four sphere. One can then find solutions of this model with two large spacetime dimensions (having de Sitter, Minkowski, or anti-deSitter geometry) and with the remaining 4 dimensions compactified on a  . We can study tunneling processes between different values of the flux number on the 4-sphere or go to the Maxwell description where the 4-form flux along the internal dimensions gets dualized to an electric field along the large spatial dimension. It is easy to see then that one can understand the tunneling between vacua in this sector as the Schwinger decay of this electric field.

We are currently investigating whether or not there is an instanton that interpolates between the two sectors in this model. This would probably be a more complicated instanton than the ones we have already studied, as it should involve a topology change to be able to interpolate between the different compactification schemes. This is an important point, since the existence of this type of instanton is necessary in order for the multiverse to explore all the sectors of the landscape.

Another interesting area for ongoing research involves bubble collisions. It is usually assumed that when two bubbles of the same vacuum collide, their domain walls annihilate in the vicinity of the collision point, with great energy release, and the two bubbles merge. At late times after the collision, the resulting configuration has the form of two expanding spheres which are joined along a circle of ever expanding radius. In the case of bubbles with different vacua, a similar configuration is formed, but now the colliding walls merge to produce a new wall that separates the two vacua inside the bubbles.

In contrast, branes separating flux vacua in different bubbles are generally localized at different points in the internal manifold and will therefore miss one another in the colliding bubbles. So the branes will not merge or annihilate, and the bubbles will simply propagate into one another, forming a new vacuum in the overlap region. This new type of behavior could have important phenomenological consequences for the observable signatures of bubble collisions.

To summarize: we set out to study bubble nucleation rates in a toy string theory landscape – the  Einstein-Maxwell model. We showed that vacuum decay can occur via the nucleation of magnetically charged 2-branes. From the  viewpoint, these branes look like expanding bubbles which have their magnetic flux on the inside reduced by one unit compared to that on the outside. We calculated the instanton action for this flux tunneling and compared it to the decompactification decay channel.

We also emphasized that the expanding bubbles resulting from flux tunneling are bounded by co-dimension  branes, which are generally localized at different points in the internal dimensions. We expect, therefore, that in bubble collisions, the branes will generally miss one another and the bubbles will continue expanding into each other’s interior, forming a new vacuum in the overlap region. This may have interesting observational implications, which we hope to explore in the future.

References to the literature can be found in our paper: Jose Juan Blanco-Pillado, Delia Schwartz-Perlov and Alex Vilenkin, “Quantum tunneling in flux compactifications” arXiv:0904.3106v1 [hep-th].

Some suggested further reading includes:

(1) the popular book by Alex Vilenkin, Many Worlds in One – the search for other universes.
(2) a Scientific American magazine article by Raphael Bousso and Joseph Polchinski, The string theory landscape, September 2004.

A bound on the speed of sound from holography?

This post is authored by Aleksey Cherman (on the left) and Abhinav Nellore (on the right). Aleksey is a graduate student in the nuclear theory group at the University of Maryland, College Park, working with Tom Cohen, and Abhi is a graduate student in Steve Gubser’s group at Princeton. Dmitry.

We all know that sound travels at about 343 m/s in air, and much faster than that in many solids. But just how much faster could sound travel if given the chance?  Could there be a medium in which the speed of sound can approach the speed of light?  Or might there be some more stringent fundamental bound on the speed of sound?

As it happens, the answer to the last question is no.  There is a medium in which the speed of sound  () can approach the speed of light :  a gas of pions (each of mass ) at a finite isospin chemical potential .  When , but is small compared to , it is possible to calculate the speed of sound using using chiral perturbation theory, and it is not hard to show that it approaches the speed of light in the chiral limit of going to zero. (see Son and Stephanov.)

So clearly there’s no general speed limit on the speed of sound for all consistent field theories.  But let’s lower our ambitions a bit.  Might there still be some broad class of theories that doesn’t include the counterexample above where there IS an interesting speed limit for the speed of sound?

