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Jellium

Eugene Wigner and Edward Teller
Eugene Wigner and Edward Teller

This tiny post is adapted from the introduction of a recent work with David García-Zelada and Paul Jung on the macroscopics and edge of a planar jellium seen as a Coulomb gas.

Potential theory.  The Coulomb kernel $g$ in $\mathbb{R}^d$, $d\geq1$, is given for all $x\in\mathbb{R}^d$ by

\[ g(x)=\begin{cases}\displaystyle\log\frac{1}{|x|}&\text{if $d=2$}\\[1em]\displaystyle\frac{1}{(d-2)|x|^{d-2}}&\text{if $d\neq2$}\end{cases}.\] The Coulomb potential at point $x$ generated by a distribution of charges, say electrons, modeled by a probability measure $\mu$ on $\mathbb{R}^d$ is defined by \[U_\mu(x)=(g*\mu)(x)=\int g(x-y)\mathrm{d}\mu(y)\in(-\infty,+\infty].\] We have $U_\mu\in\mathrm{L}^1_{\mathrm{loc}}(\mathrm{d}x)$ and the identity $-\Delta g=c_d\delta_0$, where $c_d=d\omega_d$ is the surface of the unit ball and $\omega_d$ its volume, gives the inversion formula
\[-\Delta U_\mu=c_d\mu.\]In particular $U_\mu$ is super-harmonic in the sense that $\Delta U_\mu\leq0$ since $g$ is super-harmonic. The Coulomb self-interaction energy of the distribution of charges $\mu$ is defined when it makes sense by
\[\mathcal{E}(\mu)=\frac{1}{2}\iint g(x-y)\mathrm{d}\mu(x)\mathrm{d}\mu(y)=\frac{1}{2}\int U_\mu(x)\mathrm{d}\mu(x).\] Let $V:\mathbb{R}^d\to\mathbb{R}\cup\{+\infty\}$ be a lower semi-continuous function playing the role of an external potential, producing an external electric field $-\nabla V$. If $V$ grows faster than $g$ at infinity, the Coulomb energy $\mathcal{E}_V$ with external field is defined by \[\mu\mapsto\mathcal{E}_V(\mu)=\mathcal{E}(\mu)+\int V\mathrm{d}\mu.\]

Coulomb gases. A Coulomb gas in $\mathbb{R}^d$ with $n$ particles, potential $V$, and inverse temperature $\beta\geq0$ is the exchangeable Boltzmann–Gibbs law on $(\mathbb{R}^d)^n$ with density proportional to \[\mathrm{e}^{-\beta E_n(x_1,\ldots,x_n)}\quad\text{where}\quad E_n(x_1,\ldots,x_n)=\sum_{i<j}g(x_i-x_j)+n\sum_{i=1}^nV(x_i).\] It models a gas of unit charged particles, or more precisely a random configuration of unit charged particles. We should keep in mind that we play here with electrostatics rather than with electrodynamics and that thus we do not have a magnetic field.

Wigner jellium. Let us consider $n$ unit negatively charged particles (electrons) at positions $x_1,\ldots,x_n$ in $\mathbb{C}$, lying in a positive background of total charge $\alpha>0$ smeared according to a probability measure $\rho$ on $\mathbb{C}$ with finite Coulomb energy $c=\mathcal{E}(\rho)$. We could alternatively suppose that the particles are positively charged (ions) and the background is negatively charged (electrons), this reversed choice would not affect the analysis of the model. The total energy of the system, counting each pair a single time, is given by
\[\sum_{i<j}g(x_i-x_j)-\alpha\sum_{i=1}^nU_\rho(x_i)+\alpha^2c.\]
This matches the Coulomb energy of a Coulomb gas with $V=-\frac{\alpha}{n}U_\rho$. This observation leads us to define the jellium model on $S\subset\mathbb{R}^d$ with background charge $\alpha>0$ and background distribution $\rho$ with $\mathrm{supp}\rho\subset S$ as being the Coulomb gas on $\mathbb{R}^d$, with potential $V$ given by
\[V=\begin{cases}-\frac{\alpha}{n}U_\rho&\text{on $S$}\\+\infty&\text{on $S^c$}.\end{cases}\]
We say that the system is charge neutral. when $\alpha=n$. We say that it is uniform when $\rho$ is the uniform distribution on some compact subset of $\mathbb{R}^d$. The great majority of jellium models studied in the literature are charge neutral and satisfy $S=\mathrm{supp}\rho$.

