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331. Interview with Bogzabraloff brothers: science and religion

Posted by Dmitry

at April 1, 2009

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Dear friends,

I have just returned from the two-day trip to sunny (well, compared to Helsinki) Italy, where I was kindly invited to interview Bogzabraloff brothers, two of the deepest thinkers of our time. I have also had a chance to take a closer look at rather nice facilities of the Vatican Observatory, by this is for another story. Please find the interview below, as for me – I am tired as … and go to sleep right now. I am afraid, I will be unable to answer your questions until tomorrow but in the mean time – enjoy the interview! (and pardon my English – Bogzabraloffs speak Russian, and the interview was conducted in Russian).

Interview with Bogzabraloff brothers, 30.03.2009, Castel Gandolfo, Italy

1. Bogzabraloffs

Bogzabraloff brothers are not really famous in the same sense, say, as Steven Hawking – if you are a theoretical physicist, I would estimate the probability that you have never heard those names – Ingvar and Grishka Bogzabraloff – to be about 95% (albeit you are really good and proficient in modern theoretical physics – say, you’ve graduated from Princeton). Yet, twins are the deepest, most original and unconventional thinkers I ever met in my whole life. This is one of the mortal Sins of the modern society – no physicists, biologists, philosophers, geniuses, thinkers get into the focus of public attention. Nowadays, people are more interested in life of drug-addicted porn stars.

In case you are in 95% group, allow me to introduce Bogzabraloffs. Ingvar Bogzabraloff is currently a head of the theoretical division of Vatican Observatory, chair of conformal theology at Pontificia Universit? Urbaniana, Vatican and associated member of the Permieter Institute for Theoretical Theology, Waterloo, Canada. Grishka Bogzabralov does not keep so high profile. He is a senior staff member at Vatican Observatory, spending almost all his time closely collaborating with his famous brother on various research projects.

Their career in science started back in 1973, when their father, Rurik Bogzabraloff, retired theoretical nuclear physicist, not-so-famous collaborator of Richard Feynman and Hans Bethe in Manhattan project, started to teach his 3 year sons some elementary mathematics. At that time both prodigy boys were already able to read, and the progress was fast. According to the family legend (that cannot be really proved nor disproved since no documented testimony exists), 5 year-old Grishka has once asked his father:

- If I am running faster than light, can I see my reflection in the mirror?

… and Rurik had to explain brothers some basics of special relativity. They have learned so fast and yet, the family council decided that brothers won’t live the life of unsuccessful, broken prodigies like James Sidis (they were clearly able to follow his path), trading depth of understanding for a certain boost in their progress.

Rurik Bogzabraloff has made smart decision as we will see later: both former prodigies have received their PhDs from Princeton in the age of 21, where Grishka has studied topology under Wu-chung Xiang, while Ingvar was specialized in high energy physics (his advisor at Princeton was Curtis Callan).

Their incredible understanding of the subject, their wit, their depth, their ability to perform complex calculations right in mind were so surprising that immediately after the defence the Institute for Advanced Study has offered them permanent positions.

I guess it really shouldn’t be a surprise for you if you take a look at their photo from those glorious Princeton days. Look them right into the eyes – they shine with Knowledge. Maybe, they are not yet quite completely self-confident, but that is to change soon as you’ll see

Bogzabraloffs declined.

- You see, the Institute for Advanced Study is known for breaking science careers, – Ingvar smiles. – Recall, say, Albert Einstein. Somehow, he was plain amazing before he agreed to join IAS. And he was just good afterwords, especially in bicycle driving. Clearly, we had to decline, and they hired Polyakov instead.

Grishka is quietly laughing pretending he is looking at something in the window of their spacious office in Castel Gandolfo. It is well known in some narrow circles what Grishka thinks about Polyakov’s action, Liouville field and condensation of Lagrangian multipliers.

2. Castel Gandolfo

Bogzabraloff brothers

Nowadays, you can be sure that Bogzabraloffs are not just amazingly smart kids – just take a look at the photo of them I have just made. They have became mature, and so matured their thinking. Based on my short experience, Grishka is extremely easy going and friendly person. Talking to him, you cannot help thinking that he is always smiling (although his lips don’t, his eyes do smile all the time ) Don’t ever try to dumb him down, he can get bored really quickly. Yet, he will never show this to you directly preferring instead to humour you, which he is quite professional at. Ingvar is amazingly self-confident and strong. If you ever had a chance to meet young David Gross, you have an idea what to expect.