As it turns out, the answer to that question is yes!  But conditionally: we have to lower our ambitions still further.  Working with Tom Cohen, we have been able to show is that in an a certain class of strongly coupled systems, must approach from BELOW at high temperatures.  That is, at least in this class of theories, and at least at high temperatures, there is indeed an interesting speed limit for sound.  We do not yet know whether the speed limit applies away from the high temperature limit – that’s a subject for future work.

(We’re not the only ones who have worked on this: Paul Hohler and Misha Stephanov independently got the same results we did using different methods, and our papers appeared simultaneously. )

To calculate the speed of sound, we used the gauge/gravity duality. The gauge/gravity duality is a marvellous tool:  it lets us calculate observables in some strongly coupled gauge theories just by doing classical calculations in theories with gravity. But as with most good things in life, there is a catch: there are no known gravity duals for any of the gauge theories currently used to describe nature.  This means that the duality can’t be used to make quantitative predictions for nature:  at best, one can hope to learn some interesting qualitative lessons about the behavior of strongly-coupled systems.

In our work, we considered 3+1 dimensional systems that have gravity duals with a single scalar field .   These systems can be thought of as strongly-coupled large N conformal field theories deformed by the addition of a single relevant operator , which is dual to the bulk scalar .    These single-scalar gravity theories are the simplest non-conformal gravity duals.  Different choices of potentials for the bulk scalar field correspond to different dual gauge theories.

In a 4D conformal field theory, the speed of sound is simply a constant:  , as can be seen from the general fact that ( and are the pressure and energy density of a system), and the fact that the trace of the stress-energy tensor vanishes in a CFT, so that .  In a non-conformal field theory, the speed of sound depends on the temperature and other properties of the theory.   The theories we  decided to work with are relevant deformations of a CFT, so they should look like CFTs in the far UV.   This means that at high temperatures, the should approach 1/3.  What is not so obvious is whether approaches 1/3 from above or below:  that is, is the 1/3 an upper bound?

To answer this question, we developed a high-temperature expansion for the geometry and the profile of the bulk scalar.  At very high temperatures, the geometry should look like an Schwarzschild black hole, with a vanishing scalar field – that is, the system should become approximately conformal.  Thus, our high-temperature expansion is an expansion around an Schwarzschild geometry.  It turns out that the leading correction to the geometry is almost completely insensitive to the details of the scalar potential: one only needs to know the UV scaling dimension of of .  Note that , where the upper bound is due to the restriction to relevant operators, and we use the lower bound to avoid a subtlety involving the BF bound for stability of scalars in spaces (this does not affect our conclusions).

Once we found a way to calculate the high-temperature geometry, calculating the speed of sound was easy.  In systems at zero chemical potential, the speed of sound can also be written as  , where is the entropy.  The entropy and temperature can be read off from the geometry, and we find that in the high temperature limit

where

Clearly, the correction away from 1/3 is always negative for in the allowed range.  So for all systems in the class that we considered, is the ultimate speed limit for sound, at least at high temperatures!  It would be very interesting to see whether (and under what circumstances!) this speed limit still holds for lower temperatures.

Well, at this point you might say that this is all very well, but this was in the context of a pretty limited class of theories.  That’s a fair point.   As it happens, our result holds in systems with gravity duals with more than one scalar field as well, as will be discussed in a paper we are now finalizing.

But in fact, to our knowledge,  in all 4D theories with gravity duals, at least when the systems in question are energetically stable (i.e., in a state of lowest free energy).  This is true both in fairly ad-hoc models like the ones we worked with in our paper, and in more sophisticated top-down models using various brane constructions.

So what are the next steps?  First, it makes sense to look for counterexamples, to try to figure out the right domain of validity of the sound bound.   Just how broad (or narrow!) is the class of theories to which it applies?

