24 January 2016

Sample median for a Lorentz (Cauchy) distribution - part 2

In a previous post I derived the distribution function for the median of a sample of size \(n = 2k +1\) (with \(k \geq 0\) integer) drawn from a Lorentz (or Cauchy) distribution:
g(x) = \frac{n!}{(k!)^2 \, \pi ^{n} \, \gamma} \left [ \frac{\pi ^2}{4} - \arctan ^2 \left (\frac{x - x_0}{\gamma} \right )\right ]^k \frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 }
I will now consider some of its properties.

23 January 2016

Sample median for a Lorentz (Cauchy) distribution

The Lorentz (or Cauchy) distribution
f(x) = \frac{1}{\pi \gamma}\frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 }
\end{equation}is pathological, in that it has no finite moments apart from the zero-order one (which ensures proper normalization of the density function.) Many results we usually take for granted (e.g. the central limit theorem) do not apply and, when sampled, the sample mean and sample variance are not good predictors for the position parameters \(x_0\) and \(\gamma\). This should not come as a surprise, since the distribution mean and variance do not even exist. How can we then estimate the position parameters?

19 January 2016

Scattering from a bunch of parallel wires

I got interested in this problem by trying to understand a result in [1], and also because it may be useful for some stuff I'm currently working on. Consider a collection of \(N\) very (infinitely) long objects, parallel to the \(z\)-axis and whose centers have positions \(\mathbf{R}_{j}\) in the \((x,y)\) plane. We are interested in the orientationally averaged scattering signal.

19 December 2015

What is an order parameter?

For those of us working with liquid crystals, the answer tends to be fairly automatic: "the average of the second Legendre polynomial over the orientation distribution". Only in a second step do we think to qualify the definition: it concerns the quadrupolar order parameter (call it S) in a three-dimensional system. S = 0 for isotropic orientation, S = 1 when the molecules are perfectly oriented along an axis (the director) and S = –1/2 when they are all perpendicular to the director.

Whether it is the appropriate one depends on the problem at hand: if the particles constituting the system do not have inversion symmetry we should probably use the dipolar order parameter. It is less obvious that, although S describes well the tendency of molecules to align along an axis or perpendicular to it, it is not appropriate for any preferred angle: if all molecules make the "magic angle" θm = arccos(1/√3) ≅ 54.7° with the director, S is again zero so it cannot help distinguish between this situation and a purely isotropic distribution. One should then resort to the octupolar order parameter (or possibly a combination of quadrupolar and octupolar terms, for other angles?)

It may be useful to look at order parameters pragmatically, as Landau did for his theory of phase transitions: they are constructed such as to be zero in one phase and finite in another one. It is up to us to identify the phases and to define the most convenient parameter with regard for the particular system but also for the information we want to extract.

12 December 2015

Note to writers

I do not want to read about your personal history. Unless you have led a very interesting life (literary or otherwise) or you have such an uncommon talent that everything you write is captivating, no matter the topic, do not bother. To wit, if you are not Goethe (or Feynman, or Leigh Fermor) or Proust, your recounted experience had better stand for something larger than itself.

28 November 2015

25 November 2015

The influence of intellectuals: hubris and humility

Intellectuals (and I take the term in its widest acceptance) have often believed that their ideas could change the world. Some of them, from Plato to Heidegger, tried to influence directly those in power; others hoped for a posthumous effect (or at least for cultural immortality). Criticism of this belief is as old as the belief itself and has two aspects: a pragmatic one (attempts to influence the course of history through ideas usually fail miserably) and a moral one (knowledge engenders obligations and must be used responsibly).

Elias Canetti, in his short essay The Poet’s Profession1, looks at this attitude from a very different angle: the responsibility of a writer not to humanity, but to words themselves, as tools of his trade. The true poet both understands the tremendous power of words and accepts his inability to change the world using this power.

1. In Das Gewissen der Worte [The Conscience of Words], Hanser (1975).

22 November 2015

Do nano-objects have color?

I've been reading Jim Pivarski's blog Coffeeshop Physics for some time, and I always find the topics interesting and the perspective refreshing. However, I think that his latest post "Viruses have no color" contains a number of fundamental errors, beyond the imprecisions inherent in a simplified account.

Pivarski's stated point is that objects smaller than the wavelength of light have no color, and he explains this by the uncertainty principle. Instead, he illustrates that small objects scatter less light than large ones, using a "geometrical" point of view that ignores the composition of the objects and sees them simply as opaque to the incoming light. Of course, in this approximation even large objects are colorless, since their scattering properties will not change much over the visible spectrum1.

The relevant parameter when discussing the color of an object is not the wavelength but the frequency of the incoming light. For instance, gold nanoparticles a few tens of nanometers in diameter both absorb and scatter green light more effectively than at other visible frequencies because in this range the electromagnetic field couples very effectively with the oscillation modes (plasmons) of the conduction electrons in the particle. Dispersions of such particles are therefore green when seen in reflection and red in transmission, as illustrated by the Lycurgus cup. Even atoms can be said to "have color" if we think of their characteristic transition lines (for instance, sodium lamps glow yellow).

The uncertainty principle1 only tells us that the image of the nanoparticles cannot be sharper than the wavelength used to look at them, not that this image is colorless (see such colored images here and here).

1. I neglect here the λ4 dependence in Thompson scattering, leading to the "blue-sky effect".
2. I preserve here the author's terminology, although "the uncertainty principle" is generally associated with quantum mechanics. Here the reasoning is completely classical, so we might as well call the result "the Abbe resolution limit".

20 November 2015

Why the aspect ratio? Shape equivalence for the extinction spectra of gold nanoparticles

My paper just got published in The European Physical Journal E !

In it, I argue that when describing elongated gold nanoparticles as ellipsoids (to the purpose of modelling their light extinction spectra) the natural comparison criterion is the equivalence of the various moments of mass distribution, rather than the length-to-diameter (aspect) ratio generally used in the literature. I also show that it leads to better spectral correspondence between the various shapes.