The efficiency of molecular motors

We know since the work of Sadi Carnot that the efficiency \(\eta\) (the fraction of heat converted into work) of a thermal engine cannot exceed a surprisingly simple maximum value, \(\eta_{\text{max}} = 1 - T_c/T_h\), defined in terms of the absolute temperatures of the cold and hot heat sources, \(T_c\) and \(T_h\). This limitation applies to a wide variety of devices, from combustion engines to solar cells: in the latter case, \(T_h \simeq 5800 \, \text{K}\) is that of the Sun and \(T_c\) is the ambient temperature, yielding \(\eta_{\text{max}} = 93\%\) [1].

We also know that living organisms can operate at (or even below) the temperature of their environment. In these conditions, Carnot's formula would yield zero efficiency, and thus no work production. And yet, our muscles can reach an efficiency of above 50% [2], higher than that of our cars! How can we solve this paradox?

The availability of research data

A recently published paper [1] (free preprint here) warns that research data becomes less accessible with time. The authors tried to retrieve email addresses from articles 2 to 22 years old, sent standard messages requiring the data sets and followed up on the responses. In most cases (63%) the addresses were not working or they received no response. The other outcomes were: no information on the status of the data (6%), claim of data loss (7%), refusal to share (4%) and receipt of data (19%).

Heidegger's "black books"

[UPDATE 02/03/2014] (via enowning) A short piece in The Chronicle of Higher Education covering pretty much the same ground as the articles below.
[UPDATE 27/01/2014] An interview with Peter Trawny (editor of the "black books") appeared in Die Zeit (in German). An adapted version in French was published in Le Monde (in French).

[First seen here.] The controversy around Heidegger's political position is rekindled in anticipation of the philosopher's personal notebooks being published next spring. Supposedly, they contain clear antisemitic remarks. The debate has already started in the French press [1,2] and on the radio [3].

The more general question is: to what extent is the quality of the work affected by the morals of the author? There are many possible (and partially overlapping) answers, depending on the precise failing imputed to the author. This is where Heidegger's example is instructive, since his relation to national-socialism and antisemitism is not totally clear (among other things, because his work has not yet been completely published).

Are espressos fast?

Another example of an abstract term with a very concrete Latin origin: express.

The weight of an hourglass

This seems to be a classical problem [1]: what is the weight of an hourglass? Careful consideration shows that the weight is larger while running than at rest!

The Kramers-Kronig relations - part 2

In part 1, we had stopped before going to the frequency domain  because we needed the Fourier transform of the sign function. This is where the technical difficulty appears, because we cannot simply write:

\begin{equation}
\label{eq:sgnTF}
\operatorname{sgn}(\omega) = \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t) \operatorname{sgn}(t) \tag{5}
\end{equation} as the integral does not converge. One can however define\begin{align}
\label{eq:sgnvp}
&\operatorname{sgn}(\omega) = \lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t - \epsilon |t|) \operatorname{sgn}(t) = \nonumber \\
&- \lim_{\epsilon \to 0} \left [ \frac{1}{i \omega + \epsilon} + \frac{1}{i \omega - \epsilon} \right ]= \lim_{\epsilon \to 0} \frac{2i \omega}{\omega^2 + \epsilon ^2}= \mathcal{P} \left ( \frac{2i}{\omega}\right ) \tag{6}
\end{align}

The Kramers-Kronig relations - part 1

[See part 2 for some technical aspects]
Very nice derivation of the Kramers-Kronig relations (on Wikipedia, of all places), exploiting the relation between the even and odd components of a function \(\chi (t)\) and the real and imaginary parts of its Fourier transform \(\chi (\omega) = \chi ' (\omega) + i \, \chi '' (\omega)\).

One usually invokes the analyticity of \(\chi (\omega)\) in the upper half-plane, which must first be derived from the causality: \(\chi (t) = 0\) for \(t < 0\). Complex integration along a well-chosen contour then yields the Kramers-Kronig relations in their standard form [1]:

\begin{align}
\label{eq:KK}  
\chi (\omega) &= \frac{1}{i \, \pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi (\omega ')}{\omega ' - \omega} \quad \text{or, for the components:} \nonumber \\
\chi '(\omega) &= \frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi ''(\omega ')}{\omega ' - \omega} \\
\chi ''(\omega) &= -\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi '(\omega ')}{\omega ' - \omega} \nonumber
\end{align}
where \(\mathcal{P}\) denotes Cauchy's principal value.

Statistical and subjective evidence

Are courts of law more receptive to subjective evidence (e.g. witness testimony) than to naked statistical evidence? This is the topic of a recent article [1], selected as one of last year's best philosophical papers. This "Blue Bus" problem seems to have a rather long history in the legal and psychological literature [2,3,4] and is loosely based on a real case ([2], n. 37). Wells [3] gives a particularly clear exposition.

