25 May 2015

The structure factor of a liquid - part IV

Sum rule for impenetrable systems

The hard sphere liquid is an idealized model, but some of its properties hold for a very large class of systems, those that have an impenetrable core of size \(R_c\) (\(g(r< 2 R_c = 0\)). Let us write the Fourier relation between \(g(r) -1\) and \(S(q) -1\) (the inverse of Eq. (4) in post II):

The structure factor of a liquid - part III

This is the third part in a series. In part I and part II we defined the basic concepts used in the theory of liquids, in particular the radial distribution function \(g(r)\) and the structure factor \(S(q)\).

The simplest system one can imagine is the ideal gas. There is no interaction between particles: \(u(r) = 0\), leading to \(g(r) = 1\) (the particle at the origin does not affect the position of its neighbors) and \(S(q) = 1\). The ideal gas is a trivial case, but it can be seen as the reference state for other systems. In particular, one could say that the functions \(g(r) - 1\) and \(S(q) - 1\) that appear in Equation (4) of part II quantify the difference with respect to the ideal gas (due to the interaction potential \(u(r) \neq 0\).)

23 May 2015

16 May 2015

The structure factor of a liquid - part II

[Continuing the preliminary discussion started in part I.]
We are now interested in an explicit form for \(g(\mathbf{r})\) (we return here to the general case —where \(g\) depends on the full vector \(\mathbf{r}\), and not only on its modulus— simply to avoid the radial integrals). Taking particle 0 as fixed in \(\mathbf{r}_0\), \(\rho g(\mathbf{r}) {\text{d}}^D \mathbf{r} = \text{d} n (\mathbf{r} - \mathbf{r}_0)\) is the number of particles (among the remaining \(N-1\)) found in the volume \({\text{d}}^D \mathbf{r}\) positioned at \(\mathbf{r}\) with respect to the reference particle. One can formally count these particles by writing:

The structure factor of a liquid - part I

This post only summarizes some basic concepts and results that will help understand the discussion in the following posts. For a detailed introduction to liquid theory, see one of the many books and review papers [1].

23 April 2015

Projection onto the subspace of spherical harmonics with the same degree

Recently, I've been interested in expanding an angular function over the spherical harmonics, and particularly in retrieving the amplitude of the part corresponding to a given degree \(\ell\). More precisely, let \(F(\Omega) = F(\theta,\phi) =\sum_{\ell} \sum_{m} Y_{\ell m} (\Omega)\). The projection of \(F\) onto the subspace spanned by the harmonics with a given degree \(\ell\) (I believe this space is generally denoted by \(\mathcal{H}_{\ell}\)) is:
\begin{equation}
\label{eq:proj1}
\operatorname{Proj}_{\ell} \left [F \right ] (\Omega) = \sum_{m= - \ell}^{\ell} c_{\ell m} Y_{\ell m} (\Omega)
\end{equation}

21 April 2015

Postdoc position: numerical simulations for biophysics

A post-doctoral fellowship is available at the MSC laboratory of the Paris Diderot University (Paris, France), in the framework of our ANR project.

Title: Modelling the many-body interactions between protein inclusions in cell membranes
Gross salary: ~ 2500 €/month (varies with seniority).
Duration: ~ 12 months.
Summary of the research topic: The MSC laboratory is a research unit at the very heart of Paris working on three main axes: non-linear physics, soft-matter and interface between physics, biology and medicine. The subject proposed here is part of a wider research effort, pursued in collaboration with three other laboratories from the Paris area, in the framework of a project financed by the ANR (French grant agency). In particular, the successful candidate will work in close collaboration with an experimental group in the Laboratory of Solid State Physics (LPS, Orsay).

13 April 2015

Temporary positions in science

A very interesting article in Nature on the future of postdoc positions. Well worth reading, although the solutions proposed (increasing the proportion of permanent research staff, in one way or another) are completely impractical. The comments are even more interesting than the article itself.

7 April 2015

Posting on arXiv

Since I'm on vacation these days, I decided to finally submit my published research papers to arXiv. A previous attempt was about as pleasant as a visit to the dentist, but somewhat longer (and unsuccessful). This time around I tried to submit other papers, and things went more or less smoothly, as soon as I learned to follow some guidelines (for documents produced using LaTeX):
  • Make sure that all .eps figures are of reasonable size: no files above 6MB and no more that 10MB for the whole submission. I downsized some very large files by first converting them to .pdf and cropping to remove white margins (using Acrobat) then opened the .pdf files in Photoshop and saved a copy in .eps format (without preview and using jpeg compression).
  • Check that the figures are correctly invoked in the .tex file (the name should be case-sensitive, something that is not required on Windows systems). File names should not contain special characters (more details).
  • Run LaTeX locally until the compilation is error-free.
  • Create a .zip archive with the .tex file, the .eps figures and the .bbl file (the compiling process on arXiv does not include Bibtex) and, if necessary, with supplementary material in .ps format.
  • Hope that the compilation works without errors, otherwise wade through about three screens of output, fix things and reload files. The supplementary files are simply appended to the final .pdf; this is an easy way to include additional material or tricky parts that fail to compile on the arXiv server. This is how I managed to add to one paper a page-wide table in landscape orientation, which had resisted all other methods.
The final result is here.
 
For Word documents the process should be much easier, since one can directly submit the .pdf version. I only tried this for one paper, currently on hold because it has line numbers in the margin. This restriction is not mentioned anywhere on the arXiv site (neither is the size limit, by the way.)

Overall, the entire procedure was easier than I thought. Still, the system is far from user-friendly (metadata retrieval using the DOI would be nice), the interface is firmly stuck in the 90s, some limitations seem a bit arbitrary and the help could be more detailed.