- So, how did you end up in Vatican?, – I ask.

- (Ingvar) During my thesis defence I was happy to have Robert Chambers from the University of Arizona in the auditory. At that time, the Vatican Observatory was planning to build a telescope in Arizona, and Robert was clearly one of those who knew about this Vatican intention. A couple of days after the defence, when IAS offer was already declined, I’ve got a call from Tucson – they wanted to hire me to do some research work for the forthcoming Vatican project. Honestly, I did not like the idea to go to Arizona at all, especially taking into account the fact that they wanted me to work on cosmology.

- (me) Didn’t you like cosmology at that time?

Grishka is smiling again. During this conversation brothers rarely look at each other, but each time it happens I feel like a thunderbolt passes between them. Thunderbolt of holiness.

- (Ingvar) Not really. The reason of my discomfort wasn’t quite the subject of my studies – I saw the prospects and promises cosmology gave us very clearly at that time. The problem was that Grishka, a mathematician, had to stay at Princeton, as they would hardly want to hire him as well. So, I said – I am yours under one condition.

- (me) Which is…

- (Grishka, smiling) They had to hire me as well.

- (Ingvar) You see, there is a crucial difference between scientific institutions familiar to all of us like, say, IAS or Kavli Institute in Santa-Barbara and Holy See institutions like Vatican Observatory. The latter is very generously funded and reports only to Holy See. The staff there has a lot of administrative and funding freedom. A grantee of Holy See usually has a certain sum of money at his or her disposal to be managed without any further in-depth reporting to Vatican. People in IAS can only dream of such freedom. In our case, this money turned out to be enough to fund Grishka’s position.

- (me) I see, that how you’ve got to Arizona. And Castel Gandolfo?

- (Grishka, smiling) Oh, after our famous work on AdS/CFT and conformal theology Ingvar was kindly invited by His Holiness John Paul II to head the theoretical group here at Castel Gandolfo. The offer was of the type you cannot refuse as fictional Don Corleone used to say.

3. Science and religion

Castel Gandolfo

View from the windows of Bogzabraloff’s office

I am sitting in the very old chair (18th century, I guess, but I am not an expert in antics) in their huge sunny office at Vatican Observatory. Behind huge 3 m windows of rare Bohemian glass winds chide and forest birds are singing beautifully. I can also hear that some kind of catholic mass starts in the big neighbouring hall, and monks start to sing even more beautifully than birds behind the window’s glass. Overall, the atmosphere is relaxing, the only thing you can think about in Castel Gandolfo is Science.

- (me) Before we pass to AdS/CFT, let me ask you a rather general question. Do you think that science and religion are compatible? Many famous contemporary physicists – Sean Carroll for example – deny that connection between the two can be ever established.

- (Ingvar) You see, science and religion answer to different questions. The kind of questions science tries to answer is “How”: how does the world around us work, how quarks are getting confined, how does it turn out that the Universe seems to be flat and homogeneous at cosmological scales. On the other hand, religion answers to the question “Why”. Why does our world exist? Why quarks are getting confined? Et cetera.

- (me) And why quarks are getting confined?

- (Grishka, smiling) Quarks are confined because God loves you.

I am struck for a moment by the depth of this thought which never came to my mind before.

- (me) Hmm… Let me think about it… and in the mean time ask you a related question. So, if you are saying that science and religion are incompatible, is it ethical for a scientist to apply for grants from religious organisations like, say, Templton Foundation or even the very Holy See?

Grishka clearly gets bored by this question and turns to his birds again.

- (Ingvar, frowning) I hope you are listening to me carefully, because I never said that science and religion are incompatible. In fact, I am sure that the 21st century will become the century of synthesis between science and religion. Many signs of this forthcoming, the only true Grand Unification are already present, especially to a believer’s eye, and it is wonderful that organisations like Templton Foundation help establishing a healthy dialogue between science and religion.

4. Synthesis

- (me) Speaking of synthesis between science and religion, can you remind the readers of NEQNET the history of your discovery of AdS/CFT back in 1993?