The problem statement: A bus causes some harm, and we know for sure that it necessarily belongs either to the Blue Bus Company or to the Red Bus Company. Should the Blue Bus Company be held liable?

Two scenarios are put forward:

1.  A witness testifies that the bus does indeed belong to the Blue Bus Company, but we have good reason to believe the witness is only 80% accurate.
2. The Blue Bus Company accounts for 80% of the traffic in the relevant area.

In both cases, the probability we can assign to the offending bus belonging to the Blue Company is 80%. Nevertheless, the courts are unlikely to accept the second type of evidence.

Les hautes bergères

By the lake

Coherence 4 - Space coherence

The Michelson setup is an amplitude-splitting interferometer, and hence it is well adapted for time coherence measurements since the two beams originate in the same point \(P_1 = P_2\) and the time shift \(\tau\) is simply given by the difference in arm length. To measure the space coherence, we need a wavefront-splitting interferometer, with the two beams originating in different points \(P_1 \neq P_2\) but at the same time: \(\tau = 0\). The simplest example is the Young interferometer.

Research standards, reproducibility and disclosure

The story is from a year and a half ago, but I only heard about it today, via the LA Times and Slashdot. Researchers from Amgen attempted to reproduce results from fifty-three research papers and only succeded for six of them. They published a comment in Nature, arguing for higher standards in preclinical research. 

How to prepare a presentation - style tips

In a previous post I discussed technical tips for presentations. Here I will say a few words about the style and I will conclude with a third post about the content.

At some level, every point below derives from one general principle: style is subordinated to effectiveness. Each one of your choices should make for easier communication with the audience.

How to prepare a presentation - practical tips

There are many resources on how to give a good talk, but the information is not always well structured. In this post I summarize some practical points; I'll write about style and content in future posts.

Technical issues

The safest way of avoiding technical problems is to use your own computer for the presentation, but sometimes this is not an option. Below are some points to consider, for each situation:

Coherence 3 - The Michelson interferometer

The Michelson interferometer (shown below) is an excellent setup for illustrating the concept of longitudinal coherence.  We will simplify the formalism of the previous post down to three essential ingredients:

• The beam consists of wave pulses of length l (corresponding to ξ_l in post 2).
• They are "split in half"; each half goes through one of the arms.
• The "twin" pulses arrive at the detector with a shift ΔL.

Speak, memory

...one of those blessed libraries where old newspapers

are microfilmed, as all our memories should be.

V. Nabokov

In the New Yorker, a review of a book about Henry Gustave Molaison, the man without memory (without new memories, at least) and whose ability to remember was literally sucked out of his brain.

This topic (and most anything else concerning the memory) reminds me of Borges's Funes the Memorious. whose memory was "like a garbage disposal". Finally, too much memory is as debilitating as too little...

Assembly kinetics of capsid proteins

Our paper on the self-assembly of norovirus capsids made the cover and the spotlight of the Journal of the American Chemical Society!

Coherence 2 - Time coherence

Time coherence is directly linked to the spectral width. Indeed, a perfectly monochromatic signal is coherent at all times: a maximum now ensures a maximum \(N\) periods later. Not so for a polychromatic source: two oscillations of slightly different frequency are in phase at \(t=0\) and become out of phase \(N\) periods later, as shown below.

Image from [BW].

This defines the coherence time (and the longitudinal coherence length \(\xi _l\)), via:
\begin{equation}
\label{shift}
\xi _l = N \lambda = (N - \frac{1}{2})(\lambda + \Delta \lambda) \Rightarrow N \simeq \frac{\lambda}{2 \Delta \lambda}\Rightarrow \xi _l \simeq \frac{\lambda^2}{2 \Delta \lambda}
\end{equation}

In conclusion, the larger the spectral width \(\Delta \lambda\), the shorter the longitudinal correlation length \(\xi _l\). Once again, the descriptions in the time and frequency domains are complementary.

In the next installment, we will look at time coherence in more detail, using the Michelson interferometer.

Circumspice

I usually do not link without commenting, but there is not much I could add to this:

Tom Stoppard: Information is light.

Coherence 1 - Introduction

This is the first in a series of posts where I'll try to give a simplified account of optical coherence, based on a couple of lectures I presented at the HERCULES school a few years ago. The subject is quite complex and those looking for a more rigorous and complete text should check the references below.

• What is coherence?

Let us start by putting together the "diffuse knowledge" one might have about coherence:

-  Laser light is coherent, while that emitted by thermal sources is incoherent.

-  Coherence is related to the presence (or visibility) of interference fringes.
-  Coherence "decreases as the wavelength λ decreases". This statement needs some elaboration, but it is true that is much harder to achieve coherence in the X-ray range than in the optical one.
A first attempt at a definition would be:

"Coherence" is the extent to which a field maintains a constant phase relation over time or across space.

