B. Jancovici

On the relativistic degenerate electron gas

RésuméOn étudie les propriétés d’un gaz d’électrons à température zéro et à densité assez forte pour que l’énergie de Fermi soit relativiste (de telles densités et des températures relativement faibles se rencontrent au sein des naines blanches). Le problème est étudié dans la formulation diélectrique de l’approximation des quasi-bosons: les paires électrontrou qui apparaissent à cause des interactions coulombiennes ou des interactions avec le champ des photons transverses sont traitées comme des bosons, et les processus autres que la création et l’annihilation de telles paires sont négligés; les positrons sont inclus parmi les trous possibles. On écrit et on diagonalise des hamiltoniens modèles décrivant le système à cette approximation. On calcule des constantes diélectriques longitudinale et transversale, dépendant du nombre d’onde et de la fréquence, et décrivant la réponse du système à des excitations extérieures; la renormalisation de charge nécessaire apparaît très naturellement. On étudie la propagation des ondes longitudinales et transversales; on en calcule la loi de dispersion. On calcule enfin l’énergie de l’état fondamental jusqu’au terme d’ordree4 loge2.RiassuntoSi studiano le proprietà di un gas di elettroni a temperatura zero e di densità abbastanza elevata perchè l’energia di Fermi sia relativistica (densità simili e temperature relativamente basse si riscontrano entro le nane bianche). Si studia il problema nella formulazione dielettrica dell’approssimazione di quasi-bosoni; si trattano come bosoni le coppie elettrone-buca che compaiono a causa delle interazioni coulombiane o delle interazioni con il campo dei fotoni trasversi, e si trascurano tutti i processi che non siano la creazione e l’annichilazione di tali coppie; tra le possibili buche si comprendono anche i positroni. Si scrivono e si diagonalizzano gli hamiltoniani modello che descrivono il sistema in questa approssimazione. Si calcolano le costanti dielettriche longitudinale e trasversale, che dipendono dal numero d’onda e dalla frequenza e che descrivono la risposta del sistema alle eccitazioni esterne; la necessaria rinormalizzazione della carica compare in modo molto naturale. Si studia la propagazione delle onde longitudinale e trasversa e se ne calcola la legge di dispersione. Si calcola infine l’energia dello stato fondamentale sino al termine dell’ordinee4 loge2.

The collective vibrations of a many-fermion system

Abstract The energy levels of a many-fermion system are investigated through an extension of the generator coordinate method of Peierls-Yoccoz and Wheeler-Griffin. A trial wave function Ψ for the system is taken as a superposition Ψ = ʃF(z)Φ(z) d z of all possible independent particle wave function Φ in the neighbourhood of the Hartree-Fock ground-state wave function Φ0; the functions Φ depend on many parameters z. An integral equation is established for the generator function F(z). Through appropriate approximations, this integral equation is transformed into the Schrodinger equation for a set of coupled harmonic oscillators, which can be solved. The energies and wave functions are obtained for the ground state and the low excited states of the system. The present approach is equivalent to the random phase approximation.

Nuclear reaction rate enhancement in dense stellar matter

The enhancement factor for the rate of thermonuclear reactions in dense stellar matter is calculated in the one-component, strongly coupled plasma model. The nuclear reaction rate is related to the short-range behavior of the quantum-mechanical pair correlation function. Within the range of parameters which is investigated here, the pair correlation function is expressed in terms of classical quantities, which are known from existing computer results; the classical potential of mean force is found to play an important role, from the practical point of view. Numerical results are given.

Classical Coulomb systems near a plane wall. II

The equilibrium structure of classical Coulomb systems bounded by a plane hard wall is studied near that wall. A general sum rule is derived for the asymptotic form of the charge-charge correlation function along the wall. The exact results which can be obtained for the two-dimensional one-component plasma provide a test for this new sum rule, as well as for other already known sum rules or their generalizations.

