26 June 2015

The structure factor of a liquid - part V

Finite-size effects

In previous posts, we have always considered that the liquid system was infinite and homogeneous. This may no longer be the case if:
  • the particles are confined in relatively small spaces or
  • their attractive interaction leads to the formation of dense aggregates, separated by more dilute regions.
Although physically very different, those two situations have similar effects on the structure and we will treat them together.

29 May 2015

Global cooling: is the paper really claiming it?

A recent paper in Nature finds a strong correlation between ocean circulation and oscillations in Atlantic surface temperatures which could be moving to a negative phase. According to the authors, "[t]his may offer a brief respite from the persistent rise of global temperatures."

This claim has been taken up by various sites, along a much stronger prediction: decades of global cooling by up to 0.5°C. However, I cannot find this second item anywhere in the original paper. Where does it come from?!

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}