The Dawn of Everything - I

Is anarchy¹ a realistic social organization? Based on its absence in the modern world one would be tempted to answer in the negative, at least for communities above the size of indigenous tribe (a few thousands, say.) Showing that complex and large-scale societies can function (or, even better, that they have already functioned) without a strong state and its inherent dangers would be a powerful argument in favor of attempting such a decentralized system in our time. This is exactly the argument that David Graeber and David Wengrow are trying to make in The Dawn of Everything: A New History of Humanity. Unfortunately, they are trying too hard. Below the fold, I'll give a synopsis of the first part of the book (chapters 1 to 6). A second post will summarize chapters 7-12 and a third one will conclude the review.

CNRS positions - the 2022 campaign

The 2022 campaign for permanent research positions at the CNRS (Centre national de la recherche scientifique) is open (see the submission site). The deadline is January 11th 2022 (13:00 Paris time). There are 238 open positions at the CR level, 260 for DR2 and 2 for DR1. The official texts are here: CRCN, DR2, DR1 and the evolution of these numbers over the last twenty years is shown in the graph below:

Two changes are worth mentioning this year:

• The significant increase in DR2 positions (13 in 2022, up from 3 over the last few years!) for section 50, "research management". This section's purpose is to promote (no external hire possible) CNRS researchers who undertake heavy administration tasks, such as directing a large organization (e.g. a laboratory or a large facility)¹.
  
• The emphasis on "big data" and AI, with the creation of the new interdisciplinary section 55 (5 DR2 and 5 CRCN positions), but also with targeted positions in other sections (e.g. 15, 17, 51, 53).
  

Good luck to all the candidates!

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1. This is probably part of the larger tendency of reducing the count of staff blocked at the "junior" CRCN level, by promotion to the DR rank and by the creation of the "hors-classe" CR rank in 2017. A similar strategy is announced for the universities, with more promotions from associate professor "maître de conférences" (MdCf) to full professor and to "MdCf hors classe".^↩

Conservatorismul în România şi în Occident

O discuție pe Reddit despre un mesaj al lui Dan Alexe pentru Adrian Papahagi m-a făcut să reflectez la definiția conservatorismului, la modul cum ea se aplică în România şi la eventualele diferențe față de Occident.

Dincolo de presupusul caracter universal al conservatorismului şi de eventualele lui legături cu iluminismul şi liberalismul, cred că definiția lui Alexe este corectă şi că unul din principiile fundamentale ale conservatorilor (aşa cum e formulat spre exemplu de Burke) e următorul: tradiția socială este valoroasă şi trebuie în principiu respectată dacă nu există motive foarte întemeiate de schimbare.

În țările occidentale, unde s-a instaurat de-a lungul secolului XX consensul în jurul unor valori "moderne" (din lipsă de alt termen mai bun): egalitate în drepturi pentru femei, toleranță față de anumite practici sexuale, separare între stat şi biserică şi, în general, nivelarea (măcar aparentă) a ierarhiei sociale, argumentul lui Dan Alexe este valabil: cei care vor să revină la situația dinainte de 1968, sau 1939, sau (în cazuri extreme) 1789 nu sunt conservatori, ci mai degrabă reacționari: ei vor să schimbe tradiția existentă, care le este (moral vorbind) insuportabilă, fiind motivați nu de prudență ci de indignare (ca şi progresiştii, cu care nu au nimic altceva în comun!)

Putem aplica acelaşi raționament în cazul României? Dacă acceptăm că valorile moderne de mai sus au început să fie introduse în anii 90 şi că situația socială anterioară (inclusiv sub regimul comunist) era caracterizată de conformism social, de respect față de ierarhie (a cărei structură s-a schimbat în timp, bineînțeles), de atitudini tradiționale față de sex şi de reproducere (inclusiv interzicerea avortului), de importanța bisericii (în ciuda caracterului declarat ateu al statului comunist), poziția conservatorilor ca reprezentanți ai unei "majorități tăcute" care nu a acceptat evoluția ultimelor trei decenii devine mai convingătoare.

Problema conservatorilor occidentali este că propun schimbarea, deşi principiul lor de bază este continuitatea: a celor români este că programul lor se aseamănă prea mult cu perioada dinainte de 1989, de care ar dori probabil să se distanțeze.

