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So I have an opportunity to do math research with a probabilist. I would like to propose to him a project that relates to physics, as it would be a good way to learn about another field. What are some open probabilistic problems in physics?

Thanks!

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Hi James, welcome to Physics Stack Exchange! I'm sure people will complain if I'm wrong, but I think this may not be the kind of question that is appropriate for this site - we want questions that have specific, definite answers, not just those that accumulate lists. (Or at least we try to keep the lists to a minimum) We do have one question about unsolved physics problems. –  David Z Sep 23 '11 at 20:09
    
I respect your decision. However, where do you suggest I go for an answer? I believe that closing questions of this form should only be performed once an adequate number of such questions on this site have been answered, so that people can have somewhere to look. Typically, (but maybe not in this case), "list" questions are questions that large portions of physicists will ask themselves; hence, it is useful to have them answered here. As I am unfamiliar to this site, it may be the case that a fair number of such questions have been already answered....if so, then fine. –  James Hendrix Sep 23 '11 at 20:26
    
(cont.) However, from experience, math stack exchange is much more forgiving of such questions...and to my knowledge, they are not having any problems. –  James Hendrix Sep 23 '11 at 20:27
    
I'm certainly open to keeping the question open if other site members are in favor of it. You could post on Physics Meta to ask whether people would like to have this kind of question open, if you want. (I'll give you an upvote so you can post there) But I don't really see this particular question being that useful to most physicists. If you have access to Physics Chat, that would be a great place to ask this sort of thing. Otherwise, you could go to a traditional forum like Physics Forums which is better suited to this kind of question. –  David Z Sep 23 '11 at 20:33
    
Please reopen-- it's actually pretty focused: what open problems in statistical physics are appropriate for a student to try to solve. –  Ron Maimon Sep 24 '11 at 5:38
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There are many accessible problems which are at the interface of probability and physics. There are two classes, those relating to continuous time processes (or continuous space and time statistical fields) and those relating to discrete systems, like the Ising model, Potts model, SOS model, etc. The latter problems tend to be more accessible to students, because they do not require a limiting renormalization except to extract the continuum limit.

I will describe a few problems which I find interesting. The question is open ended, so you can find hundreds, or thousands, of similar problems.

Numerical How-to

Numerical work is very tedious because there are no good programming languages. It tends to get put off by professionals, and only done by students. Nevertheless, nearly all good research nowadays begins with a computer program, and sometimes ends there.

Probabilistic systems have a rich world which can only be revealed fully by simulations. In order to do these simulations, You absolutely must first get a library which will allow you to do the following:

  • open a window of size M by N pixels (in C/C++) with a simple subroutine call, like Window(M,N), or by creating a window object (not by branching off a new thread, or by having a global window attached to the main program).
  • plot a point in this window using a color which you can change at close to maximum machine speed.
  • respond to a keystroke/mouseclick by returning the appropriate value without stopping the simulation, like a video game does. You will need this to change parameters dynamically and see what happens.

Do not pipe text output to gnuplot, do not use python or Java, you always need full machine speed and you need dynamic feedback. Sit down and write such a library in C/C++, it takes a day, and you will use it all your life. Today, it is probably easiest to wrap Qt's "QPainter" widget to do this. I am not sure how fast this is, but I suspect it is as fast as X calls. I wrote such a library for X, and it was a headache, but worth it.

Once you have this, you should explore different statistical systems by simulation, and extract interesting results by playing a physics video game, changing parameters, and finding critical points and weird behavior.

Depinning

I studied this model briefly many years ago. There are many unsolved problems, and it can be an entry point to the literature. This field is still moderately active.

There is a discrete model of friction which is used to describe situations where a membrane is moving through a sticky medium. An example of this is a charge density wave in a solid, or even a solid on solid friction problem, where one solids is sliding on top of another.

The typical situation is that every point is statistically stuck, and pulled by springs from its neighbors until it jumps. The jumping happens when the springs exceed a random threshhold force.

Alan Middleton proved the "no passing" rule in these systems, which is significant, because it proves that the transition is always second order. The result inspired Daniel Fisher to introduce overshoots. This means that when a site jumps, the pinning force on the neighbors is weakened a little. When there are overshoots, you get the following hardly studied phenomenon of phony hysteresis.

A phony hysteresis is when you have a hysteresis loop but the phase transition is still perfectly second order. The reference for this is the first arxiv version of this paper: http://prl.aps.org/abstract/PRL/v92/i25/e255502 (it's by me and a colleague). The referees required a bunch of changes which generally made the paper worse.

There is nothing known about the general class of systems with phony hysteresis. It is clear that the motion of a solid is hysteretic in this sense, because the solid starts moving with a jump, but stops moving smoothly, but it is not clear if there is a second order transition at the solid-on-solid stopping point.

