NEQNET: Non-equilibrium Phenomena

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

148. On surface growth

As Barabasi and Stanley state in the book “Fractal concepts of surface growth“,

most of our life takes place on the surface of something.

Indeed, authors of several thousands of paper citing the Randall-Sundrum seminal work take seriously the idea that our 4-dimensional world is localized on the surface of a membrane located in turn in a warped 5-dimensional universe. Many processes important for our biology are realized on the surface of cellular membranes. Phases in a first order phase transition are separated in space by an interface, with its dynamics in time being very non-trivial.

Often, interfaces that appear in different physical phenomena turn out  to be fractal. Ignite a sheet of paper from one end and follow the dynamics of interface between burned and unburned parts – you will soon realize that the interface is fractal. Look how snow is falling on the window and snowflakes are aggregating – the profile they create is also fractal. Or consider a problem close to my heart – non-critical string theory. As it turns out, the worldsheet of a string is not that nice and smooth – instead, it is actually fractal (see the famous Polyakov-Knizhnik-Zamolodchikov’s paper), and this fractal has a very non-trivial time evolution, since non-critical strings have tachyon in their spectrum

Also, in almost any physical problem involving fractal interfaces, fractal behavior is actually established only in the limit of late times. It is a kind of trivial to see for the example with snowflakes aggregation – you will only be able to observe the fractal if the number of snowflakes which fell on the window is extremely large

Let us try to understand a generic dynamics of an interface (between burned and unburned parts of paper sheet or dynamics of a cellular membrane) focusing for concreteness on non-relativistic problems.

If is the function describing a 2-dimensional surface in 3d, its dynamics is determined by a stochastic differential equation of the form

. (1)

For example, in the case of aggregating snowflakes, the first term describes incoming flow of snowflakes, the second term in the r.h.s. is given by the force of gravity, while the third, stochastic term, introduces randomness into the process of snow deposition. Generally, the stochastic term is considered to be Gaussian, because a) it is the only case when the Eq. (1) can be dealt with analytically and b) becuase in the renormalization group analysis that people usually use to study (1), non-gaussianities in introduce non-renormalizable terms in the corresponding effective action which are not important for large behavior of the function .

Anyway, the equation (1) is still too general to apply anything very concrete like RG methods for studying its solutions. How to simplify it? The whole industry was developed in recent years in order to understand how different effective terms in the r.h.s. of the Eq. (1) change the late time behavior of its solutions. For years, the general lore was that only  the following terms in (1) are relevant for late time asymptotic behavior:

(2)

(see, for example, well known Kardar-Parisi-Zhang (KPZ), Edwards-Wilkinson, Mullins-Herring etc. papers). The basic logic behind (2) is that you try to keep all possible terms (allowed by symmetries of your particular problem, of course) in the gradient expansion. Only lowest order non-linearities are important, since higher-order ones will not survive in the RG limit.

Is the form (2) complete though? The latest PRL paper by Carlos Escudero gives negative answer to this question. Indeed, let us actually try to derive (2) instead of postulating it like, say, KPZ do. We can say that (2) is the outcome of the following Langevin equation:

,

where the potential is the function of the surface curvature only:

. (3)

Taking the potential in the form (3) is reasonable – our surface is two-dimensional, and its only interesting geometrical invariant is the scalar curvature. (Though, isn’t it possible to construct some new terms out of the Gaussian curvature which would correspond to relevant operators in the RG??)

We don’t know exact form of the function , but what we can do is to expand it in power series of curvature

. (4)

This expansion is also actually reasonable. In the limit of late times only lowest orders in this expansion survive (since higher orders correspond to irrelevant operators for the RG).

Minimizing the potential (3) we find

. (5)

As we find, the non-linearities in resulting equation (5) are a bit different from those in the Eq. (2) and the RG behavior is also different from the one given by KPZ (I don’t want to present it here). The physical meaning of the corresponding RG flow is trivial:

short times are characterized by the minimization of the square of the mean curvature, during intermediate times the mean curvature itself is minimized, while in the long time limit the zeroth power of the curvature, which corresponds to the surface area, is minimized,

i.e., strong fluctuations of the mean curvature are getting suppressed at long times – take a very large piece of the Earth’s surface, start to cover it with snow or sand more or less homogeneously – of course, you will get a nearly flat surface in the end

How all this is relevant for relativistic problems?  Let us consider a relativistic bosonic string worldsheet (in d=26). Almost all string theorists like to only write the Nambu-Goto action. This would correspond to the first term in the expansion (5) – at the classical level we have to only minimize the area of the worldsheet in order to find the equations of motion for the string. But in principle nobody forces us to only keep the area term in the effective action. We could also add the Einstein-Hilbert action (but this one just gives the Euler characteristics of the  worldsheet in 2d, that is, topological invariant) or some terms constructed from the Gaussian curvature. Adding these terms is not that stupid as it may naively seem – they actually define the phase diagram of the theory of critical strings (you can find more about it in the last chapter of the Polyakov’s book).

Is there a surface growth in d=26 relativistic bosonic string, too? In a sense, yes, there is, because, as I said above, its spectrum contains tachyons. Keeping higher order terms in the effective action performing a corresponding RG analysis could probably show what happens to the worldsheet at late times – without appealing to the oscillator formalism we will probably see where tachyon condenses and how its condensate looks like.

If you liked the post, please kindly consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you want to chat with me in real time, you can find me on Twitter. The posts below are probably related to the subject of this one:

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