Fugacity Expansion Research Articles

Composite reweighting SU(2) QCD at finite temperature

The Glasgow reweighting method is evaluated for SU(2) lattice gauge theory at nonzero μ and finite T. We establish that the ‘overlap problem’ of SU(3) measurements, in which the transition points determined from thermodynamic observables have an unphysical dependence on the value of μ used to generate ensembles for reweighting, persists for SU(2). By combining the information from different lattice ensembles we alleviate sampling bias in the fugacity expansion, and identify the Lee–Yang zeros associated with the transition to a high density phase that can plausibly be associated with diquark condensation. We also confirm the existence of a line of first order transitions above a critical point in the T– μ phase plane previously predicted by effective chiral lagrangian calculations.

Composite reweighting the Glasgow method for finite density QCD

The reweighting scheme developed in Glasgow to circumvent the lattice action becoming complex at finite density suffers from a pathological onset transition thought to be due to the reweighting. We present a new reweighting scheme based on this approach in which we combine ensembles to alleviate the sampling bias we identify in the polynomial coefficients of the fugacity expansion.

Density Profiles in a Classical Coulomb Fluid Near a Dielectric Wall. I. Mean-Field Scheme

The equilibrium density profiles in a classical multicomponent plasma near a hard wall made with a dielectric material characterized by a relative dielectric constant ∈w are studied from the first Born–Green–Yvon (BGY) equation combined with Poisson equation in a regime where Coulomb coupling is weak inside the fluid. In order to prevent the collapse between charges with opposite signs or between each charge and its dielectric image inside the wall when ∈w>1, hard-core repulsions are added to the Coulomb pair interaction. The charge-image interaction cannot be treated perturbatively and the density profiles vary very fast in the vicinity of the wall when ∈w≠1. The formal solution of the associated inhomogeneous Debye–Huckel equations will be given in Paper II, together with a systematic fugacity expansion which allows to retrieve the results obtained from the truncated BGY hierarchy. In the present paper the exact density profiles are calculated analytically up to first order in the coupling parameter. The expressions show the interplay between three effects: the geometric repulsion from the impenetrable wall; the electrostatic effective attraction (∈w>1) or repulsion (∈w<1) due to its dielectric response; and the Coulomb interaction between each charge and the potential drop created by the electric layer which appears as soon as the system is not symmetric. We exhibit how the charge density profile evolves between a structure with two oppositely-charged layers and a three-layer organization when ∈w varies. (The case of two ideally conducting walls will be displayed elsewhere).

Density Profiles in a Classical Coulomb Fluid Near a Dielectric Wall. II. Weak-Coupling Systematic Expansions

In the framework of the grand-canonical ensemble of statistical mechanics, we give an exact diagrammatic representation of the density profiles in a classical multicomponent plasma near a dielectric wall. By a reorganization of Mayer diagrams for the fugacity expansions of the densities, we exhibit how the long-range of both the self-energy and pair interaction are exponentially screened at large distances from the wall. However, the self-energy due to Coulomb interaction with images still diverges in the vicinity of the dielectric wall and the variation of the density is drastically different at short or large distances from the wall. This variation is involved in the inhomogeneous Debye-Huckel equation obeyed by the screened pair potential. Then the main difficulty lies in the determination of the latter potential at every distance. We solve this problem by devising a systematic expansion with respect to the ratio of the fundamental length scales involved in the two coulombic effects at stake. (The application of this method to a plasma confined between two ideally conducting plates and to a quantum plasma will be presented elsewhere). As a result we derive the exact analytical perturbative expressions for the density profiles up to first order in the coupling between charges. The mean-field approach displayed in Paper I is then justified.

Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb Systems. I. Low Fugacity Expansion

We present an exact field theoretical representation of the statistical mechanics of classical hard-core Coulomb systems. This approach generalizes the usual sine-Gordon theory valid for point-like charges or lattice systems to continuous Coulomb fluids with additional short-range interactions. This formalism is applied to derive the equation of state of the restricted primitive model of electrolytes in the low fugacity regime up to order ρ5/2 (ρ number density). We recover the results obtained by Haga by means of Mayer graphs expansions.

Second virial coefficient for the Landau diamagnetism of a two-component plasma

This paper investigates the density expansion of the thermodynamic properties of a two component plasma under the influence of a weak constant uniform magnetic field. We start with the fugacity expansion for the Helmholtz free energy. The leading terms with respect to the density are calculated by a perturbation expansion with respect to the magnetic field. We find a new magnetic virial function for a low density plasma which is exact in quadratic order with respect to the magnetic field. Using these results we compute the magnetization and the magnetic susceptibility.