- (Grishka, somewhat excited) Both of us had a month-long vacation in Princeton on summer of 1993 invited by Witten, and a lot of wonderful stuff happened there at that time – for example, Igor Klebanov has worked on black branes. I guess the idea was just in the air. One day it has finally hit us, we checked out the symmetry group of AdS space and compared it to the symmetry group of N=4 super Yang-Mills. All pieces of the puzzle were fitting so nicely – large N limit, R-symmetry on the field theory site etc. etc. Soon we understood that the gravitational physics in the bulk is described by conformal Yang-Mills theory on the boundary, on AdS horizon.

- (Ingvar, smiling) I remember that I once said – Grishka, look! We have just explained how God establishes His Will to us! We were completely carried away by the idea! We were sleeping for 3 hours per night!

-(Grishka) And than we explained the idea of duality to Juan Maldacena and Edward Witten. They did not buy the idea at first. Juan was puzzled, Witten was angry as Hell! (He stops for a moment and cautiously looks at the office’s doors.) Sorry about that.

-(Ingvar) Of course, they were unable to get it. Once again we were proven that IAS is no good place for an actively working theoretical physicist, a deep thinker.

-(me) Can you explain the idea to NEQNET readers?

-(Ingvar) It is actually trivial. God lives on the boundary of AdS space and carries His Will to us by means of holography. Literally. There are long relativistic strings with one end attached to the stack of D3-branes where God is localized and another end – attached to us. His is a Puppeteer, and we are His puppets. We are all but reflections of His Thoughts.

-(me, impressed) Fascinating…

-(Grishka, excited) In a sense, nowadays science is back to the Old Testament and Aristotle times and – while the latter thought that there is a crystal sphere of stars on the sky, in the modern physics this sphere should be just substituted by the holographic screen. The only essential modern correction to Aristotle philosophy is relativity.

-(me) Hmm…

-(Ingvar) I think Sean Carroll deep in his heart understands the concept of holography very well, he is ready to accept this picture. Maybe, he is not quite comfortable with it – otherwise, he would not work so much on badly defined aether models. Maybe, he wants to walk one step at a time – first, back to Galileo times, then – to Aristotle.

-(me) Well, I think that I am not quite comfortable with the idea, too. Isn’t God supposed to be almighty?

-(Grishka, cautiously looking at the door of the office) Most certainly.

-(me) How can He be subject to the laws of physics then? In particular, how can He be described by conformal field theory at the boundary?

(Ingvar, seemingly fascinated) Indeed, conformally invariant God is not an idea for average minds. (He kindly smiles at me as I understand how much of a simpleton I am.) If you know the history of art sufficiently well, you will easily recall that the idea of conformal and scale invariance in religon is not new – dozens of paintings by Leonardo da Vinci, Rafael, later works of Escher devoted to religious topics – feature scale-invariant patterns. Our ancestors clearly knew the Truth. But you are right suggesting that God is not limited by conformal Ward identities for example. Later we developed many dual models without scale invariance or deviations from scale invariance.

-(Grishka) I once wrote to Joe Polchinski… Joe, I wrote, what if you think a bit about hard wall model? And then I explained him all the details.

-(Ingvar) In any event, suggesting that God is described by a conformal field theory is misleading. In the same sense you, me and Grishka are described by the Standard Model. This very fact does not contradict to another fact that your mind, conciousness, your personality exist.

-(me) Thanks so much for the explanations… NEQNET readers will be probably interested to know what are you working on now?

-(Ingvar) There are so many unanswered questions. We are trying to understand the place that Science holds with respect to Religion… Probably, Science has to answer questions that the Holy See is so far unable to answer. What is the physics of God? How exactly does his Will influence our existence? After the discovery of AdS/CFT and holographic principle we are getting closer to the ultimate answers. However, remember that, as I said, we are back to the Old Testament times right now. Therefore, the New Testament times are yet to come. And for them to come we need to understand the physics of Love. How exactly does God love us. I mean, physically. What is the physical mechanism of His Love.

I leave Bogzabraloff’s office in Castel Gandolfo enlightened and relieved. I know the Truth. I am sure that conformally invariant God cares for me. I pray to Him asking for help. Help to receive NEQNET funding from the Templton Foundation. Or even Holy See, who knows. On my way back I notice an amusing piece of paper on the heavy iron doors to Castel Gandolfo and decide to make a photo of it. Here it is:

Vatican last note

Entered Apprentice, Personal, Rants, This blog

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330. Some properties of the Burgers dynamics with Brownian or white-noise initial velocity

Posted by Patrick Valageas

at March 31, 2009

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Patrick Valageas is a permanent researcher at the IPhT (Theoretical Physics department) of CEA, Saclay. His interests include turbulence, observational cosmology (LSS formation in particular) and astrophysics. Dmitry.

