Yukawa Gas Research Articles

On the Mayer Series of Two-Dimensional Yukawa Gas at Inverse Temperature in the Interval of Collapse

A Theorem on the minimal specific energy for a system with \pm 1 charged particles interacting through the Yukawa pair potential v is proved which may stated as follows. Let v be represented by scale mixtures of d-dimensional Euclid's hat (cutoff at short scale distances) with d\geq 2. For any even number of particles n, the interacting energy U_{n} divided by n, attains an n-independent minimum at a configuration with zero net charge and particle positions collapsed altogether to a point. For any odd number of particles n, the ratio U_{n}/(n-1) attains its minimum value, the same of the previous cases, at the configuration with \pm 1 net charge and particle positions collapsed to a point. This Theorem is then used to resolve an obstructive remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which, whether the standard decomposition of the Yukawa potential into scales were adopted, would impede a direct proof of the convergence of the Mayer series of the two-dimensional Yukawa gas for the inverse temperature in the whole interval [4\pi ,8\pi ) of collapse. In the present paper, it is proven convergence up to the second threshold 6\pi and its given explanations on the mechanism that allow it to be extend up to 8\pi . The paper distinguishes the matters concerning stability from those related to convergence of the Mayer series. In respect to the latter the paper dedicates to the Cauchy majorante method applied to the density function of Yukawa gas in the interval of collapses. It also dedicates to the proof of the main Theorem and estimates of the modified Bessel functions of second kind involved in both representations of two-dimensional Yukawa potential: standard and scale mixture of the Euclid's hat function.

The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem

For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order $N$, the number of particles of the gas, with an effective error bound $N^{1-\kappa}$ for some constant $\kappa > 0$. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature. Our proof of this central limit theorem uses a loop equation for the Coulomb gas, the free energy asymptotics, and rigidity bounds on the local density fluctuations of the Coulomb gas, which we obtained in a previous paper.

Path-integral Monte Carlo simulation of time-reversal noninvariant bulk systems with a case study of rotating Yukawa gases.

We elaborate on the methodology to simulate bulk systems in the absence of time-reversal symmetry by the phase-fixed path-integral Monte Carlo method under (possibly twisted) periodic boundary conditions. Such systems include two-dimensional electrons in the quantum Hall regime and rotating ultracold Bose and Fermi gases; time-reversal symmetry is broken by an external magnetic field and the Coriolis force, respectively. We provide closed-form expressions in terms of Jacobi elliptic functions for the thermal density matrix (or the Euclidean propagator) of a single particle on a flat torus under very general conditions. We then modify the multislice sampling method in order to sample paths by the magnitude of the complex-valued thermal density matrix. Finally, we demonstrate that these inventions let us study the vortex melting process of a two-dimensional Yukawa gas in terms of the de Boer interaction strength parameter, temperature, and rotation (Coriolis force). The bosonic case is relevant to ultracold Fermi-Fermi mixtures of widely different masses under rotation.

Isothermal equation of state of three dimensional Yukawa gas

Molecular Dynamics (MD) simulations are carried out to examine the effect of particle confinement on the pressure of 3D Yukawa gas. Confinement effects are taken into account by using perfectly reflecting boundary conditions in MD simulations. An equation of state relating pressure to number density is obtained. The results of the MD simulations show that in the weak coupling regime, pressure of confined Yukawa gas is much larger than the kinetic pressure and scales quadratically with number density. The results are compared with earlier theories and experiments which show quadratic scaling of dust pressure with density.

Renormalization group study of the intrinsic finite-size effect in two-dimensional superconductors

Vortices in a thin-film superconductor interact logarithmically out to a distance on the order of the two-dimensional (2D) magnetic penetration depth $\lambda_\perp$, at which point the interaction approaches a constant. Thus, because of the finite $\lambda_\perp$, the system exhibits what amounts to an {\it intrinsic} finite size effect. It is not described by the 2D Coulomb gas but rather by the 2D Yukawa gas (2DYG). To study the critical behavior of the 2DYG, we map the 2DYG to the massive sine-Gordon model and then perform a renormalization group study to derive the recursion relations and to verify that $\lambda_\perp$ is a relevant parameter. We solve the recursion relations to study important physical quantities for this system including the renormalized stiffness constant and the correlation length. We also address the effect of current on this system to explain why finite size effects are not more prevalent in experiments given that the 2D magnetic penetration depth is a relevant parameter.

