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L. Samaj

Publications and source records attributed to L. Samaj.

At least 37 records · Page 2Linked to original sources
arXiv2010

Strong-Coupling Theory of Counter-ions at Charged Plates

We present an analytical approach to the strong coupling regime of similarly and highly charged plates in the presence of counter-ions. The procedure is physically transparent and based on an exact expansion around the ground state formed by the two-dimensional Wigner crystal of counter-ions. The one plate problem is worked out, together with the two plates situation. Unlike previous approaches, the expansion is free of divergences, and is shown to be in excellent agreement with available data of Monte-Carlo simulations under strong Coulombic couplings. The present results shed light on the like-charge attraction regime.

cond-mat.stat-mech↗
arXiv2007

A Trickiness of the High-Temperature Limit for Number Density Correlation Functions in Classical Coulomb Fluids

The Debye-Hückel theory describes rigorously the thermal equilibrium of classical Coulomb fluids in the high-temperature $β\to 0$ regime ($β$ denotes the inverse temperature). It is generally believed that the Debye-Hückel theory and the systematic high-temperature expansion provide an adequate description also in the region of small {\em strictly positive} values of $β>0$. This hypothesis is tested in the present paper on a two-dimensional Coulomb gas of pointlike $+/-$ unit charges interacting via a logarithmic potential which is equivalent to an integrable sine-Gordon field model. In particular, we apply a form factor method to obtain the exact asymptotic large-distance behavior of particle correlation functions, considered in the charge and number density combinations. We first determine the general forms of the leading and subleading asymptotic terms at strictly positive $β>0$ and then evaluate their high-temperature $β\to 0$ forms. In the case of the {\em charge} correlation function, the leading asymptotic term at a strictly positive $β>0$ is also the leading one in the high-temperature $β\to 0$ regime. On the contrary, the $β\to 0$ behavior of the {\em number density} correlation function is accompanied by an interference between the first two asymptotic terms. Consequently, the large-distance behavior of this function exhibits a discontinuity when going from strictly positive values of $β>0$ to the Debye-Hückel limit $β\to 0$. This is the crucial conclusion of the paper: the large-distance asymptotics and the high-temperature limit do not commute for the density correlation function of the two-dimensional Coulomb gas.

cond-mat.stat-mech↗
arXiv2007

A Generalization of the Stillinger-Lovett Sum Rules for the Two-Dimensional Jellium

In the equilibrium statistical mechanics of classical Coulomb fluids, the long-range tail of the Coulomb potential gives rise to the Stillinger-Lovett sum rules for the charge correlation functions. For the jellium model of mobile particles of charge $q$ immersed in a neutralizing background, the fixing of one of the $q$-charges induces a screening cloud of the charge density whose zeroth and second moments are determined just by the Stillinger-Lovett sum rules. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge $Z q$ immersed in the bulk interior of the 2D jellium with the coupling constant $Γ=βq^2$ ($β$ is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge $Z>-2/Γ$. The derivation is based on a mapping technique of the 2D jellium at the coupling $Γ$ = (even positive integer) onto a discrete 1D anticommuting-field theory; we assume that the final results remain valid for all real values of $Γ$ corresponding to the fluid regime. The generalized sum rules reproduce for arbitrary coupling $Γ$ the standard Z=1 and the trivial Z=0 results. They are also checked in the Debye-Hückel limit $Γ\to 0$ and at the free-fermion point $Γ=2$. The generalized second-moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space.

cond-mat.stat-mech↗
arXiv2007

Correlations and sum rules in a half-space for a quantum two-dimensional one-component plasma

This paper is the continuation of a previous one [L. {Š}amaj and B. Jancovici, 2007 {\it J. Stat. Mech.} P02002]; for a nearly classical quantum fluid in a half-space bounded by a plain plane hard wall (no image forces), we had generalized the Wigner-Kirkwood expansion of the equilibrium statistical quantities in powers of Planck's constant $\hbar$. As a model system for a more detailed study, we consider the quantum two-dimensional one-component plasma: a system of charged particles of one species, interacting through the logarithmic Coulomb potential in two dimensions, in a uniformly charged background of opposite sign, such that the total charge vanishes. The corresponding classical system is exactly solvable in a variety of geometries, including the present one of a half-plane, when $βe^2=2$, where $β$ is the inverse temperature and $e$ is the charge of a particle: all the classical $n$-body densities are known. For the quantum one-component plasma, two sum rules involving the truncated two-body density (and, for one of them, the density profile) have been derived, a long time ago, by heuristic macroscopic arguments: one sum rule is about the asymptotic form along the wall of the truncated two-body density, the other one is about the dipole moment of the structure factor. In the two-dimensional case at $βe^2=2$, we have now explicit expressions up to order $\hbar^2$ of these two quantum densities, thus we can microscopically check the sum rules at this order. The checks are positive, reinforcing the idea that the sum rules are correct.