• Why should we care?

Coherence is a fundamental property of light (and waves in general). Whenever we describe a phenomenon related to wave propagation we need to make an (explicit or implicit) assumption about the coherence of that field. Undergrad level treatments of physical optics and X-ray crystallography generally assume perfect coherence. Knowing the limitations of these approaches is essential for a deeper understanding of the underlying phenomena. Moreover, the concept of coherence is fundamental for modern applications such as holography.

• Definition

The mutual coherence function of the wave field V(r,t) - taken as scalar for simplicity - is defined as:

\begin{equation}
\begin{split}
\Gamma _V (P_1, P_2, \tau) &= \left \langle V(P_1,t+\tau) \tilde{V}(P_2,t) \right \rangle \\
& \quad \text{with} \quad \left \langle F(t) \right \rangle = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T F(T) \, \text{d}t
\label{Gam}
\end{split}
\end{equation}

where ~ stands for the complex conjugate; we need to integrate over long times T, since detectors are very slow compared to the radiation frequency (for the visible range), so any measurement will be averaged over a large number of periods.

Equation (1) by itself is not very eloquent, but its meaning should become more clear in the following posts, through simplifications and examples.

A first observation is that the coherence function Γ resembles an intensity. Indeed, when P1=P2 and τ=0, it is simply the average intensity at that point. We can thus renormalize by the intensity to obtain a degree of coherence γ :

\begin{equation}
\gamma _V (P_1, P_2, \tau) \equiv \frac{\Gamma _V (P_1, P_2, \tau)}{\sqrt{\Gamma _V (P_1, P_1, 0)}\sqrt{\Gamma _V (P_2, P_2, 0)}} = \frac{\Gamma _V (P_1, P_2, \tau)}{\sqrt{I(P_1)}\sqrt{I(P_2)}}
\label{degree}
\end{equation}The coherence function contains all the information we might want. However, it is quite cumbersome. In practice, it is often convenient to reduce it to only two parameters, a time (or longitudinal) coherence length, which is roughly the decay length of Γ_V with τ, and a space (or transverse) coherence length, quantifying the loss of coherence as P2 moves away from P1. We will look at longitudinal coherence in the next post.

• References

[BW]   - M. Born and E. Wolf, Principles of Optics (7th ed.),
                 Cambridge Univ. Press (1999).
[MW]  - L. Mandel and E. Wolf, Optical Coherence and Quantum Optics,
                 Cambridge Univ. Press (1995).
[G]      - J. Goodman, Introduction to Fourier Optics (3rd ed.)
                Roberts & Co Publishers (2005).
[W]    - E. Wolf, Introduction to the Theory of Coherence and Polarization of Light,
               Cambridge Univ. Press (2007).

Academic salaries - quantitative data

Very interesting international comparison (I got there by way of Inside Higher Ed). I have a couple of comments concerning France (I wrote about the salaries of French researchers before):

• The gap between entry- and midlevel salaries seems too large (almost 1:2); the authors probably took the lowest rung on the published salary scale. However, this scale is antiquated, since it starts immediately after the PhD (thirty years ago, permanent positions were available for freshly minted doctors). Nowadays, an entry-level permanent position in research or higher education requires (in practice, not legally) at least three years of experience. Since the salary is determined by seniority, the effective entry salary is 20-25% higher than the minimum listed.
• The "other benefits" for France (in the Supplementary Contract Benefits table) are marked as "5 - Not offered as part of an academic contract". The "other benefits" are:

Child benefits, subsidized of free education for children, low pro[b]ability of unemployment, salary bonuses, travelling and conference allowances, social security (incl. disability) plans, alowances for participation in research

 Some of them (the first two and social security) are generally available in France, so they can technically be seen as "not part of an academic contract". For an international comparison, however, they are relevant. Also, professors are permanent state employees, which does imply a (very) "low pro[b]ability of unemployment".

Uxorious

Beautiful English word. Apparently, the only language containing similar terms is Spanish.

Nice view

Functional derivative - intuitive presentation

Functional derivatives are very useful in physics, and they are not too difficult to manipulate; indeed, the rules are very similar to those for the "usual" derivative (of a function with respect to a variable). However, the latter has a straightforward visual interpretation, unlike the functional derivative, which might partly explain the difficulty of the concept. Another source of confusion is precisely the acquired reflex of taking the derivative "with respect to the variable".

We will consider a typical application, i.e. finding the configuration of a system via energy minimization. The system is described by the value of a field \(f(x) \) in all points \(x\) of a certain domain \(\mathcal{D}\). \(\mathcal{D}\) may be \(n\)-dimensional, in which case we will write simply \(x\) instead of \((x_1, x_2, \ldots, x_n)\). The energy \(U\) is a functional¹ of \(f\). For instance: \begin{equation} U = \int_{\mathcal{D}} \text{d}x \, f^2(x) \, . \label{eq:defU}\end{equation} In analogy with functions, an extremum of \(U\) is a configuration \(f_0(x)\) such that "small variations with respect to this position leave the energy unchanged".