A model for cloud shapes

What makes the surface of a cloud so fractal? What is the fractal dimension? This question is amenable to numerical models, and probably has a simple answer, (and would probably get a lot of press).

A cloud is formed when water condenses in the atmosphere into droplets. The droplets reflect sunlight, leading to cooling of air, and more condensation. This process is stirred by wind, but I am not sure that wind turbulence accounts for the fractal shape.

Suppose only that there is a statistical tendency for a water droplet to grow where there already are water droplets. What kind of clouds to you get for different laws of growth? Each model is important to understand independent of the application to clouds, much like diffusion limited aggregation is important.

Solid-ball/rough-surface friction

An old physics contest problem asked the following: "what is the limiting speed when you roll a perfect number-2 pencil (hexagonal cylinder) down a hill?" It is easy to see that there is a real loss of energy whenever you hit a sharp point and turn a corner, simply from conservation of angular momentum around the pivot point.

This means that when a perfect rigid marble is rolling along a rough surface, it loses energy each time the point of contact moves from one place to another. The motion of the contact point is determined by a statistical problem--- the place where a random suface (or a random 1d-surface, a walk, modelled by an Ulenbeck process) intersects a parabola tilting downward (the limiting shape of a sphere near the contact point). The probability distribution for the next contact point will reach a stable distribution very quickly, and this will determine the friction law. This distribution is just beyond the reach of analytic methods, but it is amenable to a mix of analytic and numerical methods.

The amount of energy loss is determined entirely by the statistics of the next contact point. The problem of calculating the friction from the surface parameters is interesting, and might be the dominant mechanism of rigid-ball rolling friction in some circumstances. This allows you to get a measure of the roughness of a surface just by rolling a ball over it.

Classical field turbulence

There is a class of problems which have been completely intractable, but which have simple model analogs which admit solutions. These are turbulence type problems

A turbulence problem is when you have a nonlinear field equation, like the Navier Stokes equation, and you pump it at long wavelengths. There are fewer large-wavenumber field modes than small wavenumber field modes, by counting states, so the energy will statistically leak into smaller modes in a cascade. It has been an open problem for 100 years to find the shape of this cascade.

In 2d turbulence, the cascade was essentially solved in the 1960s, by the observation that conservation of vorticity means that you can't have a normal cascade, because the vorticity and the energy determine a thermodynamic equilibrium where most of the energy is in large, not small, eddies. This is an atypical situation, it is restricted to 2d system where there is an extra conservation law with higher derivatives.

Since the turbulent phenomena is caused by the classical ultraviolet catastrophe, it should occur in any nonlinear field theory. So consider a model bosonic field with nonlinearities, for definiteness, take the nonlinear Schrodinger equation.

$$ i{d\psi\over dt} = -{1\over 2}{\nabla^2}\psi - \lambda |\psi|^2 \psi$$

For small values of $\lambda$, there will be a nonlinear cascade from long wavelength to small wavelength modes. What are the statistics of this cascade? This is the problem of superfluid turbulence. I don't know how good the simulation results are in this field, I believe they are minimal.

But the problem is essentially open for all classical nonlinear equations. So consider any nonlinear equation, and find its turbulent statistics. There are mode-counting arguments due to Kolmogorov/Onsager/Heisenberg which give you a first picture of the statistics, but these are not exact for the Navier Stokes turbulence. There must be models where they are exact, and nobody knows which models these are.

Some equations which are useful in this regard:

$$ {\partial_t^2} \phi - \nabla^2 \phi = \lambda \phi^3 $$

This is an old classic. Classically, $\lambda$ is a dimensionless parameter you can vary. The results of this type of turbulence have recently become relevant in the field called "preheating", which studies turbulent particle production in models of the end of inflation. In preheating, at the end of inflation, the inflaton is oscillating turbulently, and dumping energy to short wavelength modes, which then turn into standard model particles. The statistics of the particle production depend on the turbulent oscillation spectrum.

$$ {\partial_t} v^p_k + (V_i^{\;p} \partial_i) v^p_k + \nabla P^p = F^p(x,t) $$ $$ \partial_k v^p_k = 0 $$ $$ V^{\;p}_i = \sum_q \alpha^p_q v^q$$

The multi-component Navier Stokes equation with crazy linear advection mixing up the different components. Each different choice of $\alpha$ is a different model. Kraichnan considered something like this at one point, but I am not sure if it is exactly the same. This model is useful because it nonlinearly couples different velocity fields together, and might allow a large N limit.

There are as many such problems as there are nonlinear equations which conserve energy. You can study Yang Mills turbulence, 3d-gravity with sources turbulence (this is not hard-- 3d gravity has a description due to 'tHooft by moving point conical defects), rubber sheet oscillation turbulence (in a nonlinear regime), or any other turbulence. There is hardly any work on any of these systems, with the exception of fluid turbulence and a few scalar models. I would suggest thinking up your own.

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