Mayer Cluster Expansion Research Articles

Construction of subcritical isotherms for model and real gases on the basis of Mayer's cluster expansion.

Existing rigorous statistical approaches still cannot quantitatively describe condensation phenomena in real fluids and even model systems with some simplified interaction potential. Here, we present a method to evaluate the unlimited subcritical set of Mayer's reducible cluster integrals (the power coefficients of virial expansions) by using the information on several virial coefficients and empirical value of saturation activity. As to the requirements on the initial number of known virial coefficients, the calculations for the Lennard-Jones model indicate that only the second virial coefficient is sufficient to reproduce gas isotherms (including the flat phase-transition region) with high accuracy at low temperatures. This remarkable feature allows the simplification of the method for real fluids with an unknown interaction potential: In particular, the calculated isotherms of several real substances (including water) are in good agreement with experiments. Additionally, the obtained results indicate the existence of some important universality which needs serious future exploration.

Critical point signatures in the cluster expansion in fugacities

The QCD baryon number density can formally be expanded into a Laurent series in fugacity, which is a relativistic generalization of Mayer's cluster expansion. We determine properties of the cluster expansion in a model with a phase transition and a critical point at finite baryon density, in which the Fourier coefficients of the expansion can be determined explicitly and to arbitrary order. The asymptotic behavior of Fourier coefficients changes qualitatively as one traverses the critical temperature and it is connected to the branch points of a thermodynamic potential associated with the phase transition. The results are discussed in the context of lattice QCD simulations at imaginary chemical potential. We argue that the location of a branch point closest to the imaginary chemical potential axis can be extracted through an analysis of an exponential suppression of Fourier coefficients. This is illustrated using the four leading coefficients both in a toy model as well as by using recent lattice QCD data.

Equation of state for all regimes of a fluid: From gas to liquid

The study of Mayer's cluster expansion (CE) for the partition function demonstrates a possible way to resolve the problem of the CE nonphysical behavior at condensed states of fluids. In particular, a general equation of state is derived for finite closed systems of interacting particles, where the pressure is expressed directly in terms of the density (or system volume) and temperature-volume-dependent reducible cluster integrals. Although its accuracy is now greatly affected by the limited character of the existing data on the reducible cluster integrals and, especially, the absence of any information on their density dependence, a number of simple approximations indicate the qualitative adequacy of this equation in various regimes of a fluid: from gaseous to liquid states (including the transition region).

Cluster expansion model for QCD baryon number fluctuations: No phase transition at μB/T<π

A cluster expansion model (CEM), representing a relativistic extension of Mayer's cluster expansion, is constructed to study baryon number fluctuations in QCD. The temperature dependent first two coefficients, corresponding to the partial pressures in the baryon number $B=1$ and $B=2$ sectors, are the only model input, which we fix by recent lattice data at imaginary baryochemical potential. All other coefficients are constructed in terms of the first two and required to match the Stefan-Boltzmann limit of noninteracting quarks and gluons at $T\ensuremath{\rightarrow}\ensuremath{\infty}$. The CEM allows calculations of the baryon number susceptibilities ${\ensuremath{\chi}}_{k}^{B}$ to arbitrary order. We obtain excellent agreement with all available lattice data for the baryon number fluctuation measures ${\ensuremath{\chi}}_{2}^{B}$, ${\ensuremath{\chi}}_{4}^{B}$, ${\ensuremath{\chi}}_{6}^{B}$ and predict higher order susceptibilities, that are not yet available from Lattice QCD. The calculated susceptibilities are then used to extract the radius of convergence of the Taylor expansion of the pressure. The commonly used ratio test fails due to the nontrivial asymptotic behavior of the Taylor coefficients. At the same time, a more elaborate estimator provides finite convergence radii at all considered temperatures and in agreement with the singularities of Pad\'e approximants. The associated singularities lie in the complex ${\ensuremath{\mu}}_{B}/T$-plane and appear smoothly connected to the Roberge-Weiss transition at high temperatures and imaginary chemical potential. No evidence for a phase transition at ${\ensuremath{\mu}}_{B}/T\ensuremath{\lesssim}\ensuremath{\pi}$ and $T>135\text{ }\text{ }\mathrm{MeV}$ is found.