cond-mat.stat-mech↗
arXiv2007

Wigner-Kirkwood expansion for semi-infinite quantum fluids

For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength $λ$. This result permits one to generate an analogous $λ$-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type $\exp(-2x^2/λ^2)$ ($x$ denotes the distance from the wall boundary) which are non-analytic in $λ$. Under some condition, the analyticity in $λ$ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of $λ$, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the $ν$-dimensional (integer $ν\ge 2$) space.

cond-mat.stat-mech↗
arXiv2006

Renormalization of Hard-Core Guest Charges Immersed in Two-Dimensional Electrolyte

This paper is a continuation of a previous one [L. {Š}amaj, {\it J. Stat. Phys.} {\bf 120}:125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike $\pm$ unit charges, the latter system being in the stability-against-collapse regime of reduced inverse temperatures $0\le β<2$. In the previous work, using a sine-Gordon representation of the Coulomb gas, an exact renormalized-charge formula was derived for the special case of the {\em pointlike} guest charge $Q$, in its stability regime $β| Q| < 2$. In the present paper, we extend the renormalized-charge treatment to the guest charge with a hard core of radius $σ$, which allows us to go beyond the stability border $β| Q| = 2$. In the limit of the hard-core radius much smaller than the correlation length of the Coulomb-gas species and at a strictly finite temperature, due to the counterion condensation in the extended region $β| Q| >2$, the renormalized charge $Q_{\rm ren}$ turns out to be a periodic function of the bare charge $Q$ with period 1. The renormalized charge therefore does not saturate at a specific finite value as $| Q| \to\infty$, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits $β\to 0$ and $Q\to\infty$ with the product $βQ$ being finite, one reproduces correctly the monotonic dependence of $βQ_{\rm ren}$ on $βQ$ in the guest-charge stability region $β| Q| <2$ and the Manning-Oosawa type of counterion condensation with the uniform saturation of $βQ_{\rm ren}$ at the value $4/π$ in the region $β| Q|\ge 2$.

cond-mat.stat-mech↗
arXiv2005

Exactly solvable model of the 2D electrical double layer

We consider equilibrium statistical mechanics of a simplified model for the ideal conductor electrode in an interface contact with a classical semi-infinite electrolyte, modeled by the two-dimensional Coulomb gas of pointlike $\pm$ unit charges in the stability-against-collapse regime of reduced inverse temperatures $0\le β<2$. If there is a potential difference between the bulk interior of the electrolyte and the grounded interface, the electrolyte region close to the interface (known as the electrical double layer) carries some nonzero surface charge density. The model is mappable onto an integrable semi-infinite sine-Gordon theory with Dirichlet boundary conditions. The exact form-factor and boundary state information gained from the mapping provide asymptotic forms of the charge and number density profiles of electrolyte particles at large distances from the interface. The result for the asymptotic behavior of the induced electric potential, related to the charge density via the Poisson equation, confirms the validity of the concept of renormalized charge and the corresponding saturation hypothesis. It is documented on the non-perturbative result for the asymptotic density profile at a strictly nonzero $β$ that the Debye-Hückel $β\to 0$ limit is a delicate issue.

cond-mat.stat-mech↗
arXiv2005

"Screening" of universal van der Waals - Casimir terms by Coulomb gases in a fully-finite two-dimensional geometry

This paper is a continuation of a previous one [Jancovici and Samaj, 2004 J. Stat. Mech. P08006] dealing with classical Casimir phenomena in semi-infinite wall geometries. In that paper, using microscopic Coulomb systems, the long-ranged Casimir force due to thermal fluctuations in conducting walls was shown to be screened by the presence of an electrolyte between the walls into some residual short-ranged force. Here, we aim to extend the study of the screening (cancellation) phenomena to universal Casimir terms appearing in the large-size expansions of the grand potentials for microscopic Coulomb systems confined in fully-finite 2D geometries, in particular the disc geometry. Two cases are solved exactly: the high-temperature (Debye-Hückel) limit and the Thirring free-fermion point. Similarities and fundamental differences between fully-finite and semi-infinite geometries are pointed out.