In a similar vein, it would be great to come up with some general argument that would show whether this kind of speed limit is a general property of holographic theories, or of some interesting subclass of them.  Maybe such a speed limit can tell us something interesting about holography?  If is indeed bounded by 1/3 in all theories with gravity duals, the next obvious thing to look at would be 1/N and finite coupling corrections to the speed of sound, to see how robust the results are away from the supergravity limit where N and the ‘t Hooft coupling are infinite…

For more on all of this, see the two papers below, and the references in them.

1. A. C., T. D. C., and A. N., http://arxiv.org/abs/0905.0903

2. P. M. Hohler and M. Stephanov, http://arxiv.org/abs/0905.0900

The in-in formalism and cosmological perturbations

This post is written by two great guys Peter Adshead (his photo is on the left) and Eugene Lim (on the right). Peter is a PhD student of Richard Easther at Yale U., while Eugene is Richard’s former postdoc now working at Columbia U. Dmitry.

The discovery of the anisotropies in the cosmic microwave background (CMB) by the COBE satellite in 1992 heralded in a new era of cosmology. Instead of simply studying the gross evolution of the universe (usually termed “background evolution”), cosmologists now study the structure and evolution of the tiny perturbations about this background. The anisotropies observed in the CMB are believed to be the seeds from which all of the structure (stars, galaxies etc.) we observe today eventually grew. These perturbations, detected as an average of over/under-density of , are thought to be generated during an early period of accelerated expansion – inflation.

Until recent years, studies of cosmological perturbations have been mainly focused on the linear component – which is not so surprising given its minute amplitude. However, our instruments have advanced to a stage where we can actually begin to probe structure beyond the linear order. On the other hand, the theory of calculating perturbations beyond the linear order is an extremely challenging problem.

Cosmological perturbations are characterized by correlation functions (or moments). Given a field that lives on some spatial foliation, we can describe it by an infinite tower of N-point correlation functions. For example the power spectrum is the 2-point correlation the bispectrum is the 3-pt correlation the trispectrum is the 4-pt correlation , and so on. If these fields have initial amplitudes and phases which are drawn from a Gaussian distribution, then linear evolution (modulo any non-local interactions) will preserve this property. In this case, the statistics of the field will be entirely described by the 2-pt correlation function, or power spectrum. N-pt correlation functions beyond the 2-pt will simply be a power of the 2-pt function for N even (N/2 unconnected copies of the propagator), and will vanish if N is odd. Beyond linear order, however, interactions generally mix the modes, resulting in non-zero higher moments. These are generically called “non-Gaussianities”. Hints of a non-zero bispectrum by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite in the WMAP year 5 data release has generated enormous interest in the calculation of these correlations.

The cosmological perturbations are believed to be generated at very early times during a period inflation. During inflation, quantum fluctuations at very short scales (deep inside the Hubble horizon) are stretched by the accelerated expansion and freeze out well outside the horizon. Linear perturbation theory allows us to track the evolution of these perturbations and compute the power spectrum of fluctuations (scalar fluctuations in the curvature of the spatial hypersurfaces and gravitational waves) that is frozen outside the horizon. Beyond the linear theory, there are two important corrections to this picture: quantum corrections and non-linear couplings between modes. The former will in general correct the amplitude and running (in space) of any spectra we compute using linear theory, while the latter will lead to non-trivial higher correlation functions.

For many years, the calculation of such corrections were based on a hodge-podge of methods which were generally messy and poorly controlled. A breakthrough occurred in 2002 when J. M. Maldacena combined an approach to quantum field theory called the Schwinger-Keldysh “in-in” formalism with the Arnowitt-Deser-Misner (ADM) formulation of Einstein’s general relativity, to exactly calculate the 3-pt correlation function of perturbations predicted by standard slow roll inflation (arXiv:astro-ph/0210603). This formalism was further refined and extended to quantum corrections by S. Weinberg in 2005 (arXiv:hep-th/0506236). As the name “Schwinger-Keldysh” indicates, this method is not new – it was formulated in 1960’s to calculate finite time matrix elements. It has been used by condensed matter physicists and finite temperature field theorists for many years. However, its application to cosmology is limited, groundbreaking papers were written by R. D. Jordan (1986) and E. Calzetta and B. L. Hu (1987), and it was not applied to cosmological perturbations until Maldacena and Weinberg.

