One-Dimensional Coulomb Systems

Choquard, Ph.; Kunz, H. R.; Martin, Ph. A.; Navet, M.

doi:10.1007/978-3-642-81592-8_39

Cited by 14 publications

(24 citation statements)

References 6 publications

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“…On the other hand if a single fractional charge is placed in a system of integer charges then the resulting potential can never be screened because the ensemble of integer charges can do no more than cancel integer parts of the corresponding electric field. One dimensional Coulomb systems have been reviewed in (Choquard, Kunz, Martin & Navet 1981). Since then, highly non-trivial results have been obtained on the correlations of classical charges confined to a circle and interacting with a logarithmic potential (Forrester 1992, Forrester 1993b, Forrester 1993a.…”

Section: B Theorems On Debye Screeningmentioning

confidence: 99%

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Brydges

Martin

1999

Journal of Statistical Physics

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140

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Results on the correlations of low density classical and quantum Coulomb systems at equilibrium in three dimensions are reviewed. The exponential decay of particle correlations in the classical Coulomb system -Debye-Hückel screening -is compared and contrasted with the quantum case where strong arguments are presented for the absence of exponential screening. Results and techniques for detailed calculations that determine the asymptotic decay of correlations for quantum systems are discussed. Theorems on the existence of molecules in the Saha regime are reviewed. Finally, new combinatoric formulas for the coefficients of Mayer expansions are presented and their role in proofs of results on Debye-Hückel screening is discussed.Typeset using REVT E X III-3 C. The scaling limit of the lattice dipole gas III-5 IV. Semi-classical Coulomb gas IV-1 A. The Feynman-Kac representation IV-1 B. The gas of charged filaments IV-3 C. Quantum fluctuations destroy exponential screening IV-7 D. Origin of the van der Waals forces IV-9 E. Semi-classical analysis of Coulombic correlations IV-11 F. Breakdown of exponential screening in the Quantum Sine-Gordon representation IV-15 V. The emergence of thermodynamics and of various physical laws from the more fundamental levels of atomic theory and statistical mechanics were high points in our education in physics. But by the time we reached textbook treatments of the complex world of the Coulomb interaction, for example, formation of atoms and molecules and equilibria between them, we learned to demand less from theory and to be more content with reasoning by analogy consistent with thermodynamics. We know experimentally that atoms and molecules form, that they can behave like mixtures of ideal gases, so phenomenologically they have chemical potentials and we no longer insist that this emerge in some limit from N-body quantum Coulomb systems. We also know that Coulomb systems can have screening and dielectric phases. But when screening is expected to occur, it is quite a common practice to rather bluntly replace the Coulomb potential by a screened potential inherited from meanfield theories without inquiring too deeply into the legitimacy of this change. Likewise, there does not exist a first principle theory of the dielectric constant that does not presuppose the existence of atoms and molecules. At best, assuming that atoms form, one uses several laws, such as the Clausius-Mosotti formula, whose microscopic foundations have to be elucidated. In all these cases, this is a sensible attitude since most of non-relativistic physics is reputed to be hidden within the N-particle Coulomb Hamiltonian. There are however basic facts that can be cleanly formulated as limiting theories and shown to persist near the limit as well.This review is devoted to these types of results on Coulomb systems at low density. At low density the most famous properties of Coulomb systems are related to screening. The classical Debye-Hückel theory and its quantum analogue the random phase approximation have...

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Section: B Theorems On Debye Screeningmentioning

confidence: 99%

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Brydges

Martin

1999

Journal of Statistical Physics

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140

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“…(3) instead of log |x i − x j |. This well known model of 1d Coulomb gas has been studied earlier in the context of 1d charged plasma [51]. It is known as the one dimensional one component plasma (1d OCP) or the "jellium" model, where N charges of the same sign interact in the presence of a uniform background of opposite charges, assuring charge neutrality.…”

Section: Introductionmentioning

confidence: 99%

Extreme statistics and index distribution in the classical 1dCoulomb gas

Dhar¹,

Kundu²,

Majumdar³

et al. 2018

J. Phys. A: Math. Theor.

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We consider a one-dimensional gas of N charged particles confined by an external harmonic potential and interacting via the one-dimensional Coulomb potential. For this system we show that in equilibrium the charges settle, on an average, uniformly and symmetrically on a finite region centred around the origin. We study the statistics of the position of the rightmost particle x max and show that the limiting distribution describing its typical fluctuations is different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also compute the large deviation functions which characterise the atypical fluctuations of x max far away from its mean value. In addition, we study the gap between the two rightmost particles as well as the index N + , i.e., the number of particles on the positive semiaxis. We compute the limiting distributions associated to the typical fluctuations of these observables as well as the corresponding large deviation functions. We provide numerical supports to our analytical predictions. Part of these results were announced in a recent Letter, Phys. Rev. Lett. 119, 060601 (2017).

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“…Indeed this is also the 1dOCP or "jellium" model, where N charges of the same sign interact in the presence of a uniform background of opposite charges, assuring charge neutrality. This model is a paradigm for 1d charged plasma [39] as several observables can be calculated analytically [40][41][42][43][44].…”

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Exact Extremal Statistics in the Classical 1D Coulomb Gas

et al. 2017

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We consider a one-dimensional classical Coulomb gas of N like-charges in a harmonic potential -also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position xmax of the rightmost charge in the limit of large N . We show that the typical fluctuations of xmax around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of xmax for the Dyson's log-gas. We also compute the large deviation functions of xmax explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. Our theoretical predictions are verified numerically.The Tracy-Widom (TW) distribution has emerged ubiquitously in diverse systems in the recent past [1, 2]. It was originally discovered as the limiting distribution of the top eigenvalue x max of an N × N Gaussian random matrix [3]. Since then, it has appeared in various areas of physics [4,5], mathematics [6,7], and information theory [8]. For example, in physics it has appeared in stochastic growth models and related directed polymer in 1 + 1 dimensional random media belonging to the Kardar-Parisi-Zhang (KPZ) universality class [9-15], non-intersecting Brownian motions [16], non-interacting fermions in a one-dimensional trapping potential [17][18][19], disordered mesoscopic systems [20] and even in the YangMills gauge theory in 2-dimensions [16]. It has also been measured experimentally in several systems including liquid crystals [21], coupled fiber lasers [22] or disordered superconductors [23]. The TW distribution describes the probability of typical fluctuations of x max around its mean. In contrast, the atypical fluctuations of x max to the left and right, far from its mean, are described respectively by the left and right large deviation tails. These tails have been computed explicitly [24][25][26][27] and shown to correspond to two different thermodynamic phases separated by a third order phase transition [28,29]. Similar third order phase transitions have also been found in a variety of other systems [29][30][31][32][33][34][35].For Gaussian ensembles in random matrix theory (RMT), the joint probability distribution function (PDF) of the N real eigenvalues {x 1 , · · · , x N } is known explicitly [36,37] P({x i }) = B N e

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One-Dimensional Coulomb Systems

Cited by 14 publications

References 6 publications

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Extreme statistics and index distribution in the classical 1dCoulomb gas

Exact Extremal Statistics in the Classical 1D Coulomb Gas

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