Bernard Henri-Lévy is the anti-Forrest Gump

They both get their photo taken at major historical events, but FG has short hair, buttons his shirts all the way up and is wise without being smart.

The Pareto distribution and Price's law

As detailed in the previous post, the ratio \(f\) of the top authors that publish a fraction \(v\) of all publications is independent from the total number of authors \(N_0\). Of course, this result is incompatible with Price's law (that for \(v=0.5\), \(f = 1/\sqrt{N_0}\)). This issue has been discussed by Price and co-workers [1], but I will take here a slightly different approach.

I had assumed in my derivation that he domain of the distribution was unbound above (\(H = \infty\)), and that the exponent \(\alpha\) was higher than 1. One can relax these assumptions and check their effect on \(f\) by:

1. imposing a finite upper bound \(H\) and
2. by setting \(\alpha = 1\). Note that 2. also requires 1. 

Role of the upper bound

In the finite \(H\) case one must use the full expressions (containing \(H\) and \(L\)) for the various quantities. In this section, we will continue to assume that \(\alpha > 1\). Since \(L\) acts everywhere as a scale factor for \(x\) (and \(H\)) I will set it to 1 in the following. It is also reasonable to assume that the least productive authors have one publication (why truncate at a higher value?!) Consequently, all results will also depend on \(H\), but presumably not explicitly on \(N_0\), which is a prefactor for the PDF and should cancel out of all expectation calculations. It is, however, quite likely that \(H\) itself will depend on \(N_0\), since more authors will lead to a higher maximum publication number!

In my opinion, the most reasonable assumption is that there is only one author with \(H\) publications, so that \(N_0 p(H) = 1 \Rightarrow H \simeq (N_0 \alpha)^{\frac{1}{\alpha + 1}}\), neglecting the normalization prefactor of \(p(x)\).

The threshold number \(x_f\) is easy to obtain directly from \(S(x)\):

\[x_f = \left [ f + (1-f) H^{-\alpha}\right ]^{-1/\alpha}\]

From its definition, the fraction \(v\) is given by: \(v = \dfrac{\alpha}{\mu} \dfrac{1}{1-H^{-\alpha}} \dfrac{1}{\alpha - 1} \left ( x_f^{1-\alpha} - H^{1-\alpha} \right )\). Note that we need here the complete expression for the mean [2]:

\[\mu = \dfrac{\alpha}{\alpha - 1} L \dfrac{1-H^{1-\alpha}}{1-H^{-\alpha}}\]

Plugging \(x_f\) and \(\mu\) in the definition of \(v\) and setting \(v = 1/2\) yields:

\begin{equation} f = f_{\infty} \dfrac{\left ( 1 + H^{1-\alpha}\right )^{\frac{\alpha}{\alpha - 1}} - 2^{\frac{\alpha}{\alpha - 1}}H^{-\alpha}}{1-H^{-\alpha}}, \quad \text{with } f_{\infty} = \left( \dfrac{1}{2} \right )^{\frac{\alpha}{\alpha - 1}},\end{equation}

and we assume that the upper bound is given by:

\begin{equation} H = (N_0 \alpha)^{\frac{1}{\alpha + 1}}. \end{equation}

Exponent \(\alpha = 1\)

Let us rewrite the PDF, CDF and survival function in this particular case:

\[p(x) = \dfrac{1}{1 - H^{-1}} \dfrac{1}{x^2}; \, F(x) = \dfrac{1- x^{-1}}{1 - H^{-1}} ; \, S(x) = 1 - F(x) = \dfrac{x^{-1}- H^{-1}}{1 - H^{-1}}\]

\[x_f = S^{-1}(f) = \dfrac{1}{f + (1-f) H^{-1}}\]

\[v = \dfrac{1}{2} = 1 - \dfrac{\ln(x_f)}{\ln(H)} \Rightarrow x_f = \sqrt{H} \quad \text{and, since } H = \sqrt{N_0}, \, x_f = N_0^{1/4}\]

Putting it all together yields \(f = \dfrac{N_0^{1/4} - 1}{N_0^{1/2} - 1}\) and, in the high \(N_0\) limit, \(f \sim N_0^{-1/4}\), so the number of "prolific" authors \(N_p = f N_0 = N_0^{3/4}\), a result also obtained by Price et al. [1] using the discrete distribution. They also showed that other power laws (from \(N_0^{1/2}\) to \(N_0^{1}\)) can be obtained, depending on the exact dependence of \(H\) on \(N_0\).

