Fourth moment sum rule for the charge correlations of a two-component classical plasma

We consider an ionic fluid made with two species of mobile particles carrying either a positive or a negative charge. We derive a sum rule for the fourth moment of equilibrium charge correlations. Our method relies on the study of the system response to the potential created by a weak external charge distribution with slow spatial variations. The induced particle densities, and the resulting induced charge density, are then computed within density functional theory, where the free energy is expanded in powers of the density gradients. The comparison with the predictions of linear response theory provides a thermodynamical expression for the fourth moment of charge correlations, which involves the isothermal compressibility as well as suitably defined partial compressibilities. The familiar Stillinger-Lovett condition is also recovered as a by-product of our method, suggesting that the fourth moment sum rule should hold in any conducting phase. This is explicitly checked in the low density regime, within the Abe-Meeron diagrammatical expansions. Beyond its own interest, the fourth-moment sum rule should be useful for both analyzing and understanding recently observed behaviours near the ionic critical point

Quantum Coulomb systems : screening, recombination and van der Waals forces

The study of quantum Coulomb systems at equilibrium is important for understanding properties of matter in many physical situations. Screening, recombination and van der Waals forces are basic phenomena which result from the interplay of Coulomb interactions, collective effects and quantum mechanics. Those phenomena are introduced in the first part of this lecture, through various physical examples. Their treatment within mean-field theories and phenomenological approaches is also exposed, while related predictions are discussed. This sheds light on fundamental issues, which must be analyzed without any \textsl{a priori} approximations or modelizations. The second part of this lecture is precisely devoted to the presentation of various exact results for the quantum proton-electron hydrogen plasma. Such results are derived within the Screened Cluster Representation, which is constructed by combining the path integral representation of the Coulomb gas with Mayer-like diagrammatical techniques. They illustrate the breakdown of Debye exponential screening by quantum fluctuations, as well as the emergence of familiar chemical species in suitable low-temperature and low-density limits. Also, the amplitude of van der Waals forces is shown to be reduced by free charges

Approach to a stationary state in an external field

International audienceWe study relaxation towards a stationary out of equilibrium state by analizing a one-dimensional stochastic process followed by a particle accelerated by an external field and propagating through a thermal bath. The effect of collisions is described within Botlzmann's kinetic theory. We present analytical solutions for the Maxwell gas and for the very hard particle model. The exponentially fast relaxation of the velocity distribution toward the stationary form is demonstrated. In the reference frame moving with constant drift velocity the hydrodynamic diffusive mode is shown to govern the distribution in the position space. We show that the exact value of the diffusion coefficient for any value of the field is correctly predicted by Green-Kubo autocorrelation formula generalized to the stationary state

Thermodynamics of atomic and ionized hydrogen : analytical results versus OPAL tables and Monte Carlo data

We compute thermodynamical properties of a low-density hydrogen gas within the physical picture, in which the system is described as a quantum electron-proton plasma interacting via the Coulomb potential. Our calculations are done using the exact Scaled Low-Temperature (SLT) expansion, which provides a rigorous extension of the well known virial expansion -- valid in the fully ionized phase -- into the Saha regime where the system is partially or fully recombined into hydrogen atoms. After recalling the SLT expansion of the pressure [A. Alastuey et al, J. Stat. Phys. {\bf 130}, 1119 (2008)], we obtain the SLT expansions of the chemical potential and of the internal energy, up to to order $\exp(|E_H|/kT)$ included ($E_H \simeq -13.6$ eV). Those truncated expansions describe the first five non-ideal corrections to the ideal Saha law. They account exactly, up to the considered order, for all effects of interactions and thermal excitations, including the formation of bound states (atom $H$, ions $H^-$ and $H_2^+$, molecule $H_2$, ...) and atom-charge and atom-atom interactions. Among the five leading corrections, three are easy to evaluate, while the remaining ones involve well-defined internal partition functions for molecule $H_2$ and ions $H^-$ and $H_2^+$, for which no closed-form analytical formula exist currently. We provide accurate low-temperature approximations for those partition functions by using known values of rotational and vibrational energies. We compare then the predictions of the SLT expansion with the OPAL EOS and data of path integral quantum Monte Carlo (PIMC) simulations. In general, a good agreement is found. At low densities, the simple analytical SLT formulae reproduce the values of the OPAL tables up to the last digit in a large range of temperatures, while at higher densities ($\rho\sim10^{-2}$ g/cm$^3$), some discrepancies between the SLT, OPAL and PIMC results are observed

Analysis of path integrals at low temperature : Box formula, occupation time and ergodic approximation

We study the low temperature behaviour of path integrals for a simple one-dimensional model. Starting from the Feynman-Kac formula, we derive a new functional representation of the density matrix at finite temperature, in terms of the occupation times of Brownian motions constrained to stay within boxes with finite sizes. From that representation, we infer a kind of ergodic approximation, which only involves double ordinary integrals. As shown by its applications to different confining potentials, the ergodic approximation turns out to be quite efficient, especially in the low-temperature regime where other usual approximations fail

Generalized Stillinger-Lovett conditions for the one-component plasma

We derive a new set of sum rules for the equilibrium correlations of the classical one-component plasma. These sum rules can be viewed as generalizations of the usual Stillinger-Lovett rule refering to the two-point correlations. They characterize the long-wavelength behaviour of the Fourier transform (with respect to one point) of the ( n + 1)-point charge correlation function for any n. Like the Stillinger-Lovett rule, the new rules express perfect screening conditions in a plasma phase.Nous établissons un nouvel ensemble de règles de somme concernant les corrélations d'équilibre du plasma classique à une composante. Ces règles peuvent être vues comme des généralisations de la règle usuelle de Stillinger-Lovett relative aux corrélations à deux points. Elles caractérisent le comportement aux grandes longueurs d'onde de la transformée de Fourier (par rapport à un seul point) de la fonction de corrélation de charge à (n + 1) points (n quelconque). Comme la règle de Stillinger-Lovett, les nouvelles règles traduisent des conditions d'écran parfait en phase plasma

Pollution algébrique des corrélations de charge d'un électrolyte classique au point critique de la transition liquide-gaz

We consider a classical Two-Component Plasma analog of the Restricted Primitive Model of electrolyte, where the hard-core interaction is replaced by a soft differentiable potential. Within the Born-Green-Yvon hierarchy for the equilibrium distribution functions, we shed light on an infection mechanism where the charge correlations are polluted by the density correlations at the critical point of the liquid-gas transition. This implies an algebraic decay of critical charge correlations. Such breakdown of exponential clustering should provide dielectric rather than conducting properties at the critical point, leading to the violation of certain charge-charge sum rules. This is in agreement with Monte Carlo simulations.Nous considérons un plasma à deux composantes analogue au modèle primitif restreint d'un électrolyte, où l'interaction de sphères dures est remplacée par un potentiel régulier différentiable. Dans le cadre des équations de la hiérarchie Born-Green-Yvon , nous mettons en évidence un mécanisme d'infection dans lequel les corrélations de charge sont polluées par les corrélations de densité au point critique de la transition liquide-gaz. Ceci entraîne une décroissance algébrique des corrélations de charge critiques. Une telle disparition de l'écran exponentiel devrait conduire à des propriétés diélectriques plutôt que conductrices au point critique, en lien avec la violation de certaines règles de somme pour les corrélations de charge. Cette prédiction est en accord avec les simulations de Monte-Carlo

Approximations simples d'intégrales de chemins à température finie

LYON-ENS Sciences (693872304) / SudocSudocFranceF

Van der Waals forces in presence of free charges: an exact derivation from equilibrium quantum correlations

International audienc