cond-mat.stat-mech↗
arXiv2005

Anomalous Effects of "Guest" Charges Immersed in Electrolyte: Exact 2D Results

We study physical situations when one or two "guest" arbitrarily-charged particles are immersed in the bulk of a classical electrolyte modelled by a Coulomb gas of positive/negative unit point-like charges, the whole system being in thermal equilibrium. The models are treated as two-dimensional with logarithmic pairwise interactions among charged constituents; the (dimensionless) inverse temperature $β$ is considered to be smaller than 2 in order to ensure the stability of the electrolyte against the collapse of positive-negative pairs of charges. Based on recent progress in the integrable (1+1)-dimensional sine-Gordon theory, exact formulas are derived for the chemical potential of one guest charge and for the asymptotic large-distance behavior of the effective interaction between two guest charges. The exact results imply, under certain circumstances, anomalous effects such as an effective attraction (repulsion) between like-charged (oppositely-charged) guest particles and the charge inversion in the electrolyte vicinity of a highly-charged guest particle. The adequacy of the concept of renormalized charge is confirmed in the whole stability region of inverse temperatures and the related saturation phenomenon is revised.

cond-mat.stat-mech↗
arXiv2005

Saturation of Electrostatic Potential: Exactly Solvable 2D Coulomb Models

We test the concepts of renormalized charge and potential saturation, introduced within the framework of highly asymmetric Coulomb mixtures, on exactly solvable Coulomb models. The object of study is the average electrostatic potential induced by a unique "guest" charge immersed in a classical electrolyte, the whole system being in thermal equilibrium at some inverse temperature $β$. The guest charge is considered to be either an infinite hard wall carrying a uniform surface charge or a charged colloidal particle. The systems are treated as two-dimensional; the electrolyte is modelled by a symmetric two-component plasma of point-like $\pm e$ charges with logarithmic Coulomb interactions. Two cases are solved exactly: the Debye-Hückel limit $βe^2\to 0$ and the Thirring free-fermion point $βe^2 = 2$. The results at the free-fermion point can be summarized as follows: (i) The induced electrostatic potential exhibits the asymptotic behavior, at large distances from the guest charge, whose form is different from that obtained in the Debye-Hückel (linear Poisson-Boltzmann) theory. This means that the concept of renormalized charge, developed within the nonlinear Poisson-Boltzmann theory to describe the screening effect of the electrolyte cloud, fails at the free-fermion point. (ii) In the limit of an infinite bare charge, the induced electrostatic potential saturates at a finite value in every point of the electrolyte region. This fact confirms the previously proposed hypothesis of potential saturation.

cond-mat.stat-mech↗
arXiv2004

Screening of classical Casimir forces by electrolytes in semi-infinite geometries

We study the electrostatic Casimir effect and related phenomena in equilibrium statistical mechanics of classical (non-quantum) charged fluids. The prototype model consists of two identical dielectric slabs in empty space (the pure Casimir effect) or in the presence of an electrolyte between the slabs. In the latter case, it is generally believed that the long-ranged Casimir force due to thermal fluctuations in the slabs is screened by the electrolyte into some residual short-ranged force. The screening mechanism is based on a "separation hypothesis": thermal fluctuations of the electrostatic field in the slabs can be treated separately from the pure image effects of the "inert" slabs on the electrolyte particles. In this paper, by using a phenomenological approach under certain conditions, the separation hypothesis is shown to be valid. The phenomenology is tested on a microscopic model in which the conducting slabs and the electrolyte are modelled by the symmetric Coulomb gases of point-like charges with different particle fugacities. The model is solved in the high-temperature Debye-Hückel limit (in two and three dimensions) and at the free fermion point of the Thirring representation of the two-dimensional Coulomb gas. The Debye-Hückel theory of a Coulomb gas between dielectric walls is also solved.