Kinetic field theory: exact free evolution of Gaussian phase-space correlations

In recent work we developed a description of cosmic large-scale structure formation in terms of non-equilibrium ensembles of classical particles, with time evolution obtained in the framework of a statistical field theory. In these works, the initial correlations between particles sampled from random Gaussian density and velocity fields have so far been treated perturbatively or restricted to pure momentum correlations. Here we treat the correlations between all phase-space coordinates exactly by adopting a diagrammatic language for the different forms of correlations, directly inspired by the Mayer cluster expansion. We will demonstrate that explicit expressions for phase-space density cumulants of arbitrary n-point order, which fully capture the non-linear coupling of free streaming kinematics due to initial correlations, can be obtained from a simple set of Feynman rules. These cumulants will be the foundation for future investigations of perturbation theory in particle interactions.

One-loop diagrams in the random Euclidean matching problem.

The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e., the optimal matching problem between points randomly distributed on a d-dimensional Euclidean space, where the cost to minimize depends on the points' pairwise distances. Using Mayer's cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation. We give the diagrammatic rules for each term of the expansion, and we analyze in detail the one-loop diagrams. A characteristic feature of the theory, when diagrams are perturbatively computed around the mean field part of the action, is the vanishing of the mass at zero momentum. In the non-Euclidean case of uncorrelated costs instead, we predict and numerically verify an anomalous scaling for the sub-sub-leading correction to the asymptotic average cost.

Universal power-law decay of electron-electron interactions due to nonlinear screening in a Josephson junction array

Josephson junctions are the most prominent nondissipative and at the same time nonlinear elements in superconducting circuits allowing Cooper pairs to tunnel coherently between two superconductors separated by a tunneling barrier. Due to this, physical systems involving Josephson junctions show highly complex behavior and interesting novel phenomena. Here, we consider an infinite one-dimensional chain of superconducting islands where neighboring islands are coupled by capacitances. We study the effect of Josephson junctions shunting each island to a common ground superconductor. We treat the system in the regime where the Josephson energy exceeds the capacitive coupling between the islands. For the case of two offset charges on two distinct islands, we calculate the interaction energy of these charges mediated by quantum phase slips due to the Josephson nonlinearities. We treat the phase slips in an instanton approximation and map the problem onto a classical partition function of interacting particles. Using the Mayer cluster expansion, we find that the interaction potential of the offset charges decays with an universal inverse-square power law behavior.

Mayer expansion of the Nekrasov prepotential: The subleading ε2-order

The Mayer cluster expansion technique is applied to the Nekrasov instanton partition function of N=2SU(Nc) super Yang–Mills. The subleading small ε2-correction to the Nekrasov–Shatashvili limiting value of the prepotential is determined by a detailed analysis of all the one-loop diagrams. Indeed, several types of contributions can be distinguished according to their origin: long range interaction or potential expansion, clusters self-energy, internal structure, one-loop cyclic diagrams, etc. The field theory result derived more efficiently in [1], under some minor technical assumptions, receives here definite confirmation thanks to several remarkable cancellations: in this way, we may infer the validity of these assumptions for further computations in the field theoretical approach.

The virial expansion re-visited: A new interpretation

A central topic in statistical mechanics, a research area Professor Hao Bailin devoted himself throughout the years, is the study of the state of matter. For a gas system the study begins with an understanding of the equation of state relating the pressure [Formula: see text], volume [Formula: see text] and the temperature [Formula: see text]. In 1901 Onnes introduced the virial expansion [Formula: see text] as an empirical formula expressing [Formula: see text] in a power series of particle density [Formula: see text], where [Formula: see text] is the number of particles. A first-principle understanding of the virial expansion was provided years later by the advent of the Mayer cluster expansion in statistical mechanics in the 1930s. However, following Onnes the virial expansion has since been generally regarded as an expansion in density. Here we re-visit the virial expansion using the Mayer expansion, and show that the virial expansion should be considered as an expansion in specific volume, the ratio of the effective volume of a gas molecule and its allotted mean volume. This consideration is illustrated in the case of the hard sphere gas.