Conversely, a Coulomb gas with sub-harmonic potential $V$ (meaning $\Delta V\geq0$) can be seen as a jellium with background \[\rho=\frac{\Delta V}{c_d\alpha}\mathrm{d}x\] on $S=\mathbb{R}^d$. When $V$ is not sub-harmonic then $\rho$ is no longer a positive measure but we can still interpret it as a background with opposite charge on $\{\Delta V<0\}$.

The complex Ginibre ensemble is a famous Coulomb gas with \[d=2\quad\text{and}\quad V=\left|\cdot\right|^2,\] for which $\Delta V$ is constant, leading to an interpretation of this Coulomb gas as a degenerate jellium on the full space with Lebesgue background. This Coulomb gas or Wigner jellium describes the eigenvalues of a Gaussian random complex $n\times n$ matrix $A$ with density proportional to $\exp(-\mathrm{Trace}(A\bar{A}^\top))$. Equivalently, the entries of $A$ are independent and identically distributed with independent real and imaginary parts having a Gaussian law of mean $0$ and variance $1/(2n)$. This gas  or jellium also appears in various other places in the mathematical physics literature, for instance as the modulus of the wave function in Laughlin’s model of the fractional quantum Hall effect, in the description of the vortices in the Ginzburg-Landau model of superconductivity, and in a model of rotating trapped fermions.

The Forrester-Krishnapur spherical ensemble is another remarkable Coulomb gas with \[d=2\quad\text{and}\quad V=\Bigr(1+\frac{1}{n}\Bigr)\log(1+\left|\cdot\right|^2),\] for which $\Delta V=4(1+1/n)/(1+\left|\cdot\right|^2)^2$, leading to an interpretation of this Coulomb gas as a jellium on the full space with a heavy tailed background. The name of this gas comes from the fact that it is the image by the stereographical projection of the Coulomb gas on the sphere, with constant potential, onto to the complex plane. This Coulomb gas describes the eigenvalues of $AB^{-1}$ where $A$ and $B$ are two independent copies of complex Ginibre random matrices. We can loosely interpret $AB^{-1}$ as a sort of matrix analogue of the Cauchy distribution since when $A$ and $B$ are $1\times 1$ matrices, this is precisely a Cauchy distribution.

We can also consider such a background-potential inverse problem for the one-dimensional log-gases of random matrix theory, which can be seen as two-dimensional Coulomb gases confined to the real line, such as the Gaussian Unitary Ensemble. For instance it follows from the discussion in Peter Forrester book that the logarithmic potential of the density
\[
x\mapsto\frac{\sqrt{2n}}{\pi}\sqrt{1-\frac{x^2}{2n}}\mathbf{1}_{|x|<\sqrt{2n}}
\]
is given on the interval $S=[-\sqrt{2n},\sqrt{2n}]$ by
\[
x\mapsto \frac{x^2}{2}+\frac{n}{2}\Bigr(\log\frac{n}{2}-1\Bigr).
\]

A bit of history. The jellium model was used around 1938 by Eugene P. Wigner in a famous article for the modeling of electrons in metals, more than ten years before his renowned works on random matrices. This model was inspired from the Hartree-Fock model of quantum mechanics, see GV, LN, LS, S1, and LLS1, LLS2. The term jellium was apparently coined by Conyers Herring since the smeared charge could be viewed as a positive jelly, see H1. The model is also known as a one-component plasma with background. As already mentioned, usually charge-neutral jellium models are studied, and this is done typically after restricting the electrons to live on some  compact support of positive background. The restriction ensures integrability of the energy and the interest is usually focused on the distribution/behavior of electrons in the bulk of the limiting system when the volume of the compact set goes to infinity (thermodynamic limit). There are some exceptions where the edge has been considered, for instance in CFTW. Also, the edge of Laughlin states has been considered in CFTW and GJA.