cond-mat.stat-mech↗
arXiv2004

Translation Symmetry Breaking in the One-Component Plasma on the Cylinder

The two-dimensional one-component plasma, i.e. the system of point-like charged particles embedded in a homogeneous neutralizing background, is studied on the surface of a cylinder of finite circumference, or equivalently in a semiperiodic strip of finite width. The model has been solved exactly by Choquard et al. at the free-fermion coupling $Γ=2$: in the thermodynamic limit of an infinitely long strip, the particle density turns out to be a nonconstant periodic function in space and the system exhibits long-range order of the Wigner-crystal type. The aim of this paper is to describe, qualitatively as well as quantitatively, the crystalline state for a larger set of couplings $Γ=2γ$ ($γ=1,2,...$ a positive integer) when the plasma is mappable onto a one-dimensional fermionic theory. The fermionic formalism, supplemented by some periodicity assumptions, reveals that the density profile results from a hierarchy of Gaussians with a uniform variance but with different amplitudes. The number and spatial positions of these Gaussians within an elementary cell depend on the particular value of $γ$. Analytic results are supported by the exact solution at $γ=1$ ($Γ=2$) and by exact finite-size calculations at $γ=2,3$.

cond-mat.stat-mech↗
arXiv2004

Is the Two-Dimensional One-Component Plasma Exactly Solvable?

The model under consideration is the two-dimensional (2D) one-component plasma of pointlike charged particles in a uniform neutralizing background, interacting through the logarithmic Coulomb interaction. Classical equilibrium statistical mechanics is studied by non-traditional means. The question of the potential integrability (exact solvability) of the plasma is investigated, first at arbitrary coupling constant Γvia an equivalent 2D Euclidean-field theory, and then at the specific values of Γ=2*integer via an equivalent 1D fermionic model. The answer to the question in the title is that there is strong evidence for the model being not exactly solvable at arbitrary Γbut becoming exactly solvable at Γ=2*integer. As a by-product of the developed formalism, the gauge invariance of the plasma is proven at the free-fermion point Γ=2; the related mathematical peculiarity is the exact inversion of a class of infinite-dimensional matrices.

cond-mat.stat-mech↗
arXiv2003

Microscopic Calculation of the Dielectric Susceptibility Tensor for Coulomb Fluids II

For a Coulomb system contained in a domain Λ, the dielectric susceptibility tensor χ_Λ is defined as relating the average polarization in the system to a constant applied electric field, in the linear limit. According to the phenomenological laws of macroscopic electrostatics, χ_Λ depends on the specific shape of the domain Λ. In this paper we derive, using the methods of equilibrium statistical mechanics in both canonical and grand-canonical ensembles, the shape dependence of χ_Λ and the corresponding finite-size corrections to the thermodynamic limit, for a class of general ν-dimensional (ν\ge 2) Coulomb systems, of ellipsoidal shape, being in the conducting state. The microscopic derivation is based on a general principle: the total force acting on a system in thermal equilibrium is zero. The results are checked in the Debye-Hückel limit. The paper is a generalization of a previous one [L. Šamaj, J. Stat. Phys. 100:949 (2000)], dealing with the special case of a one-component plasma in two dimensions. In that case, the validity of the presented formalism has already been verified at the exactly solvable (dimensionless) coupling Γ= 2.

cond-mat.stat-mech↗
arXiv2003

Universal behavior of quantum Green's functions

We consider a general one-particle Hamiltonian H = - Δ_r + u(r) defined in a d-dimensional domain. The object of interest is the time-independent Green function G_z(r,r') = < r | (z-H)^{-1} | r' >. Recently, in one dimension (1D), the Green's function problem was solved explicitly in inverse form, with diagonal elements of Green's function as prescribed variables. The first aim of this paper is to extract from the 1D inverse solution such information about Green's function which cannot be deduced directly from its definition. Among others, this information involves universal, i.e. u(r)-independent, behavior of Green's function close to the domain boundary. The second aim is to extend the inverse formalism to higher dimensions, especially to 3D, and to derive the universal form of Green's function for various shapes of the confining domain boundary.

math-ph↗
arXiv2002

The statistical mechanics of the classical two-dimensional Coulomb gas is exactly solved

The model under consideration is a classical 2D Coulomb gas of pointlike positive and negative unit charges, interacting via a logarithmic potential. In the whole stability range of temperatures, the equilibrium statistical mechanics of this fluid is exactly solvable via an equivalence with the integrable 2D sine-Gordon field theory. The exact solution includes the bulk thermodynamics, special cases of the surface thermodynamics, and the large-distance asymptotic behavior of the two-body correlation functions.

cond-mat.stat-mech↗
arXiv2002

Evolution of quantum systems with a scaling type of time-dependent Hamiltonians

We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.

quant-ph↗