The equation of state of a magnetized plasma

Results for thermodynamic functions of a low-density quantum Coulomb system in a uniform magnetic field are reviewed. We start with a fugacity expansion of the Helmholtz free energy. Results for the leading terms with respect to the fugacity are considered. A new magnetic virial function is presented which is exact in quadratic order with respect to the magnetic field. Possible applications are discussed.

XYmodels with disorder and symmetry-breaking fields in two dimensions

The combined effect of disorder and symmetry-breaking fields on the two-dimensional $\mathrm{XY}$ model is examined. The study includes disorder in the interaction among spins in the form of random phase shifts as well as disorder in the local orientation of the field. The phase diagrams are determined and the properties of the various phases and phase transitions are calculated. We use a renormalization-group approach in the Coulomb gas representation of the model. Our results differ from those obtained for special cases in previous works. In particular, we find a changed topology of the phase diagram that is composed of phases with long-range order, quasi-long-range order, and short-range order. The discrepancies can be ascribed to a breakdown of the fugacity expansion in the Coulomb gas representation. Implications for physical systems such as planar Josephson junctions and the faceting of crystal surfaces are discussed.

Frustration in finite density QCD

We present a detailed analysis of the QCD partition function in the Grand Canonical formalism. Using the fugacity expansion we find evidence for numerical instabilities in the standard evaluation of its coefficients. We discuss the origin of this problem and propose an issue to it. The correct analysis shows no evidence for a discontinuity in the baryonic density in the strong coupling limit. The moderate optimism that was inspired by the Grand Canonical Partition Function calculations in the last years has to be considered ill-founded.

Finite density QCD in the chiral limit

We present the first results of an exact simulation of full QCD at finite density in the chiral limit. We have used a MFA (Microcanonical Fermionic Average) inspired approach for the reconstruction of the Grand Canonical Partition Function of the theory; using the fugacity expansion of the fermionic determinant we are able to move continuously in the (β − μ) plane with m = 0.

Relativistic Quantum Statistical Aharonov-Bohm Effect of Many-Boson System in Mesoscopic Ring**The project supported by National Natural Science Foundation of China, and by XIAN Weijian through the Foundation of Zhongshan University Advanced Research Center in Hong Kong

The relativistic quantum statistical properties of charged many-boson system with rest mass for each boson confined in a mesoscopic ring are investigated. The small ring is threaded by Aharonov–Bohm magnetic flux . At low temperature, , we have performed analytical calculations starting from evaluating the logarithm of grand partition function by using Mellin transform. The pair-production problem is excluded from our consideration for the temperature is low. The number density, total energy, specific heat and persistent current are obtained in general. These thermodynamic functions oscillate periodically in with period , and they are related to the modified Bessel functions of the second kind The chemical potential is restricted by to ensure the positive number density of particles and the convergence of fugacity expansions. We find that these functions decay exponentially with temperature, and they also decrease as the circumference L increases. The non-relativistic and ultra-relativistic behaviors of these functions are discussed by considering the asymptotic behaviors of . There is no Bose–Einstein condensation in this system.

Ursell Operators in Statistical Physics II: Microscopic Properties of a Dilute Quantum Gas

We apply the formalism of Ursell operators, introduced in a previous article, to the calculation of reduced density operators (one and two-body density operators) in a dilute quantum gas ; the calculation is not a fugacity expansion and is therefore not limited to low degrees of degeneracy. We obtain quantum corrections for the one-particle density operator as a function of the second Ursell operator. For the two-body density operator, we examine how statistics and interactions combine their effects on the correlations between particles ; in particular we discuss in detail how hard cores potentials affect the short range correlations, a non-perturbative effect.

Screening in two-dimensional non-abelian vortex systems

We study charge screening in a system of two-dimensional non-abelian vortices, at finite temperature. Such vortices are generated after an SO(3) global symmetry group is spontaneously broken to a discrete subgroup Q 8 , where the latter is isomorphic to the non-abelian group of quatemions. Poisson-Boltzmann like equations are derived for the various inter-vortex potentials, and the solutions to lowest order in a small fugacity expansion, are shown to behave much like those in abelian or Coulomb charge systems. The consequences for the phase structure of the system are discussed, where the fugacities associated to the thermal production of the various types of non-abelian vortices, are shown to play a key role.

Yang-Lee theory and the conductor-insulator transition in asymmetric log-potential lattice gases

A feature of a conducting phase at low density is that there is a singularity in the fugacity expansion of the pressure, whereas the same expansion in the insulating phase gives an analytic series. The Yang-Lee characterization of a phase transition thus implies that in the conducting phase the zeros of the grand partition function must pinch the real axis in the complex scaled fugacity (ξ) plane at ξ=0, whereas in the insulating phase a neighborhood of ξ=0 must be zero free. Exact and numerical calculations are presented which suggest that for two-component log-potential lattice gases in one dimension with dimensionless couplingΓ, the zeros pinch the point ξ=0 forΓ<2, while forΓ⩾2 a neighborhood of ξ=0 is zero free. The conductor-insulator transition therefore takes place atΓ=2 independent of the density and other parameters in the model.

