Convergence of Mayer and Virial expansions and the Penrose tree-graph identity

We establish new lower bounds for the convergence radius of the Mayer series and the Virial series of a continuous particle system interacting via a stable and tempered pair potential. Our bounds considerably improve those given by Penrose (J Math Phys 4:1312, 1963) and Ruelle (Ann Phys 5:109–120, 1963) for the Mayer series and by Lebowitz and Penrose (J Math Phys 7:841–847, 1964) for the Virial series. To get our results, we exploit the tree-graph identity given by Penrose (Statistical mechanics: foundations and applications. Benjamin, New York, 1967) using a new partition scheme based on minimum spanning trees.     

On the Convergence of Cluster Expansions for Polymer Gases

We compare the different convergence criteria available for cluster expansions of polymer gases subjected to hard-core exclusions, with emphasis on polymers defined as finite subsets of a countable set (e.g. contour expansions and more generally high- and low-temperature expansions). In order of increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via a direct combinatorial handling of the terms of the expansion. We show that for subset polymers our sharper criterion can be proven both by a suitable adaptation of Dobrushin inductive argument and by an alternative—in fact, more elementary—handling of the Kirkwood-Salzburg equations. In addition we show that for general abstract polymers this alternative treatment leads to the same convergence region as the inductive Dobrushin argument and, furthermore, to a systematic way to improve bounds on correlations.     

Quantum Statistical Second Virial Coefficient for Real Gases and Plasmas

Using the resolvent representation of the binary Boltzmann operator the second virial coefficient is expressed by a contour integral over the Jost function of the two-particle scattering problem. This new formula is useful for many general investigations and also for practical calculations if the Jost functions for the scattering problem are known. As special examples real gases with hard core potentials and fully ionized plasmas with Coulomb interactions are treated. For the plasma case the second virial coefficient is derived without approximations with respect to the interaction parameter. The contribution of the bound states to the free energy is discussed and compared with the effect of ionic association to the classical free energy and the conductivity.     

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.     

Screened Cluster Expansions for Partially Ionized Gases

We consider a partially ionized gas at thermal equilibrium, in the Saha regime. The system is described in terms of a quantum plasma of nuclei and electrons. In this framework, the Coulomb interaction is the source of a large variety of phenomena occurring at different scales: recombination, screening, diffraction, etc. In this paper, we derive a cluster expansion adequate for a coherent treatment of those phenomena. The expansion is obtained by combining the path integral representation of the quantum gas with familiar Mayer diagrammatics. In this formalism, graphs have a clear physical interpretation: vertices are associated with recombined chemical species, while bonds describe their mutual interactions. The diagrammatical rules account exactly for all effects in the medium. Applications to thermodynamics, van der Waals forces and dielectric versus conductive behaviour will be presented in forthcoming papers.     

Convergence properties of multiple reflection expansions

The convergence properties of the multiple reflection expansion for massless scalar and Dirac propagators in a general convex cavity are studied. A sufficient condition for convergence is arrived at and the regions in the complex frequency plane are examined where this condition is satisfied. For the case of a spherical cavity, a necessary and sufficient condition for convergence is obtained by making use of a partial wave expansion for the propagator. The corresponding convergence regions in the complex frequency plane are also determined in more detail.     

Cluster expansions for correlated finite nuclei: The normalization integral

Cluster expansions of various kinds are studied with regard to the calculation of the normalization integral in ¹⁶O. The factor Aviles-Hartog-Tolhoek cluster expansion is shown to be very satisfactory at third order. A diagrammatic analysis of this cluster expansion is also included and numerically tested.     

Optimal Convergence Rates of Classical Solutions for Vlasov-Poisson-Boltzmann System

The dynamics of charged dilute particles can be modeled by the two species Vlasov-Poisson-Boltzmann system when the particles interact through collisions in the self-induced electric field. By constructing the compensating function for multi-species particle system, the optimal time decay of global classical solutions to this system near a global Maxwellian is obtained through a refined energy method.     

Convergence of spherical harmonic expansions for the evaluation of hard-sphere cluster integrals

ForN particles (N>2), by means of a spherical harmonic expansion of Silverstone and Moats, a 3N-dimensional cluster may be reduced to 2N+1 trivial integrals andN–1 interesting integrals. For hard spheres, theN–1 interesting integrals are products of polynomials integrated between binomial bounds. With simple clusters, closed forms are obtained; for more complex clusters, infinite series inl (ofY _lm ) appear. It is here shown for representative cases that these series converge exponentially rapidly, the leading pair of terms accounting for all but a few tenths of a percent of the total cluster integral.     

Virial Coefficients from Unified Statistical Thermodynamics of Quantum Gases Trapped under Generic Power Law Potential in d Dimension and Equivalence of Quantum Gases

From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases, trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in d=3 and d=2. But more importantly we found that, the virial coefficients of Bose and Fermi gases become identical (except the second virial coefficient, where the sign is different) when the gases are trapped under harmonic potential in d=1. This result suggests the equivalence between Bose and Fermi gases established in d=1 (J. Stat. Phys. DOI 10.1007/s10955-015-1344-4). Also, it is found that the virial coefficients of two-dimensional free Bose (Fermi) gas are equal to the virial coefficients of one-dimensional harmonically trapped Bose (Fermi) gas.     