The “in-in” formalism is just standard quantum field theory rigged to compute correlation functions instead of transition amplitudes between states (“in-out”). In other words, we want to calculate the expectation value of any number of products of fields at some time t, given initial conditions at an earlier time . Mathematically, one begins with the action of general relativity coupled to a slowly rolling scalar field which is driving the inflationary expansion

The ADM formalism then separates out the dynamical degrees of freedom from the gauge degrees of freedom and gives an action containing only dynamical variables. Maldacena’s insight was to realize that one could express the action entirely in terms of tensor fluctuations (gravitational waves) and, depending on the choice of gauge, either the Bardeen curvature, or fluctuations in the scalar field. This is achieved by solving constraint equations for the auxiliary fields, that appear in the action as Lagrange multipliers, and substituting the solutions back into the action.

The Hamiltonian is then constructed in the usual way, defining canonical momenta, performing the Legendre transformation and eliminating time derivatives of the dynamical quantities in favor of the canonical momenta. One then separates the Hamiltonian into a piece that is quadratic in the fluctuations , and a higher order part and works in an interaction picture. In this scheme, drives the evolution of the operators while drives the evolution of the states. The correlation of an operator, , is then

,

where is the time ordering symbol.

One can then use Feynman-diagram like expansions to calculate the correlation to any order. The power spectrum can be thought of as the propagator and so lowest order results look like trees while corrections appear as loops. The key difference between these diagrams and the standard Feynman diagrams is that there is no time-flow. While each vertex is associated with both a momentum (or space) and a time integration, the propagators carry only 3-momentum.

In our recent paper (arXiv:0904.4207) we clarify and extend Weinberg’s approach (which is markedly different from the original “doubling of fields” approach, see for example Calzetta and Hu (1987)). In particular, we explore a subtlety that occurs in “in-in” calculations. To project out the correct initial conditions, one allows a small amount of evolution in imaginary time in the time integrals appearing above. This can be understood as insisting that the fields start off in the Bunch-Davies vacuum at the initial time. The subtlety is that the operators on the left and right of the expression are conjugates of each other. While the fields themselves are of course Hermitian, the deformation of the integration contour is not. This contour information has to be carefully treated throughout the calculation.

In our paper, we argue that the most efficient method for performing such calculations is to explicitly define contours and carry them around in the time integrals. There are alternative ways of tracking this information, for example one can analytically continue the integration variable to contain the appropriate imaginary piece. However, this prescription allows manipulations which can lead to erroneous results. Specifically, if one uses analytic continuation, the above correlation can be written as

.

This formulation splits diagrams up into pieces which, considered alone, are unphysical (if one compares the expressions at second order, the term has been split into two pieces). If one is not careful to keep track of the imaginary pieces of the various time variables, one can run into spurious divergences or obtain erroneous results. The divergences manifest themselves when one is working at second order or above (order here refers to the number of vertices), and appear if one considers a limit where the momenta flowing through the vertices is not distinct. In reality, the small imaginary pieces introduced to select the vacuum prevent this divergence.

While this expression may appear to be a simplification of the previous one, it actually makes the computation much more algebraically intensive. To obtain results one must first combine all terms to cancel any divergences.

With the impending launch of the PLANCK satellite, we will be eventually be able to beat down the noise: PLANCK’s signal-to-noise limit in mode space is expected to be at least 5-10 times more sensitive than WMAP, allowing us to probe the CMB map to 10 times the current resolution. This will allow us to construct higher correlation functions and probe the running of the power spectrum, greatly increasing our ability to distinguish between different models of inflation. Thus the development of a controlled and consistent theory of calculating cosmological perturbations beyond linear order is very timely.

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