I would like to thank Dmitry for giving me the opportunity to present two recent papers of mine (arXiv:0810.4332 and arXiv:0903.0956), on the Burgers equation, from the point of view of a cosmologist. They consider the one-dimensional Burgers dynamics for Brownian and white-noise initial velocity, and expand some previous results on the probability distributions of velocity and Lagrangian increments, as well as on the distribution of the density and the shock mass function.

First, let me recall that the Burgers equation,

$\partial_t {\bf v} + ({\bf v}.\nabla){\bf v}=\nu \Delta {\bf v}$ ,

was originally introduced as a simplified model for fluid turbulence, as it shares the same hydrodynamical (advective) nonlinearity and several conservation laws with the Navier-Stokes equation. However, it also appears in many physical problems, such as the propagation of nonlinear acoustic waves or the formation of large-scale structures (filaments, clusters of galaxies) in cosmology. In the latter context, where one considers the inviscid limit ( $\nu\rightarrow 0^+$ ), it is known as the “adhesion model” and it provides a good description of the large-scale filamentary structure of the cosmic web (this involves a rescaling of time and velocities, so that is actually the linear growing mode of density perturbations, proportional to the scale factor in a critical-density universe, ${\bf x}$ is a comoving coordinate, and ${\bf v}$ is – up to a time-dependent factor – the peculiar velocity, i.e. the mean Hubble flow associated with the expansion of the universe has been subtracted).

In this context, one is interested in the statistical properties of the dynamics, as described by the density and velocity fields, starting with a random Gaussian initial velocity at t=0 and a uniform density (the density evolving through the usual continuity equation). These initial conditions are the signature of quantum fluctuations generated in the primordial Universe and agree with the small Gaussian fluctuations observed on the cosmic microwave background. In the hydrodynamical context, this setup corresponds to “decaying Burgers turbulence” (whereas stationary “Burgulence” would be obtained by adding an external stochastic forcing). Thus, the evolution is deterministic and the stochasticity arises from the random initial conditions.

In the inviscid limit, the one-dimensional Burgers dynamics can be understood in very simple terms through a discrete model (as the Burgers dynamics also appears as the continuum limit of a ballistic aggregation process). Indeed, if we set $\nu=0$ , we obtain the equation of motion of free particles, ${\rm d} v/{\rm d}t=0$ , that keep forever their initial velocity and can cross each other. This is actually the well-known Zeldovich approximation in the cosmological context (which is exact at linear order).

Parabolic construction of the solution

Fig. 1. The parabolic construction of the solution. At time t₁ , x₁ is a regular point, while there is a shock at time t₂ at position x₂ ( t₂>t₁ ).

Then, adding an infinitesimal viscosity prevents such particle crossings, so that when particles collide they stick together with conservation of momentum (but with a loss of energy). In the hydrodynamical continuum case, this corresponds to the formation of shocks, that lead to negative jumps of the velocity field (as faster particles overtake slower ones). This is actually the reason why the Burgers equation was introduced in cosmology, as an improvement over the Zeldovich approximation to prevent particles from crossing each other and escaping to infinity. The sticking is intended to mimic the trapping of particles in the gravitational potential wells built by the dynamics.

Most theoretical studies of the Burgers dynamics focus on the one-dimensional case, with power-law initial energy spectra, $E_0(k)\propto k^n$ , where E_0(k) is the Fourier transform of the initial velocity correlation. Then, for -3<n<1 , the evolution is self-similar and the characteristic scale grows as $L(t)\propto t^{2/(n+3)}$ .