Spherical Hecke algebra in the Nekrasov-Shatashvili limit

The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of $\mathcal{N}=2$ gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter $\epsilon_2$. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.

Confinement and Mayer cluster expansions

In this paper, we study a class of grand-canonical partition functions with a kernel depending on a small parameter ϵ. This class is directly relevant to Nekrasov partition functions of 𝒩 = 2 SUSY gauge theories on the 4d Ω-background, for which ϵ is identified with one of the equivariant deformation parameter. In the Nekrasov–Shatashvili limit ϵ→0, we show that the free energy is given by an on-shell effective action. The equations of motion take the form of a TBA equation. The free energy is identified with the Yang–Yang functional of the corresponding system of Bethe roots. We further study the associated canonical model that takes the form of a generalized matrix model. Confinement of the eigenvalues by the short-range potential is observed. In the limit where this confining potential becomes weak, the collective field theory formulation is recovered. Finally, we discuss the connection with the alternative expression of instanton partition functions as sums over Young tableaux.

Notes on Mayer expansions and matrix models

Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of N=2 4d gauge theories. The associated canonical model involves coupled integrations that take the form of a generalized matrix model. It can be studied with the standard techniques of matrix models, in particular collective field theory and loop equations. In the first part of these notes, we explain how the results of collective field theory can be derived from the cluster expansion. The equalities between free energies at first orders is explained by the discrete Laplace transform relating canonical and grand canonical models. In a second part, we study the canonical loop equations and associate them with similar relations on the grand canonical side. It leads to relate the multi-point densities, fundamental objects of the matrix model, to the generating functions of multi-rooted clusters. Finally, a method is proposed to derive loop equations directly on the grand canonical model.

The Equation of State for Non-ideal Quark Gluon Plasma

The mass spectra of quarkonium systems at T= 0 are analyzed by solving the nonrelativistic radial wave equation using the internal energy potential. The QGP matter is studied through the dissociations of quarkonium states. A modified form of the internal energy potential function is used to determine the EoS at different number of quark flavors by using Mayer's cluster expansion theory and phenomenological thermodynamic model. The thermodynamic model gives a good agreement with the lattice results rather than Mayer’s cluster expansion theory. One can conclude that, the Mayer’s cluster expansion theory may be more suitable to study a weakly coupled plasma while, the QGP may be considered as a strongly interacting plasma.

Equation of State Beyond the Radius of Convergence of the Virial Expansion

The well-known problem of the virial expansion low-density limitation is considered within Mayer's cluster expansion method. The expression for the configuration integral and the corresponding equation of state are presented based on this approach but not limited by the convergence radius of the series for density and activity. When taking into account any number of irreducible integrals at the thermodynamic limit, this equation of state demonstrates the exact coincidence with the virial one inside the domain of its convergence but specifies the condensation process directly outside that domain. Thus, the assumption of some researchers that the condensation should appear in the domain where the proof of the virial expansion is limited may now be regarded as confirmed, exclusively using the classical Gibbs statistics.

Thermodynamics of d-dimensional hard sphere fluids confined to micropores

We derive an analytical expression of the second virial coefficient of d-dimensional hard sphere fluids confined to slit pores by applying Speedy and Reiss' interpretation of cavity space. We confirm that this coefficient is identical to the one obtained from the Mayer cluster expansion up to second order with respect to fugacity. The key step of both approaches is to evaluate either the surface area or the volume of the d-dimensional exclusion sphere confined to a slit pore. We, further, present an analytical form of thermodynamic functions such as entropy and pressure tensor as a function of the size of the slit pore. Molecular dynamics simulations are performed for d = 2 and d = 3, and the results are compared with analytically obtained equations of state. They agree satisfactorily in the low density regime, and, for given density, the agreement of the results becomes excellent as the width of the slit pore gets smaller, because the higher order virial coefficients become unimportant.