Fraction \(f\) of the most prolific authors that contribute \(v = 1/2\) of the total output, as a function of the total number of authors, \(N_0\), for various exponents \(\alpha\). The unbound limit \(f (H \rightarrow \infty)\), calculated in the previous post is also shown for \(\alpha > 1\). With my choice for the relation between \(N_0\) and \(H\), this also corresponds to \(N_0 \rightarrow \infty\). The particular value \(\alpha = 1.16\) yields the 20/80 rule, but also the 0.6/50 rule shown as solid black line. Note that the curve for \(\alpha = 1\) is computed using a different formula than the others and does not reach a plateau: its asymptotic regime \(f \sim N_0^{-1/4}\) is shown as dotted line.

The graph above summarizes all these results: for \(\alpha = 1\), \(f\) reaches the asymptotic regime \(f \sim N_0^{-1/4}\) very quickly (\(N_0 \simeq 100\)). For \(\alpha > 1\), \(f\) leaves this asymptote and saturates at its unbound limit \(f (H \rightarrow \infty)\), calculated in the previous post. This regime change is very slow for \(\alpha  < 2\): the plateau is reached for \(N_0 > 10^6\).

In conclusion, an attenuated version of Price's law is indeed obtained for \(\alpha = 1\)(where it holds for any \(N_0\)) but also for reasonably low \(\alpha > 1\), in particular for \(\alpha = 1.16\) (of 20/80 fame) where it applies for any practical number of authors! As soon as \(\alpha\) exceeds about 1.5, the decay is shallow and saturates quickly so \(f\) is relatively flat.

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¹ Allison, P. D. et al., Lotka's Law: A Problem in Its Interpretation and Application Social Studies of Science 6, 269-276, (1976).↩

² https://en.wikipedia.org/wiki/Pareto_distribution ↩

The Pareto distribution and the 20/80 rule

I mentioned in the previous post Pareto's 20/80 rule. Here, I will discuss Pareto's distribution, insisting on how (and in what conditions) it gives rise to this result. I had some trouble understanding the derivation as presented in various sources, so I will go through it in detail.

The functional form of the Pareto distribution is a power law, over an interval \((L,H)\) such that \(0<L<H\leq \infty\). I will use the notations of the Wikipedia page unless stated otherwise. Its probability density function (PDF) \(p(x)\) and cumulative distribution function (CDF) \(F(x)\) are (\(\alpha\) is real and strictly positive):

\[p(x) = \dfrac{\alpha}{1-(L/H)^{\alpha}} \dfrac{1}{x} \left ( \dfrac{L}{x} \right ) ^{\alpha}\quad ; \quad F(x) =  \dfrac{1-(L/x)^{\alpha}}{1-(L/H)^{\alpha}}\]

One often uses the complementary CDF (or survival function) defined as:

\[S(x) = 1 - F(x) = \dfrac{1}{1-(L/H)^{\alpha}}\left [ \left ( \dfrac{L}{x} \right )^{\alpha} - \left ( \dfrac{L}{H} \right )^{\alpha}\right ]\]

Note that the survival function is very similar to the PDF multiplied by \(x\): \(S(x) \simeq \dfrac{x}{\alpha} p(x)\), the difference being due only to the final truncation term. However, this is only true for power laws, as one can easily check by writing \(p(x) = F'(x)\) and solving the resulting ODE. We should therefore carefully distinguish \(x p(x)\) (which is, for instance, the integrand to use for computing the mean of the distribution) and \(S(x)\) which "has already been integrated", so to speak.

Let us use this continuous model to describe the distribution of publications (neglecting for now its intrinsically discrete character). \(x\) stands for the number of publications by one author, bounded by \(L\) and \(H\). The number of authors that published \(x\) books is given by \(N_0 \, p(x)\). \(N_0\) is the total number of authors. 