The case $d=3$ is considered by Lieb and Narnhofer, and quoting them: “It is also possible to consider the one- and two-dimensional versions of this problem, where the Coulomb potential $|x|^{-1}$ is replaced by $-|x|$ and $-\log|x|$, respectively. In the one-dimensional, classical case, Baxter calculated the partition function exactly. For that case, Kunz showed that the one-particle distribution function exists and that it has crystalline ordering, i.e., the Wigner lattice exists for all temperatures. Brascamp and Lieb showed the same to be true in the quantum mechanical case for one-component fermions when $\beta$ is large enough. Although we do not deal with the one-dimensional problem here, our methods would apply in that case. In two dimensions there are difficulties connected with the long-range nature of the $-\log|x|$ potential, and we shall not discuss this here.” For more background literature on the jellium, see also AJ, JLM, F, AJJ, JJ, CDFLV. See in particular LWL for the fluctuations of non-neutral jelliums.

Coulomb gas models appeared naturally in statistics around 1920-1930 in the study of the spectrum of empirical covariance matrices of Gaussian samples. Nowadays we speak about the Laguerre ensemble and Wishart random matrices. This was almost ten years before the introduction of the jellium model by Wigner. In the 1950’s, Wigner rediscovered, by accident, these works by reading a statistics textbook, and this motivated him to use random matrices for the modeling of energy levels of heavy nuclei in atomic physics, see this former blog-post. We refer to Bohigas and Weidenmüller for these historical aspects. The work of Wigner was amazingly successful, and he received in 1963 a Nobel prize in Physics “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”. The term Coulomb gas is explicitly used by Dyson in his first seminal 1962 paper and by Ginibre in his 1965 paper. The term Fermi-gas is also used.

 

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Financement des revues scientifiques

Le Tricheur à l'as de carreau, par Georges de La Tour vers 1635Au sujet des modes de financement des publications académiques, les choses avancent, lentement, par à-coups. L’INSMI du CNRS a œuvré par exemple pour la création du Centre Mersenne à Grenoble. La montée en puissance et la conduite du changement ne sont pas simples. Idem pour le projet MathOA.  Plus récemment, sur le plan des actions macroscopiques structurantes, la France s’engage sur la science ouverte, tandis que l’Europe met en avant le diamond open access dans son Plan S pour 2021 !

La guerre continue sur d’autres fronts. En France, en mars 2019, le tribunal de grande instance de Paris a ordonné aux fournisseurs d’accès à Internet français de bloquer l’accès aux sites Sci-Hub et Library Genesis. La décision fait suite à une plainte déposée par Elsevier et Springer Nature. Mais ce blocage est contournable avec un simple VPN ou un serveur DNS alternatif. Cette décision de justice n’est qu’un  épisode de plus au long feuilleton judiciaire international. Toujours en France, Elsevier est aussi partie prenante d’un projet d’accord controversé avec le consortium Couperin.

En matière de publications, les universitaires sont à la fois les producteurs, les évaluateurs, et les consommateurs, au bout du compte rackettés et pris en otages par les mastodontes Elsevier et Springer. Le numérique n’a fait que forcer le trait. Il n’y a pas vraiment besoin d’être de gauche pour être contre la position de Elsevier et Springer: ils sont devenus des parasites, et il faudrait à minima les remettre à leur juste place de prestataires de services en concurrence.

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Kernels

Charles-Augustin de Coulomb
Charles-Augustin de Coulomb (1736-1806)

The Coulomb or Newton kernel in \( {\mathbb{R}^d} \), \( {d\geq1} \), is often defined for all \( {x\neq0} \) as

\[ g(x):=\begin{cases} -|x| & \mbox{if }d=1,\\ \log\frac{1}{|x|} & \mbox{if }d=2,\\ \frac{1}{|x|^{d-2}} & \mbox{if }d\geq3, \end{cases} \]

where \( {|x|:=\sqrt{x_1^2+\cdots+x_d^2}} \) is the Euclidean norm. It is the fundamental solution of the Laplace or Poisson equation, in the sense that

\[ \Delta g=-c_d\delta_0 \quad\mbox{where}\quad c_d:= \begin{cases} 2 & \mbox{if }d=1,\\ 2\pi & \mbox{if }d=2,\\ (d-2)|\mathbb{S}^{d-1}| & \mbox{if }d\geq 3, \end{cases} \]

where \( {\mathbb{S}^{d-1}:=\{x\in\mathbb{R}^d:|x|=1\}} \), and, denoting \( {\Gamma} \) the Euler Gamma function,

\[ |\mathbb{S}^{d-1}| =2\frac{\pi^{d/2}}{\Gamma(d/2)}. \]