Hopf-Cole solution and geometrical construction

In my papers I mostly focussed on the cases n=-2 (Brownian initial velocity) and n=0 (white-noise initial velocity), which are two standard cases where many exact results can be obtained. I now describe the general method used in these works. They rely on the well-known Hopf-Cole transformation, that provides the explicit solution of the Burgers equation. Indeed, introducing the velocity potential, $v=\partial\psi/\partial x$ , and making the change of variable $\psi(x,t)=-2\nu\ln\theta(x,t)$ , transforms the nonlinear Burgers equation into the linear heat equation. This gives the explicit solution

$\psi(x,t)=-2\nu \ln \int_{-\infty}^{\infty} \frac{{\rm d} q}{\sqrt{4\pi\nu t}} \; \exp\left[-\frac{(x-q)^2}{4\nu t}-\frac{\psi_0(q)}{2\nu}\right]$ ,

where $\psi_0(q)$ is the initial potential. Then, in the inviscid limit, $\nu\rightarrow 0^+$ , a steepest-descent method gives

$\psi(x,t)=\min_q \left[ \psi_0(q) + \frac{(x-q)^2}{2t} \right] \hspace{0.5cm} \mbox{and} \hspace{0.5cm} v(x,t)=\frac{x-q(x,t)}{t}$ ,

where q(x,t) is the point where the minimum in the first expression is reached. Outside of shocks, this is the initial Lagrangian position of the particle that is located at the Eulerian position at time . If there are two solutions, $q_-{}<q_+$ , there is a shock at position that contains all the mass coming from $\left[{}q_-,q_+{}\right]{}$

This explicit solution has several well-known geometrical interpretations. For instance, let us consider the downward parabola ${\cal P}_{x,c}(q)$ , centered at and of maximum ,

${\cal P}_{x,c}(q)=- \frac{(q-x)^2}{2 t} + c$ .

Then, starting from below with a large negative value of , such that the parabola is everywhere well below $\psi_0(q)$ , we increase until the two curves touch one another. Then, the abscissa of the point of first-contact is the Lagrangian coordinate q(x,t) and the potential is given by $\psi(x,t)=c$ . At early times, the parabolas have a large curvature so that the contact point is close to , while at late times the parabolas are almost flat, so that many shocks have formed and particles have travelled over a large distance. For the non-smooth, power-law initial conditions, $E_0(k)\propto k^n$ with -3<n<1 , shocks actually form as soon as t>0 . Moreover, shocks are dense in Eulerian space for -3<n<-1 and isolated for -1<n<1 .

Let us first consider the case of white-noise initial velocity ( n=0 ), that is, the initial potential $\psi_0(q)$ is a Brownian motion. Then, the process $q\mapsto\psi_0$ is Markovian, and from the previous geometrical construction we can see that a key quantity is the conditional probability density $K_{x,c}(q_1,\psi_1;q_2,\psi_2)$ for the Markov process $\psi_0(q)$ , starting from $\psi_1$ at q_1 , to end at $\psi_2$ at $q_2 \geq q_1$ , while staying above the parabolic barrier, $\psi_0(q)>{\cal P}_{x,c}(q)$ , for $q_1\leq q\leq q_2$ . Since $\psi_0(q)$ is a Brownian motion, this kernel $K_{x,c}$ obeys the usual diffusion equation but with an absorbing parabolic boundary at ${\cal P}_{x,c}$ , and it is possible to obtain its explicit expression in terms of Airy functions. In the case n=-2 , $\psi_0(q)$ is the integral of a Brownian motion, and the process $q\mapsto\{\psi_0,v_0\}$ is Markovian. Thus, we are now led to consider the conditional probability density $K_{c,x}(q_1,\psi_1,v_1;q_2,\psi_2,v_2)$ . It obeys an advective-diffusion equation with parabolic absorbing barrier and we can again derive its explicit expression.

Then, from the geometrical construction in terms of parabolas, the properties of the system can be expressed in terms of these kernels $K_{x,c}$ . For instance, in the case n=0 (white-noise), restricted to the range [q_-,q_+] , the probability $p_x(q_-\leq q'\leq q)$ that the Lagrangian coordinate q'(x,t) is in the range [0,q] , factorizes into 2 terms, i) $\psi_0$ stays above ${\cal P}_{x,c}$ but crosses ${\cal P}_{x,c+{\rm d} c}$ over $q_-\leq q'\leq q$ , ii) $\psi_0$ stays above ${\cal P}_{x,c}$ for $q{}<{}q'{}<{}q_+$ , which can both be expressed in terms of $K_{x,c}$ , and we eventually integrate over the height and take the limit $q_\pm\to\pm\infty$

Thus, the reason why the two cases n=-2 and n=0 can be explicitly solved is that one obtains Markovian processes, so that the geometrical construction for point distributions can be broken into such pieces, each involving a simple kernel $K_{x,c}$ , which are joined by matching at the boundaries. This is no longer possible for generic initial conditions, where the behavior of the initial potential $\psi_0(q)$ over some range [q_1,q_2] does not only depend on a few values at the boundaries but on the full curve $\psi_0(q)$ on both sides.