Pair correlation function in a dense plasma and pycnonuclear reactions in stars

The short-range behavior of the pair correlation function in a dense onecomponent plasma (jellium) is investigated. As an intermediate step, the short-range behavior of the classical pair correlation function is obtained. Actually, although the temperature and the density are assumed to be such that the thermodynamic properties are almost classical, quantum mechanics (tunnel effect) always dominates the pair correlation function at short distances. The quantum pair correlation function is calculated by treating the many-body quantum effects by a perturbation theory, and by using a semiclassical approximation based on path integrals. The results are applied to the computation of the nuclear reaction rate in dense stellar matter (pycnonuclear reactions).

The electrical double layer: A solvable model

In classical equilibrium statistical mechanics, the two‐dimensional two‐component Coulomb gas is exactly solvable at the special value of the reduced inverse temperature Γ=2. This is used for building an exactly solvable model of the electrical double layer. A charged hard wall (primitive electrode), a polarizable interface, an ideal conductor electrode, a semipermeable membrane are studied: the density profiles and correlation functions are computed. The differential capacity and the surface tension are also obtained.

On the two-dimensional Coulomb gas

This is a sequel to a recent work of Gaudin, who studied the classical equilibrium statistical mechanics of the two-dimensional Coulomb gas on a lattice at a special value of the coupling constantГ such that the model is exactly solvable. This model is briefly reviewed, and it is shown that the correlation functions obey the sum rules that characterize a conductive phase. A related model in which the particles are constrained to move on an array of equidistant parallel lines has simpler mathematics, and the asymptotic behavior of its correlation functions is studied in some detail. In the low-density limit, the lattice model is expected to have the same properties as a system of charged, hard disks; the correlation functions, internal energy, and specific heat of the latter are discussed.

Coulomb Systems Seen as Critical Systems: Finite-Size Effects in Two Dimensions

It is known that the free energy at criticality of a finite two-dimensional system of characteristic sizeL has in general a term which behaves like logL asL→∞; the coefficient of this term is universal. There are solvable models of two-dimensional classical Coulomb systems which exhibit the same finite-size correction (except for its sign) although the particle correlations are short-ranged, i.e., noncritical. Actually, the electrical potential and electrical field correlationsare critical at all temperatures (as long as the Coulomb system is a conductor), as a consequence of the perfect screening property of Coulomb systems. This is why Coulomb systems have to exhibit critical finite-size effects.

CLASSICAL COULOMB SYSTEMS SCREENING AND CORRELATIONS REVISITED

From the laws of macroscopic electrostatics of conductors (in particular, the existence of screening), taken as given, one can deduce universal properties for the thermal fluctuations in a classical Coulomb system at equilibrium. The universality is especially apparent in the long-range correlations of the electrical potentials and fields. The charge fluctuations are derived from the field fluctuations. This is a convenient way to study the surface charge fluctuations on a conductor with boundaries. Explicit results are given for simple geometries. The potentials and the fields have Gaussian fluctuations, except for a short-distance cutoff.From the laws of macroscopic electrostatics of conductors (in particular the existence of screening) taken for granted, one can deduce universal properties for the thermal fluctuations in a classical Coulomb system at equilibrium. The universality is especially apparent in the long-range correlations of the electrical potentials and fields. The charge fluctuations are derived from the field fluctuations. This is a convenient way for studying the surface charge fluctuations on a conductor with boundaries. Explicit results are given for simple geometries. The potentials and the fields have Gaussian fluctuations, except for a short-distance cutoff.

Large charge fluctuations in classical Coulomb systems

The typical fluctuation of the net electric chargeQ contained in a subregionΛ of an infinitely extended equilibrium Coulomb system is expected to grow only as √S, whereS is the surface area ofΛ. For some cases it has been previously shown thatQ/√S has a Gaussian distribution as ¦Λ¦→∞. Here we study the probability law for larger charge fluctuations (large-deviation problem). We discuss the case when both ¦Λ¦ andQ are large, but now withQ of an order larger than √S. For a given value ofQ, the dominant microscopic configurations are assumed to be those associated with the formation of a double electrical layer along the surface ofΛ. The probability law forQ is then determined by the free energy of the double electrical layer. In the case of a one-component plasma, this free energy can be computed, for large enoughQ, by macroscopic electrostatics. There are solvable two-dimensional models for which exact microscopic calculations can be done, providing more complete results in these cases. A variety of behaviors of the probability law are exhibited.