• The first question is: who are the first \(f\) more prolific authors (in Pareto's case, \(f = 0.2 = 20\)%)? More precisely, what is the threshold number of publications \(x_f\) separating them from the less prolific ones?

This is quite easy: if we go through the list of authors (ordered by increasing \(x\)) when we reach \(x_f\) we will have counted the lower fraction, so \(\int_{L}^{x_f} p(x) \text{d}x = F(x_f) = 1-f\). Thus, the survival function is \(S(x_f) = \int_{x_f}^{H} p(x) \text{d}x = f\) and we can simply invert this dependency to get \(x_f =S^{-1}(f)\).

•  The second question is: how many publications did these top \(f\) authors contribute?

We need to count the authors again, but with an additional factor of \(x\), since there are \(N_0 \, p(x)\) authors with exactly \(x\) publications, for a total contribution of \(x \, N_0 \, p(x)\). The fraction of publications contributed by the top \(f\) authors \(v\) is then:

\[v = \dfrac{\int_{x_f}^{H} x \, N_0 \, p(x) \text{d}x}{\int_{L}^{H} x \, N_0 \, p(x) \text{d}x} = \dfrac{\int_{x_f}^{H} x \, p(x) \text{d}x}{ \mu}\]

where \(\mu\) is the mean of the distribution and \(N_0 \mu\) is the total number of publications.

In the simple case \(H = \infty\) (which requires \(\alpha > 1\)), one has:

\[p(x) = \dfrac{\alpha}{x} \left ( \dfrac{L}{x} \right )^{\alpha}, \quad \text{with} \quad \mu = \dfrac {\alpha}{\alpha-1} L\]

\[f  = S(x_f) =\left ( \dfrac{L}{x_f} \right )^{\alpha} \Rightarrow x_f = L f^{-1/\alpha}\]

 

Plugging the above into the equation for \(v\) yields:

\[v = \dfrac{1}{\mu} \int_{x_f}^{\infty} x \, p(x) \text{d}x = \dfrac{\alpha}{\mu} \int_{x_f}^{\infty}  \left ( \dfrac{L}{x} \right ) ^{\alpha} \text{d}x = \left ( \dfrac{L}{x_f} \right ) ^{\alpha-1} =f^{\frac{\alpha-1}{\alpha}} \Rightarrow f = v^{\frac{\alpha}{\alpha-1}}\] 

Pareto's rule \(f=0.2\) and \(v=0.8\) requires \(\alpha \simeq 1.161\): a power law with this exponent will obey the rule, irrespective of the values of \(L\) and \(N_0\). Despite the neat coincidence in the established statement of the principle, there is absolutely no need that \(f+v=1\)! For instance, the same \(\alpha\) implies that, for \(v=0.5\), \(f \simeq 0.065\), a result I have already used in the previous post.

Price's law is not intensive

Price's law was proposed in the context of scientific publishing: 

The square root of the total number of authors contribute half of the total number of publications.

It is a more extreme version of Pareto's 20/80 rule, which would state that 20% of authors contribute 80% of the total number of publications (see next post for the relation between the two). Unlike Pareto's rule, however, Price's law is not stable under extension. This is a trivial observation, but I have not yet seen it in the literature, just like I have not seen much empirical evidence for Price's law.

Let us denote by \(N\) the total number of authors and by \(N_p\) the number of "productive" authors (the top authors that provide half of all publications). As the ratio of two extensive quantities, \(p\) should be independent of the system size \(N\): consider ten systems (e.g. the research communities in different countries, different subjects, etc.), each of size \(N\), with the same publication distribution and hence the same \(N_p\). Half of the total number of publications is published by \(10 \, N_p\) contributors, so the overall productivity is \(p' = \dfrac{10 N_p}{10 N} = p\). According to Price's law, it should however be \(p' = p/\sqrt{10}\) ! The situation is similar to having ten identical vessels, all under the same pressure \(p\). If we connect them all together the pressure does not change, although both the volume and energy increase by a factor of ten.

Price's law does have a "convenient" feature: simply by selecting the representative size \(N\) one can obtain any productivity, since \(p = 1/\sqrt{N}\). For instance, the same Pareto distribution that yields the 20/80 rule predicts that 0.7% of causes yield half of the effects. This result is reproduced by Price's law with \(N \simeq 23000\).