Brownian initial velocity

In the case n=-2 , one obtains the peculiar property that Lagrangian increments, $\Delta q=q(x_2)-q(x_1)$ , are exactly independent and homogeneous for non-overlapping cells (far from any boundaries). Then, point distributions factorize. Next, these increments show for small Eulerian distance, $\Delta x=|x_2-x_1|\rightarrow 0$ , the bifractal behavior

$\nu>\frac{1}{2}: \langle (\Delta q)^{\nu}\rangle \sim \Delta x , \;\; \nu<\frac{1}{2}: \langle (\Delta q)^{\nu}\rangle \sim (\Delta x)^{2\nu}$ .

As is well-known, the scaling obtained for large $\nu$ is due to shocks, that are associated with finite jumps for $\Delta q \sim 1$ (the factor \Delta x arises from the probability to encounter a massive shock in the interval of size $\Delta x$ ). This also leads to the small-scale scaling $\langle [v(x+\ell)-v(x)]^p\rangle \propto \ell$ ( $p\geq 2$ ) for the velocity structure functions. These exponents are universal, in the sense that they appear as soon as shocks have formed, independently of the initial conditions. They are different from those obtained in Navier-Stokes turbulence, where the relevant structures are more varied and less singular (because of pressure effects).

Probability distribution of the overdensity for the Brownian case

Fig. 2. The probability distribution $P_X(\eta)$ of the overdensity, $\eta=m/({\overline \rho} \Delta x)$ , for Brownian initial velocity. Smaller corresponds to more nonlinear scales or times.

A quantity of interest, within the cosmological context, is the distribution, $P_X(\eta)$ , of the mean overdensity, $\eta=\rho/{\overline \rho}$ , within a cell of size $\Delta x$ , where $X\propto \Delta x/t^2$ is the rescaled length that expresses the self-similar evolution. As shown in the figure, at large scales (or early times) it goes to a sharp Gaussian peaked around the mean, $\langle \eta\rangle=1$ , as we recover the Gaussian initial conditions. At smaller nonlinear scales (or late times), where shocks govern the dynamics, an intermediate power-law regime develops. This is quite similar to the behavior observed in numerical simulations of cosmological gravitational clustering. In fact, it is possible to obtain its exact expression as

$P_X(\eta)=\sqrt{\frac{X}{\pi}} \, \eta^{-3/2} \, e^{-X(\sqrt{\eta}-1/\sqrt{\eta})^2}$ .

Moreover, the ratios of the density cumulants are scale-invariant,

$\frac{\langle\eta^p\rangle_c}{\langle\eta^2\rangle_c^{p-1}}=(2p-3)!!$ .

In the cosmological context, this property is known as the “stable-clustering ansatz”, which is a reasonable approximation in the nonlinear regime. Thus, the 1d Burgers dynamics with Brownian initial velocity provides a dynamical model where this property is exactly fulfilled. Finally, it is interesting to note that the mass function of shocks obtained in this system, $n(M)=1/\sqrt{\pi} M^{-3/2} e^{-M}$ , with $M \propto m/t^2$ , happens to be identical to the prediction of the Press-Schechter ansatz, that is also widely used in cosmology.

White-noise initial velocity

In the case n=0 , a qualitative difference from the previous case is that socks are no longer dense but isolated. Then, at small scales most cells are empty. One recovers the scaling $\langle(\Delta q)^{\nu}\rangle \propto \Delta x$ at small scales, now for all $\nu>0$ , that is associated with shocks, and the similar scaling for the velocity structure functions.

Probability distribution of the overdensity for white-noise

Fig. 3. The probability distribution $P_X(\eta)$ of the overdensity for white-noise initial velocity.

The system is now dominated by shocks, even at large scales, so that the distribution of the overdensity no longer goes to a Gaussian in the limit of large scales (or early times). Thus, at large densities it always shows a cubic exponential cutoff, $P_X(\eta) \sim e^{-X^3\eta^3/12}$ , with $X \propto x/t^{2/3}$ . At small nonlinear scales, apart from the Dirac contribution, $\delta(\eta)$ , associated with empty cells, the regular contribution again shows a power-law regime for low and moderate densities (but there is no longer a cutoff at very low densities).