This partial differential equation is in the sense of Schwartz distributions \( {\mathcal{D}'(\mathbb{R}^d)} \). Note that on \( {\mathbb{R}^d\setminus\{0\}} \), the function \( {g} \) is \( {\mathcal{C}^\infty} \) and harmonic in the sense that \( {\Delta g(x)=\partial_1^2g(0)+\cdots+\partial_d^2g(0)=0} \) as a function for all \( {x\neq0} \). The behavior at the origin makes \( {g} \) super-harmonic (its Laplacian is \( {\leq0} \)), which is a trace analogue of concavity.

The case \( {d=1} \) is intuitive: the derivative of \( {|x|} \) in the sense of distributions is the Heaviside function \( {-\mathbf{1}_{x<0}+\mathbf{1}_{x>0}} \) (essentially a jump at zero of height \( {2} \)), and the second derivative twice the Dirac mass, \( {2\delta_0} \). The case \( {d=2} \) appears also as special: it blows up at infinity.

The physical interpretation of \( {g(x)} \), up to physical constants, is the potential generated at point \( {x} \) by a charge at the origin, the field being \( {-\nabla g(x)} \). The electric field if we model electrostatics (Coulomb), and the gravitational field if we model gravity (Newton). Note that \( {-g(x)\nabla g(x)} \) vanishes at infinity if \( {d\geq2} \), but not if \( {d=1} \), which makes a difference for integration by parts.

For a probability measure \( {\mu} \) on \( {\mathbb{R}^d} \), the potential generated by \( {\mu} \) at point \( {x} \) is

\[ U_\mu(x)=(g*\mu)(x)=\int g(x-y)\mu(\mathrm{d}y). \]

In dimension \( {d=1} \) or \( {d=2} \), this is well defined as soon as \( {\mu} \) integrates \( {g} \) at infinity. Note that \( {g} \) is Lebesgue locally integrable. The convolution operator \( {\mu\mapsto U_\mu} \) is the inverse of the Laplacian, and we have the inversion formula

\[ \Delta U_\mu=(\Delta g)*\mu=-c_d(\delta_0*\mu)=-c_d\mu, \]

in \( {\mathcal{D}'(\mathbb{R}^d)} \). The potential \( {U_\mu} \) is harmonic outside the support of \( {\mu} \).

Alternative formulation. The following alternative definition is simpler:

\[ g(x):=\begin{cases} \frac{1}{(d-2)|x|^{d-2}} & \mbox{if }d=1\mbox{ or } d\geq3,\\ \log\frac{1}{|x|} & \mbox{if }d=2, \end{cases} \]

which satisfies

\[ \Delta g=-c_d\delta_0 \quad\mbox{where}\quad c_d:=|\mathbb{S}^{d-1}| \]

with the convention \( {|\mathbb{S}^0|:=|\{-1,1\}|=2} \). Indeed if \( {d=1} \) then \( {\frac{1}{(d-2)|x|^{d-2}}=-|x|} \).

This alternative formulation makes the equilibrium measure nicer in the quadratic confinement case. Namely, the electrostatic energy of a distribution of charges modeled by a probability measure \( {\mu} \) on \( {\mathbb{R}^d} \) with external field generated by a potential \( {V:\mathbb{R}^d\rightarrow\mathbb{R}} \) is

\[ \mathcal{E}_V(\mu) :=\iint g(x-y)\mu(\mathrm{d}x)\mu(\mathrm{d}y)+\int V(x)\mathrm{d}\mu(x). \]

This functional is strictly convex and lower semi-continuous with respect to the narrow convergence of probability measures. Its minimizer, \( {\mu_*=\arg\inf\mathcal{E}_V} \), is called the equilibrium measure. The Euler-Lagrange equation gives that \( {\mu_*} \) has density

\[ \frac{\Delta V}{2c_d} \]

on its support. When \( {V} \) is strong enough at infinity then this support is compact. For instance when \( {V(x)=|x|^2} \) then we find that the equilibrium measure \( {\mu_*} \) is the uniform distribution on the unit ball of \( {\mathbb{R}^d} \). Indeed, in this case

