Given a set of point-like “seeds” in $d$-dimensional Euclidean space, the associated Voronoi tessellation is the partitioning of space into cells such that each point of space is in the cell of the seed to which it is closest. In the case of independent and uniformly distributed seeds one speaks of a Poisson-Voronoi tessellation. We will discuss the statistical properties of rare or “extreme” cells: those (in $d=2$) that have a large number $n$ of sides; or (in $d=3$) that have a large number $n_F$ of faces, or that have a face with a large number $n_E$ of edges. Calculating the probability of such events reveals the entropic forces that are at play. Whereas microscopically we are facing a stochastic many-body problem, we will show that macroscopically there arise deterministic laws when $n$, $n_F$, or $n_E\to\infty$.
The properties of ordinary macroscopic matter are almost entirely determined by Coulomb interactions. Here I will give a brief survey of some exact results for such systems.
We prove a Lieb-Oxford-type inequality on the indirect part of the Coulomb energy (also known as the exchange-correlation energy) of a general many-particle quantum state, with a lower constant than the original one, but involving an additional gradient correction. The result is similar to an inequality of Benguria, Bley and Loss, except that our correction term is purely local, which is more usual for density functional theory. No previous knowledge of the subject will be assumed; a very short tutorial on the subject and its importance to quantum chemistry will be presented. (Joint work with Mathieu Lewin)
In the classical theory of fluids the derivatives with respect to particle density of correlation functions of order $k$ are related to spatial integrals of correlation functions of order $(k+1)$. We study the corresponding infinite hierarchy of equations and the resulting generalized compressibility equations. We show that the critical behaviour requires taking into account correlation functions of arbitrary order. Critical exponents corresponding to integrated $k$-particle spatial correlations are shown to grow linearly with order $k$. We prove that the application of the superposition approximation expressing higher order correlations in terms of lower order ones precludes the existence of a critical point. We also comment on some predictions emerging from the YGB hierarchy.
Large colloids, when immersed in a polar solvent like water, become charged by a fixed surface charge density, compensated by mobile “counterions”. Such sparse classical systems of particles exhibit poor screening properties, so the validity of standard Coulomb sum rules is questionable. The counterions mediate an effective interaction between like-charged colloids, which may be attractive at low enough temperatures. I shall concentrate on two-dimensional models with the logarithmic interaction potential, going from the finite disc to the semi-infinite planar geometry of the confining domain. For any coupling constant being a positive even integer, using an anticommuting representation of the partition function and many-body particle densities, a sequence of sum rules is derived. The amplitude function, which characterizes the asymptotic inverse-power law behavior of the two-body density along the wall, is found to be related to the particle density profile. The dielectric susceptibility tensor has the anticipated disc value in the thermodynamic limit, in spite of zero contribution from the bulk region.
We discuss recent developments on the one dimensional KPZ equation and its universality. We first show the exact solutions for the height distribution and stationary two point correlation functions. Then we explain various generalizations and applications. We introduce a few new models in the KPZ class with some possible applications, and also discuss wide applicability of the KPZ universality in various contexts.
Inspired from H-theorem requirements, a novel class of exact solutions to the quantum or classical Boltzmann equation is uncovered. These solutions, valid for arbitrary collision laws, hold for time-dependent confinement. We exploit them, in a reverse-engineering perspective, to work out a protocol that shortcuts any adiabatic transformation between two equilibrium states in an arbitrarily short time span, for an interacting system. Particle simulations corroborate the analytical predictions.
Collaboration with David Guéry-Odelin (LCAR, Toulouse).