Outside of bibliometry, Price's law has been invoked in economics, for instance by Jordan Peterson in (at least) one of his videos. What I find amusing is that it seems to contradict the principle of economies of scale: if there is a connection between the productivity \(p\) and the economic efficiency (and this is the more likely the higher the personnel costs are) then an increase in the size of a company decreases its efficiency. For instance, a chain of ten supermarkets would be less effective than ten independent units, which would be less effective than many small shops. Since the market is supposed to select for efficiency, we should witness fragmentation, rather than consolidation. 

References:

https://subversion.american.edu/aisaac/notes/pareto-distribution.pdf Clear derivation of the 20/80 principle from the general Pareto distribution.

Heuristic derivation of physical laws - II

In the previous post I presented the main result of Trachenko et al. [1] concerning the speed of sound in solids and a possible fundamental upper bound for this parameter. Here, I will add a couple of observations. 

Clean derivation

The heuristic formula for the speed of sound in elemental solids, \( \frac{v_{\text{est}}}{c} = \alpha \sqrt{\frac{m_e}{2 m_p} } A^{-1/2}\) arises quite naturally; in particular, the ratio \(\frac{m_e}{m_p}\) intervenes because the Ry contains the mass of the electron, but the density is given by that of nucleons. In contrast, Press and Lightman [2] put this factor in "by hand", noting that the electron is bound, but the whole molecule vibrates (see the paragraph above Eq. (9)) and neglect the mass number dependence. This line of reasoning is also presented by Ref. [1] as a second option.

Large experimental scatter

The experimental values in Fig. 1 are rather scattered around the theoretical prediction; this is to be expected for such a simple approach, and even agreement within a factor of two for all points is remarkable; however, this should be taken into account in the following discussion. For instance, when mentioning the excellent agreement (within 3%) between the theoretically predicted maximum and the fitted value I would have expected the authors to give the uncertainty on the latter, as well as on the exponent of the variation with the mass number. How close is it to \(-1/2\)?

At what pressure?

The most serious difficulty of the universality claim has to do with the conditions under which the speed of sound is measured (or evaluated numerically). The upper value \( v_{u} = \alpha \sqrt{\frac{m_e}{2 m_p} \, c} \) should apply for solid hydrogen, and the authors further limit this to metallic hydrogen, but this putative phase only occurs at high pressure, above 400 GPa (note that the speed of sound in solid hydrogen at a few GPa is much lower than \( v_{u}\), see e.g. [3]). Simulations then yield good agreement with the \( v_{u}\), but one is now confused: why compare hydrogen at 600 GPa with all the other elements at standard pressure? Could the speed of sound in high-pressure diamond exceed \( v_{u}\)?

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¹ Trachenko, K. et al., Speed of sound from fundamental physical constants Science Advances 6, eabc8662, (2020).↩

² Press, W. H. and Lightman, A. P., Dependence of macrophysical phenomena fundamental constants Phil. Trans. R. Soc. Lond. A 310, 323-336, (1983); two lines above Eq. (10).↩
³ Guerrero, C. L. and Perlado, J. M., Speed of sound in solid molecular hydrogen-deuterium: Quantum Molecular Dynamics Approximation Journal of Physics: Conference Series 717, 012018, (2016).↩

Heuristic derivation of physical laws - I

From time to time I find myself fascinated by the idea of deducing the relations that describe a certain phenomenon not by solving the full relevant equations but via a simplified model. I am not talking about purely dimensional analysis based on Buckingham's Pi theorem [1], but of more complicated situations, involving several parameters with the same dimensions. When presented by a gifted author (such as Weisskopf [2]), the process seems very straightforward, and one would even be tempted to teach it to undergrads. When going into the details, however, things soon become more complicated and good numerical agreement is sometimes due to the fortunate compensation of two opposite errors.

A recent paper on the speed of sound in solids [3] provides a good illustration. The authors propose a remarkably simple expression for the maximum speed of sound and support it with simulations of metallic hydrogen.