\[ \frac{\Delta V}{2c_d} =\frac{2d}{2|\mathbb{S}^{d-1}|} =\frac{2d}{2d|\mathbb{B}^d|} =\frac{1}{|\mathbb{B}^d|} \]

where \( {\mathbb{B}^d=\{x\in\mathbb{R}^d:|x|=1\}} \) is the unit ball of \( {\mathbb{R}^d} \). Recall that if \( {s(r)} \) and \( {v(r)} \) are respectively the surface of the sphere of radius \( {r} \) and the volume of the ball of radius \( {r} \) in \( {\mathbb{R}^d} \) then \( {v(r)=r^dv(1)} \) and \( {s(r)=v'(r)=dr^{d-1}v(1)} \) hence \( {s(1)=dv(1)} \).

An even more compact definition. If we think the dimension \( {d} \) as being a real positive number, we may observe that for all \( {x\neq0} \),

\[ \lim_{d\rightarrow2}\frac{\frac{1}{|x|^{d-2}}-1}{d-2} =\partial_{s=0}\frac{1}{|x|^{s}} =\partial_{s=0}\mathrm{e}^{-s\log|x|} =-\log|x|. \]

This means that the formula \( {\frac{1}{(d-2)|x|^{d-2}}} \), already valid for \( {d=1} \) and \( {d\geq3} \), is actually also valid for \( {d=2} \) provided that we remove a singularity. This suggests to define the kernel, for all integers \( {d\geq1} \) and even all real numbers \( {d\geq1} \), and all \( {x\neq0} \), as

\[ g(x):=\frac{1}{(d-2)|x|^{d-2}}-\frac{1}{d-2}, \]

which satisfies

\[ \Delta g=-2\frac{\pi^{d/2}}{\Gamma(d/2)}\delta_0. \]

Note that physically, the potential is important only for defining the field, and as a consequence, the potential is defined up to an additive constant. In other words, what matters is the differences of potential values rather than the values themselves.

Riesz kernels. The Riesz kernel in dimension \( {d\geq2} \) and parameter \( {s\geq0} \) with \( {0<s<d} \) is defined for all \( {x\neq0} \) by the formula

\[ g(x):=\frac{1}{|x|^{d-s}}. \]

We recover the Coulomb/Newton kernel when \( {s=2} \). The Riesz kernel is the fundamental solution of the fractional Laplacian, which is a Fourier multiplier, and a non-local operator when \( {d\neq2} \). Its inverse is a convolution operator.

Conclusion. All in all, if we would like to incorporate all cases in a compact formula, we could consider in \( {\mathbb{R}^d} \), \( {d\geq1} \), for all \( {s\in\mathbb{R}} \) and \( {x\neq0} \) the kernel

\[ g(x):=\frac{1}{s|x|^s}, \]

with the convention \( {g(x)=-\log|x|} \) if \( {s=0} \). The Coulomb/Newton case corresponds to taking \( {s=d-2} \) as we have seen. Indeed, this is more or less the choice often made by my colleague Mathieu Lewin for instance, see for example arXiv:1905.09138.

Entropy. The logarithm appears as a derivative of power functions also in relation with entropy and hypercontractivity, as explained in a previous post. The same for the derivation of the logarithmic Sobolev inequality from the Beckner inequalities.

Further reading.

Final word. Mathematics is also revealing common structures among apparently different things. In Physics, going beyond integers and the apparent physical meaning has advantages, as in the replica trick. You may already know this aphorism by Henri Bouasse, French Physicist from Toulouse of the 19-th century: Le physicien traite les problèmes du véhicule à une roue (la brouette), à deux roues (tilbury ou bicyclette), à trois, à quatre roues. Le mathématicien traite le problème général du véhicule à \( {n} \) roues, \( {n} \) étant entier ou fractionnaire, positif ou négatif, réel ou imaginaire.

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Enseigner

MatrixTrois idées basiques pour enseigner différemment :

  • Demander au département d’installer un serveur JupyterHub pour faire du Julia/Python/R
  • Utiliser le Google colaboratory ou rstudio.cloud pour faire du Python ou du R en ligne (1)
  • Demander aux étudiants de rendre leur projet sur github pour apprendre à s’en servir (2)

(1) signalé par mon collègue Jamal Atif. L’analogue chez Microsoft est Azure Notebooks.

(2) signalé par mon collègue Robin Ryder.

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