Kosterlitz-Thouless transitions on a fluctuating surface of genus zero.

We investigate the Kosterlitz-Thouless transition for hexatic order on a fluctuating spherical surface of genus zero and derive a Coulomb gas Hamiltonian to describe it. In the Coulomb gas Hamiltonian, charge densities arises from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strength is determined by the hexatic rigidity. We then convert it into the sine-Gordon Hamiltonian and find a linear coupling between a scalar field and the Gaussian curvature. After integrating over the shape fluctuations, we obtain the massive sine-Gordon Hamiltonian, which corresponds to a neutral Yukawa gas, and the interaction between the disclinations is screened. We find, for $K_{A}/\kappa \gg 1/2$ where $K_{A}$ and $\kappa$ are hexatic and bending rigidity, respectively, the transition is supressed altogether, much as the Kosterlitz-Thouless transition is supressed in an infinite 2D superconductor. If on the other hand $K_{A}/\kappa \ll 1/2$, there can be an effective transition.

Free energy density for continuous systems with and without superstability assumptions

The thermodynamic limit of the free energy density for a large class of continuous, charged systems with stable and sufficiently regular many-body forces is studied. The main effort is placed on demonstrating rigorously that thermodynamic functions do not depend on the typical boundary conditions. Among the results the most important and new ones are the following. (1) For any superstable or superstable in an extended sense and sufficiently regular many-body interactions, the thermodynamic limit of the conditioned free energy density exists and is equal to that corresponding to the free boundary condition case. (2) A new criterion for the nondependence of the limiting free energy density on the typical boundary condition is formulated and proved. This criterion does not require any superstability type of assumptions. Among several applications of the results we list only two: (a) to the case of two-dimensional, neutral Yukawa gas in the monopole phase, which is not even stable according to the standard notion of stability, and (b) the uniqueness of a limiting Gibbs grand canonical state (modulo some technical conjecture) is proved for a class of continuous systems with two-body potentials of positive type.

Mayer expansions and the Hamilton-Jacobi equation

We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal “integration out” is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for “connected parts” and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past theΒ=4π threshold and another derivation of some earlier results of Gopfert and Mack.

Convergence of Mayer expansions

The tree graph bound of Battle and Federbush is extended and used to provide a simple criterion for the convergence of (iterated) Mayer expansions. As an application estimates on the radius of convergence of the Mayer expansion for the two-dimensional Yukawa gas (nonstable interaction) are obtained.

An iterated Mayer expansion for the Yukawa gas

We construct a convergent expansion for the Yukawa gas at small activity and inverse temperatureΒ < 4π/e 2.

On the massive sine-Gordon equation in the higher regions of collapse

We prove that the renormalization program discussed in [1] can also be developed beyond the\(\bar \alpha ^2 = 2\pi (\sqrt {17} - 1)\) threshold found in the preceding work. This result, as a byproduct, also allows a simplification in the technical part of the proof of ultraviolet stability in the ϕ34-theory [2]. In the last section of this work we discuss, heuristically, but in some detail the interpretation of the sine-Gordon theory as a two-dimensional Yukawa gas for βe2=α2>4π.

On the massive sine-Gordon equation in the first few regions of collapse

We prove that the renormalization program discussed in [1] can also be developed beyond the\(\bar \alpha ^2 = 2\pi (\sqrt {17} - 1)\) threshold found in the preceding work. This result, as a byproduct, also allows a simplification in the technical part of the proof of ultraviolet stability in the ϕ34-theory [2]. In the last section of this work we discuss, heuristically, but in some detail the interpretation of the sine-Gordon theory as a two-dimensional Yukawa gas for βe2=α2>4π.

Two dimensional gases of weight charges and SU( N) fermino theories

We analyze two dimensional gases composed of particles interacting via a Coulomb or Yakawa potential through their “non-Abelian” charges. These charges are taken to be elementary weight or root vectors of SU( N). The grand partition function of these gases is shown to be equivalent to the generating functional of sine-Gordon models with weight vectors and hence to that of SU( N) fermion models. The fermion field creates or annihilates topological solitons which have elementary weight vectors as topological quantum numbers. Then, we discuss the confinement of fermions in the SU( N) Higgs models, where instantons ( Z N vortices) constitute a Yukawa gas of weight charges. We prove that fermions are confirmed by the effects of instantons in the SU( N) Higgs models in contrast with the Abelian Higgs model.