Closed virial equations for hard parallel cubes andsquares

A correlation between maxima in virial coefficients (Bn), and “kissing” numbers for hard hyper-spheres up to dimension D = 5, indicates a virial equation and close-packing relationship. Known virial coefficients up to B7, both for hard parallel cubes and squares, indicate that the limiting differences Bn – Bn-1 behave similar to spheres and disks, in the respective expan-sions relative to maximum close packing. In all cases, the difference Bn – Bn-1 is approaching a negative constant with similar functional form in each dimen-sion. This observation enables closed-virial equa-tions-of-state for cubes and squares to be obtained. In both the 3D and 2D cases, the virial pressures begin to deviate from MD thermodynamic pressures at densities well below crystallization. These results consolidate the general conclusion, from previous papers on spheres and disks, that the Mayer cluster expansion cannot represent the thermodynamic fluid phases up to freezing as commonly assumed in statistical theories.

Soft disks in a narrow channel

The pressure components of ‘soft’ disks in a two-dimensional narrow channel are analyzed in thedilute gas regime using the Mayer cluster expansion and molecular dynamics. Channels with eitherperiodic or reflecting boundaries are considered. It is found that when the two-body potential,u(r), is singular atsome distance r0, the dependence of the pressure components on the channel width exhibitsa singularity at one or more channel widths which are simply related tor0. In channels with periodic boundary conditions and for potentials which are discontinuous atr0, the transverse and longitudinal pressure components exhibit a1/2 anda 3/2 singularity, respectively. Continuous potentials with a power-law singularity result inweaker singularities of the pressure components. For channels with reflecting boundaryconditions the singularities are found to be weaker than those corresponding to periodicboundaries.

Equation of state of a quark-gluon plasma using the Cornell potential

The equation of state (EOS) of quark-gluon plasma (QGP) using the Cornell potential based on Mayer's cluster expansion is presented. The string constant and the strong coupling constant for QGP are calculated. The EOS developed could describe the lattice EOS for pure gauge, two-flavor and three-flavor QGP qualitatively.

Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards’ field theory

Edwards’ functional integral approach to the statistical mechanics of polymer liquids is amenable to a diagrammatic analysis in which free energies and correlation functions are expanded as infinite sums of Feynman diagrams. This analysis is shown to lead naturally to a perturbative cluster expansion that is closely related to the Mayer cluster expansion developed for molecular liquids by Chandler and co-workers. Expansion of the functional integral representation of the grand-canonical partition function yields a perturbation theory in which all quantities of interest are expressed as functionals of a monomer–monomer pair potential, as functionals of intramolecular correlation functions of non-interacting molecules, and as functions of molecular activities. In different variants of the theory, the pair potential may be either a bare or a screened potential. A series of topological reductions yields a renormalized diagrammatic expansion in which collective correlation functions are instead expressed diagrammatically as functionals of the true single-molecule correlation functions in the interacting fluid, and as functions of molecular number density. Similar renormalized expansions are also obtained for a collective Ornstein–Zernicke direct correlation function, and for intramolecular correlation functions. A concise discussion is given of the corresponding Mayer cluster expansion, and of the relationship between the Mayer and perturbative cluster expansions for liquids of flexible molecules. The application of the perturbative cluster expansion to coarse-grained models of dense multi-component polymer liquids is discussed, and a justification is given for the use of a loop expansion. As an example, the formalism is used to derive a new expression for the wave-number dependent direct correlation function and recover known expressions for the intramolecular two-point correlation function to first-order in a renormalized loop expansion for coarse-grained models of binary homopolymer blends and diblock copolymer melts.

Analysis of the ordering transition of hard disks through the Mayer cluster expansion

The available virial coefficients for the two-dimensional hard-disks model are transformed into a matrix representation of the thermodynamic potentials, which allows for an accurate description of the whole fluid phase, up to the phase transition. We find that the fluid phase terminates at the transition point, implying a second-order phase transition in accordance with the Kosterlitz-Thouless-Halperin-Nelson-Young scenario of a transition into a hexatic phase. The density and pressure at the transition are calculated from the available first ten virial coefficients, and are found to be in excellent agreement with recent Monte-Carlo calculations. Finally, we calculate the equation of state in the critical region.