These are not variations in the variable \(x\), but rather changes of \(f(x) \) over the entire domain \(\mathcal{D}\). Let us write \(f(x) = f_0(x) + \delta f (x)\) and limit ourselves to terms linear in \(\delta f\). More rigorously, we can write \(\delta f (x) =\epsilon g(x) \) and work in the limit \(\epsilon \to 0\). \(\delta f \) and \(g\) are functions defined on \(\mathcal{D}\) and can be subject to certain constraints (on the boundary, in particular). The resulting change in \(U\) is:\begin{equation} \delta U = \int_{\mathcal{D}} \text{d}x \, [f_0(x)+\delta f (x)]^2(x) - \int_{\mathcal{D}} \text{d}x \, f_0(x)^2(x) = \int_{\mathcal{D}} \text{d}x \, \delta f (x) 2 f_0(x) \, . \label{eq:deriv}\end{equation}to first order in \(\delta f\). We say that \(2 f\) is the functional derivative of \(U\), \(\frac{\delta U}{\delta f}\) and write:\begin{equation} \delta U = \int_{\mathcal{D}} \text{d}x \, \delta f (x) \frac{\delta U}{\delta f} \quad \text{which is similar to} \quad \text{d}f = \text{d}x \cdot \nabla f \label{eq:diff}\end{equation}with the identification: \(\delta f \equiv \text{d}x\), \(\frac{\delta U}{\delta f} \equiv \nabla f\), and \(\int_{\mathcal{D}} \text{d}x \equiv \, \cdot \, \) plays the role of the scalar product. Ensuring that \(\delta U = 0\) for any variation \(\delta f\) requires that \(\frac{\delta U}{\delta f} = 0 \Rightarrow f_0(x) =0\) for our example \eqref{eq:defU}, exactly as \( \text{d}f = 0\) for all \( \text{d}x\) implies \( \nabla f =0\).

To develop the similarity: \(\text{d}x \) and \(\nabla f\) are \(n\)-dimensional vectors, and their product yields the variation of \(f\) along \(\text{d}x \). \(\delta f\) and \(\frac{\delta U}{\delta f}\) are (infinite-dimensional) vectors, and their "product" yields the variation of \(U\) "in the direction of" \(\delta f\). To belabor the point, this "direction" is defined in the function space, and not in the domain \(\mathcal{D}\) of the integration variable \(x\).

We glossed here over the fact that \(\text{d}x \) and \(\nabla f\) do not belong to the same vector space (the contravariant/covariant distinction). In the same vein, \(\frac{\delta U}{\delta f}\) need not belong to the same family of functions as \(\delta f\) (or \(g\)) and must more generally be defined as a distribution.

---

^{1. A functional is a mathematical object that acts upon a function and yields a scalar. We assume that it can always be written as an integral over \(\mathcal{D}\) (as in \eqref{eq:defU}), although sometimes this integral will symbolize operating with a distribution upon \(f\). ↩}

Choice of terms

A Google search yields:

• 19 million results for "islamist president mohammed morsi"
•   2 million results for "democratically-elected president mohammed morsi"

DFW: This Is Water

A video by The Glossary brought renewed attention to David Foster Wallace's 2005 commencement speech This Is Water. Fortunately for me, since I had managed to miss the text all these years.

What is the relation between Wallace's default setting and Heidegger's the One (das Man)? Both of them are inauthentic modes of existence, but the former is self-centered while the latter stems from conformity.

A secondary question is how familiar DFW was with Heidegger. Aside from an ironic allusion in Infinite Jest, the only serious reference comes from a 2005 interview: "Heidegger’s the guy most people think got us into this bind, but when I was working on Broom of the System I saw Wittgenstein as the real architect of the postmodern trap." Some authors have also tried to interpret Infinite Jest and The Pale King in terms of Heidegger's The Question Concerning Technology.

Google+: Browser no longer supported

Trying to go to my Google+ account from Google's main page yields the following warning:

Your Browser is no longer supported

This is a bit annoying, since I'm using Firefox 23... Fortunately, there is an easy workaround: use your full profile link: https://plus.google.com/[profile number] and from the Profile menu on the left you will be able to access all your data.

To retrieve your profile number, once logged in go to the profile menu (top right), which should still be accessible and click on Account. On this page, find item Edit your profile. This link contains the profile number. Click on it.

P.S. Chrome 28 works just fine.

UPDATE: Everything is OK again, without any intervention on my part.

Loire Valley

More photos.

Château de Chambord