The “stable-clustering ansatz” is no longer satisfied, but the ratios $\langle\eta^p\rangle_c/\langle\eta^2\rangle_c^{p-1}$ still have finite limits for $\Delta x\rightarrow 0$ . This is again an universal property due to shocks. As with the comparison with Navier-Stokes turbulence, the cosmological gravitational clustering dynamics appears more complex, as the relevant structures are extended halos rather than point-like singularities (which could explain why the “stable-clustering ansatz” is not exactly realized in cosmology).

Conclusion

I have given here a very brief introduction to studies of the Burgers equation, focussing on analogies with gravitational clustering. Although such works do not provide quantitative predictions for actual turbulence or gravitational dynamics, the Burgers dynamics still shows a great interest per se, as it appears in many other contexts. Moreover, it can serve as a useful benchmark to test approximation schemes devised for turbulence or gravitational dynamics, as the nonlinearities are similar, and one can compare with exact results. Readers who are interested in such topics can have a look at the recent review arXiv:0704.1611, which contains all references I have not given above.

Some literature

1. S. Gurbatov, A. Malakhov and A. Saichev, “Nonlinear random waves and turbulence in nondispersive media: waves, rays and particles”, Manchester University Press, 1991.
2. M. Vergassola and B. Dubrulle and U. Frisch and A. Noullez, “Burgers’equation, Devil’s staircases and the mass distribution for large-scale structures”, Astron. Astrophys. 289 (1994) 325 – one nice paper which also makes the link with cosmology.
3. S. N. Gurbatov, S. I. Simdyankin, E. Aurell, U. Frisch and G. Toth, “On the decay of Burgers turbulence”, J. Fluid Mech. 344 (1997) 339 – and one paper for the turbulence point of view.

Cosmology, Hydrodynamics, Journal club, Master Postdoc, Turbulence

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329. Human Activity in the Web

Posted by Filippo Radicchi

at March 30, 2009

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Filippo Radicchi is a research scientist in Complex Systems Lagrange Lab, ISI Foundation, Turin. He is interested in non-equilibrium diagrammatic methods, RG group analysis of complex networks and community detection. Dmitry.

We use to spend a relevant part of our time surfing the Web: we read news, make posts in blogs, share photos and music, buy books or other goods, etc. The Web offers great possibilities to communicate and retrieve information and none of the precedent technologies can be compared to the Web in terms of globality and velocity of communication.

The Web represents also an important source of information for scientific purposes. Actions performed in the Web are generally stored in electronic databases. Think for example about NEQNET: when we you make a post or leave a comment, meaningful information about your action are stored in the database present on the computer which hosts the service: in addition to the content of the message, also your nickname and the time stamp of your message are saved. Electronic databases collecting information about human activity in the Web can be therefore used in order to understand how people behave and interact.

Former studies have already focused on computer related human activities. Particular attention has been addressed to the activity patterns of humans. Interesting information can be extracted from the statistical analysis of the so called inter-event times. Imagine we know the the instants of time $t_1 \leq t_2 \leq \ldots \leq t_{n_i}$ in which a user i has performed n_i actions. From such information, we can calculate n_i -1 inter-event time gaps: $\tau_1=t_2 -t_1, \ldots, \tau_{n_i-1}=t_{n_i}-t_{n_i-1}$ . Then we can compute the inter-event time probability distribution function (pdf) of the i-th user as

$P_i(\tau)=x_i(\tau) / (n_i-1)$ ,

where $x_i(\tau)$ is the total number of consecutive actions performed by the user i which differ by $\tau$ units of time. The global (calculated over the whole population) pdf can be then calculated as

$P(\tau)=\sum_i (n_i-1) P_i(\tau) / \sum_i (n_i-1)$

which is basically the weighted average of the pdfs of single users: each user contributes to the global pdf linear proportionally to her/his global activity. Global inter-event pdfs have been studied in the case of e-mail communication [Nature 435, 207-211 (2005)], Web surfing [Phys. Rev. E 78, 026123 (2008)], etc. In all these cases, it has been shown that the global inter-event time pdf can be well fitted by a power-law

$P(\tau) \sim \tau^{-\beta}$ ,

where the exponent $\beta$ ranges from 1 to 2, depending on the case of study. This finding is particularly relevant because human activity seems to be characterized by a bursty behavior: long periods of inactivity followed by short periods of intense activity. Some models have been introduced in order to explain this emergent behavior [Nature 435, 207-211 (2005)]. More recently, in [Proc. Natl. Acad. Sci. USA 105, 18153-18158 (2008)] it has been shown that the power-law decay could be explained as the superposition of non-homogeneous poissonian processes.