Let us try some dimensional analysis: neglecting the contribution of the shear modulus, the longitudinal speed of sound \(v = \sqrt{\frac{K}{\rho}}\), where \(K\) is the bulk modulus of the material and \(\rho\) its mass density. \(K\) is in units of \(\text{Pa} = \text{J/m}^3\), so a first estimate \(K_{\text{est}} = 1\, \text{Ry}/a_0^3\), where the Rydberg \(\text{Ry} = \frac{\alpha^2}{2} m_e c^2= 13.6\text{ eV} = 1313\text{ kJ/mol}\) is the binding energy of the electron in the hydrogen atom, which can be elegantly expressed in terms of the fine structure constant \(\alpha\), the mass of the electron \(m_e\) and the speed of light in vacuum \(c\). The length scale \(a_0 = 0.053 \text{ nm}\) is the Bohr radius. This estimation fails miserably, because the resulting \(K_{\text{est}} = 13000\, \text{GPa}\), while the experimental values for (elemental) solids are all below 500 GPa, and most are even below 100 GPa (see the middle panel in Figure 1 of Ref. [4]). The reason is obvious if we look at the other panels of the same Figure: The Ry overestimates the cohesion energies \(E_c\) by a factor between 2 and 20, while the molar volume estimated using \(a_0\) as an interatomic distance, \(V_{\text{est}} = N_A \, a_0^3\), is almost two orders of magnitude below the real-life data. Of course, \(a_0\) is the radius so the distance between atoms is at least \(2a_0\), reducing the discrepancy by a factor of 8. This is still not enough and, furthermore, trying the estimate the numerical prefactors kind of defeats the whole purpose of dimensional analysis.

Surprisingly, estimating the speed of sound works much better! I'll follow here the reasoning in [3], although a very similar formula (diferring only by a factor of \(\sqrt{2}\)) was obtained by Press and Lightman [5]. Let us denote the (unspecified) interatomic distance by \(d\): then \(K_{\text{est}} = E_c/d^3\) and \(\rho = M_{\text{atom}}/d^3 \simeq A \, m_p/d^3\), where \(A\) is the mass number of the atom in question and \(m_p\) is the mass of the proton. Taking once again \(E_c = 1 \, Ry\), we finally obtain:

\begin{equation}
\label{eq:vA}
v_{\text{est}} = \sqrt{\frac{1 \, Ry/d^3}{A \, m_p/d^3}} = \sqrt{\frac{Ry}{A \, m_p}} = \alpha \, c \sqrt{\frac{m_e}{2 A \, m_p}} \Longrightarrow \frac{v_{\text{est}}}{c} = \alpha \sqrt{\frac{m_e}{2 m_p} } A^{-1/2}
\end{equation}

This is the first result of Ref. [3], and it works quite well, with the experimental points scattered around it within half a decade (factors of 0.6 to 2.4), see their Figure 1.
As mentioned above, Eq. \eqref{eq:vA} is not exactly new; the authors supplement it by taking the lightest element (H, with A = 1) and claiming that the corresponding value is an upper bound for the speed of sound in condensed phases:

\begin{equation}
\label{eq:vu}
v_{u} = \alpha \sqrt{\frac{m_e}{2 m_p} } c \simeq 36 \, \text{km/s}
\end{equation}

Remarkably, their DFT calculations for metallic hydrogen are in excellent agreement (within 3%) with Eq. \eqref{eq:vu}. This is a very strong conclusion: an upper limit for a physical parameter is given in terms of fundamental constants and is supported by numerical results. I do however have some reservations, which I will detail in the next post.

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¹ Barenblatt, G. I. Scaling, self similarity, and intermediate asymptotics, Cambridge University Press (1996).↩

² Weisskopf, V. F. Search for Simplicity Am. J. Phys. 53, 19, (1985).↩
³ Trachenko, K. et al., Speed of sound from fundamental physical constants Science Advances 6, eabc8662, (2020).↩
⁴ Brazhkin, V. V. et al., Harder than diamond: dreams and reality Phil. Mag. A 82, 231-253, (2002); cited in [3] as Ref. (15).↩
⁵ Press, W. H. and Lightman, A. P., Dependence of macrophysical phenomena fundamental constants Phil. Trans. R. Soc. Lond. A 310, 323-336, (1983); two lines above Eq. (10).↩