A Significant Enhancement of Cluster Formation of Krypton and Xenon Gases

Clusters of rare gases have currently become a kind of widely used targets in the study of high-intensity laser interaction with matter［1,2］. However, a good understanding of the cluster media is vital if results from laser-cluster interaction experiments are to be interpreted correctly. During the course of investigation of the time history of a cluster jet generated in supersonic expansion of rare gases into vaccum through a supersonic conical nozzle (26 mm long, opening angle 5°) by time resolving Rayleigh scattering measurements in which a 532 nm 0.3 mJ laser beam was used, we find that the general trend of argon clusters is similar to that reported by R.A. Smith et al.［3］. However, the picture for Krypton and Xenon gases is completely different. A two-peak structure of the scattered light intensity, instead of a single-peak one, has been revealed. It is interesting to note that the second peak is much more stronger than the first one. As an example, Fig.1 shows the time history of Xen clusters produced at a gas backing pressure P0=3 atm. In the Figure, the peak intensity of the second peak is surprisingly 62 times higher than the first one, indicating that the average cluster size c in the latter case is increased about 62 times under the assumption that during the expansion process all the atoms in the gas condensate into clusters. The result, which is believed to be related to a double gas ejection via the pulsed valve′s two adjacent openings, is of significant importance since this effect would be promising for applications in which very large size clusters are required while a relatively modest vacuum maintains. Meanwhile, this finding may provide us with the opportunity of a further understanding into the nucleation mechanism of expanding gases in a nozzle.     

Virial Theorem for Angular Displacement and Torque

The usual Virial theorem is expressed through the coordinate and the force, 2áT? = áX\fracdVdX? = -áXF?2\langle T\rangle =\langle X\frac{dV}{dX}\rangle =-\langle XF\rangle , F=-\fracdVdXF=-\frac{dV}{dX}, XF is the work done by the force F, T is the kinetic energy. In this paper we extend the usual discussion on the Virial theorem about coordinate-force variables to the case of angular displacement-torque variables. By virtue of introducing the entangled state representation and the bosonic operator realization of the Hamiltonian of quantum pendulum system we derive the Virial theorem for angular variable and torque.     

The Convergence of Cluster Expansion for Continuous Systems with Many-Body Interaction

The existence of a unique thermodynamic state for dilute classical systems is proved for a class of multi-particle potentials under ordinary assumptions of stability and integrability. Thus we do not use the cumbersome conditions of regularity needed in previous publications for the many-body analysis. The method relies on the Poisson measure representation and cluster expansion for distribution functions.     

Virial expansion for charged colloids and electrolytes

Using a field-theoretic approach, we derive the first few coefficients of the exact low-density (“virial”) expansion of a binary mixture of positively and negatively charged hard spheres (two-component hard-core plasma, TCPHC). Our calculations are nonperturbative with respect to the diameters d₊ and d_- and charge valences q₊ and q_- of positive and negative ions. Consequently, our closed-form expressions for the coefficients of the free energy and activity can be used to treat dilute salt solutions, where typically d ₊∼d _- and q ₊∼q _-, as well as colloidal suspensions, where the difference in size and valence between macroions and counterions can be very large. We show how to map the TCPHC on a one-component hard-core plasma (OCPHC) in the colloidal limit of large size and valence ratio, in which case the counterions effectively form a neutralizing background. A sizable discrepancy with the standard OCPHC with uniform, rigid background is detected, which can be traced back to the fact that the counterions cannot penetrate the colloids. For the case of electrolyte solutions, we show how to obtain the cationic and anionic radii as independent parameters from experimental data for the activity coefficient. Received 6 September 2001 / Received in final form 20 May 2002 Published online 24 September 2002     

Numeration as a Factor Relating the Quantum and Classical Mechanics of Ideal Gases

We show that the statistical approach to quantum mechanics allows to define a factor related to numeration theory in mathematical logic, and to apply this factor to the study of the nucleus of helium-5 and other light nuclei. In particular, the use of this hidden factor of numbering gives us, instead of the quantum picture of the Bohr orbits for small energies, the purely classical behavior of particles corresponding to the rotation of wave packets around the nucleus.     

Adaptive Cluster Expansion for the Inverse Ising Problem: Convergence,Algorithm and Tests

We present a procedure to solve the inverse Ising problem, that is, to find the interactions between a set of binary variables from the measure of their equilibrium correlations. The method consists in constructing and selecting specific clusters of spins, based on their contributions to the cross-entropy of the Ising model. Small contributions are discarded to avoid overfitting and to make the computation tractable. The properties of the cluster expansion and its performances on synthetic data are studied. To make the implementation easier we give the pseudo-code of the algorithm.