In our paper, we study three very large databases. We considered a big set of inquiries performed on the search’s engine of America On Line, all logging actions performed by users on the English website of Wikipedia and a big set of feedback messages sent by users on the Ebay (EB) website. The global inter-event time pdf calculated for the EB dataset is shown in Figure 1.

329. Human Activity in the Web

As one can clearly see, the global pdf is characterized by a power-law decay modulated by periodic (daily) oscillations. It should be noticed that the definition of the global pdf is meaningful only in the hypothesis that all users behaves in the same way, which means that each $\tau$ is a random variable extracted from the same pdf (the global one) independently of the considered user. This assumption is however wrong. If we calculate the statistical significance of the global $P(\tau)$ to describe the activity pattern of single users we see that it significantly violates the null hypothesis. A simple Kolmogorov-Smirnov (KS) test which systematically compare the global $P(\tau)$ with each of the single users’pdf (see Figure 2), shows that fraction of users whose activity pattern is describe by P(tau) within a significant level at least equal to Q is much less than expected.

329. Human Activity in the Web

The main reason of this discrepancy is due to the heterogeneity of the population in terms of number of operations performed. Not all users perform the same number of actions, but instead the number of users who have performed a operations equal to n follows a broad distribution. Interestingly, users performing the same number of operations have similar activity patterns. We first define $P^{(n)}(\tau)$ as the inter-event pdf averaged only over users who have performed n total actions. We first see that the statistical significance of $P^{(n)}(\tau)$ is much better than the one of $P(\tau)$ (Figure 3).

329. Human Activity in the Web

Each panel reports the fraction of users R(Q) whose activity patterns are described by the pdf P⁽ⁿ⁾( $\tau$ ) with a probability at least equal to Q for different values of n. The qualitative comparison with Figure 2 tells us that the P⁽ⁿ⁾( $\tau$ ) can describe the activity patterns of single users much better than P⁽ⁿ⁾( $\tau$ ). The reliability of P⁽ⁿ⁾( $\tau$ ) decreases however as n increases.

In addition, we can see $P^{(n)}(\tau)$ depends on in the sense that the decay exponent of this pdf varies as a function of n (see Figure 4).

329. Human Activity in the Web

Each panel reports the inter-event time pdfs P⁽ⁿ⁾( $\tau$ ) for the same values of n considered in Figure 3. P⁽ⁿ⁾( $\tau$ ) can be well fitted by a power-law (dashed lines), but the decay exponent varies with n. In the presented cases we have: $\beta$ ≈ 1.1, 1.2, 1.8 and 2.3.

The importance of the result is twofold. First, it is important to stress that the study of the global $P(\tau)$ is meaningless. The global pdf is defined on the basis of wrong hypothesis and therefore every results obtained by its analysis are biased. Instead of a global pdf valid for every user, it is better to focus on the study of many different pdfs each corresponding to users with similar activity. Second, the finding opens a new direction for the modelling the process. New models are required in order to understand how and why the number of operations influences the decay exponent of the inter-event time pdfs.

People interested in this topic is invited to read our manuscript and visit the homepage, where all data can be freely downloaded.

Condensed Matter, Fun, Journal club, Master Mason

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Posted by Dmitry

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If you were unable to figure out the physics behind the a treasury yield curve, here is a very nice explanation:

By the way, if you are interested in economics and are familiar with YouTube, I highly recommend subscribing to khanacademy. The guy is just great.

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The following quantities are plotted here – total US GDP, investments, SP500 index and so called treasury yield curve,

Yield curve

By definition, the yield curve is the relation between the interest rate and the time to maturity of the debt, and the yield curve above is the one for US treasury securities and US dollar interest rates.

Would you like to discuss a bit the physics of the yield curve? The reason why it should be interesting is surprising anti-correlation with SP500 data as you can see from the plot, that is, the yield curve can be used to predict the time of crisis’ take-off. What is the nature of this anti-correlation?

Via Schegloff.

Entered Apprentice, Fun, Markets and quantitative analysis