Simplified virial expansions in the canonical ensemble

Abstract For a classical monatomic real gas the well-known virial expansion of the equation of state and the reduced distribution function are rederived within the canonical ensemble formalism. In particular, two simplified methods are presented which lead directly to the desired result without making use of fugacity or perturbation expansions. The first method generalizes an argument given by Brout. This generalized argument is also used in the second method, which is based on cluster expansions and which is directly related to van Kampen's approach.

Two-Hilbert space formulations of the quantum statistical mechanics of reactive fluids: Dimer formation and decay

A two-Hilbert space formalism is first used to develop a general class of representations for the quantum mechanics of N-particle reactive systems. Here the N-particle Hilbert space ℋN is supplemented by a larger arrangement channel space 𝒞N of vectors with Hilbert space valued components for each N-particle clustering, and an injection mapping of ℋN vectors into ‘‘physical’’ 𝒞N vectors. Such representations, for which components of the latter vectors carry an appropriate physical clustering interpretation, provide a rigorous and flexible basis for describing the statistical mechanics of reactive fluids, where atoms and molecules are treated on an equal footing (the molecular picture). Corresponding equilibrium multispecies fugacity or virial expansions follow immediately. Here we focus on analysis of the (previously derived) arrangement channel BBGKY hierarchy for a system where recombination and dissociation, as well as exchange reactions, occur. This formulation (coupled with a corresponding scattering theory) automatically suggests a reactive Boltzmann ansatz which incorporates (standard) noninteracting asymptotic dynamics only for two-molecule nonreactive and reactive exchange collisions. In contrast, e.g., with three molecule recombination, two-molecule dynamics for all three pairs is included (as required for a description of recombination via gradual stabilization of metastables). Finally we compare the resulting reduced form of appropriate channel space hierarchy equations, for a process involving dimer formation and decay, with the corresponding kinetic equations of Lowry and Snider.

Fluids with highly directional attractive forces. III. Multiple attraction sites

We derive a reformulation of statistical thermodynamics for fluids of molecules which interact by highly directional attraction. The molecular model consists of a repulsive core and several sites of very short-ranged attraction. We explore the relationship between graph cancelation in the fugacity expansion and three types of steric incompatibility between repulsive and attractive interactions involving several molecules. The steric effects are used to best advantage in a limited regrouping of bonds. This controls the density parameters which appear when articulation points are eliminated in the graphical representation. Each density parameter is a singlet density for a species consisting of molecules with a specified set of sites bonded. The densities satisfy subsidiary conditions of internal consistency. These conditions are equivalent to a minimization of the Helmholtz free energyA. Graphical expressions forA and for the pressurep are derived. Analogs of thes-particle direct correlation functions and of the Ornstein-Zernike equation are found.

Fluids of hard diatomic molecules. I

The problem addressed is that of finding the complete orientation-dependent pair distribution function of a fluid of hard diatomic molecules. The treatment decomposes the Mayer f function into several terms and uses a graph-theoretical treatment. The basic idea is the explicit use of molecular geometry in constructing approximations. The fugacity expansion is found to be superior to the density expansion for this purpose. The neglect of large classes of graphs is shown to be exact for extremely asymmetric diatomics and is introduced as a new approximation for the less asymmetric case. Topological reduction of the reduced expansion leads to a result that resembles the density expansion, but contains an additional parameter. An integral equation of Ornstein–Zernike type is derived, and an approximation for closing the equation is suggested. The resulting equations are calculationally tractable and can be used to obtain both the centers correlation function and the higher harmonics of the pair distribution function. The relation to RAM perturbation theory is discussed briefly.

Fugacity and Density Expansion for Systems with Bound States

AbstractThe fugacity expansion in different approximations for low densities and for systems with bound and scattering states is discussed. The meaning of a mixed expansion and the possibility to introduce a chemical picture for a system in which bound states dominate is proved. The possibility of a thermodynamical instability is shown.

Effective two particle potential for adsorbed monolayers

We consider the fugacity expansion for the two-particle correlation functions for a dilute fluid being adsorbed onto a plane structureless surface. For a sharply peaked adsorption potential, as asymptomatic expansion of each graph may be performed. Adding the first two terms of the expansions gives the two-particle correlation function for a two-dimensional system with an effective two-body potential. The effective potential is evaluated explicitly.