Borel-summability of the high temperature expansion for classical continuous systems

It is shown that for classical gases with stable, bounded and absolutely integrable pair interactions, the Taylor expansions in of the correlation functions and the pressure are Borel-summable at =0.     

The high temperature expansion of the classical XYZ chain

The β$\beta$-expansion of the Helmholtz free energy (HFE) up to order β¹²${\beta }^{12}$ of the classical XYZ model with a single-ion anisotropy term and external magnetic field is calculated and compared to the numerical solution of Joyce's [Phys. Rev. Lett. 19 (1967) 581] for the XXZ   classical model, with neither single-ion anisotropy term nor external magnetic field. This comparison shows that the derived analytical expansion is valid for intermediate temperatures such as kT/_Jx≈0.5$\mathit{kT}/J_x\approx 0.5$. The specific heat and magnetic susceptibility of the S=2$S=2$ antiferromagnetic chain can be approximated by their respective classical results, within an error of 2.5%$2.5\%$, up to kT/J≈0.8$\mathit{kT}/J\approx 0.8$. For a vanishing external magnetic field the ferromagnetic and antiferromagnetic chains are shown to have the same classical HFE; their behaviour in a non-vanishing external magnetic field is also described.     

Topology on interactions for classical continuous systems

For classical continuous systems with interactions without hard core a metric space of interactions is formed. To each type of thermodynamic limit there exist classes of interactions for which the pressure exists and is a continuous functional. Metric spaces are created with respect to each kind of ensemble, such that the sets of local thermodynamic functions (pressure and the densities of free energy and entropy) are equicontinuous.     

On the solutions of correlation equations for classical continuous systems

A method for solving the finite-volume Kirkwood-type correlation equations for tempered boundary conditions is developed. The central idea is an analytic continuation in the activity of the resolvent formulas for the solutions. The uniqueness theorem is proved for activities in a larger domain of the complex plane than the “standard” circle of analyticity¹). A connection with the eigenvector problem for the corresponding Kirkwood-type operators is discussed. We compare also the correlation equation method with the “equilibrium equation” one handling directly with the Gibbs probability measure.     

The cluster expansion for classical and quantum lattice systems

We develop a high-temperature expansion for general lattice systems which can be applied to classical as well as quantum systems. Applying the expansion we prove analyticity of correlation functions, uniqueness of equilibrium states, and cluster properties for classical and quantum lattice systems in the high-temperature region.     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Phase transitions and the high temperature expansion for the reggeon calculus

We study a recently proposed lattice version of the reggeon field theory. First a simple one-dimensional system possessing many of the features of the reggeon calculus (Ising model in an imaginary magnetic field) is solved. Surprisingly, the system is found to undergo a phase transition at a non-zero critical temperature, which, although of first order obeys the universality and scaling laws previously postulated for the reggeon calculus. Returning to the full lattice theory, the machinery for performing a high temperature expansion is set up, and initial calculations carried out to order T^?3. In this order, best estimates for the critical indices η and v in the asymptotic elastic amplitude A(s,t) ～ is (ln s)^ηf(t(lns)^v) yield $\text{η = 0 (}\text{1}\text{2}\text{～ 1), v = 0(1}\text{1}\text{2}\text{～ 2)}$. Check on the method, including comparisons with known Ising model results, are also discussed.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

Analysis of thermal expansion coefficients under the effect of high temperature for minerals

A simple and straightforward method for the determination of thermal expansion is investigated and applied for four minerals of geophysical importance. The results obtained for four minerals such as Mg₂SiO₃, Al₂O₃, Grossular garnet and Pyrope garnet at different temperatures are found to be in excellent agreement with the experimental data. The simplicity of the method is discussed in the light of another method in high temperature research on minerals.     

Strong coupling expansion for classical statistical dynamics

We discuss the simple, randomly driven systemdx/dt = –x –x³ +f(t), wheref(t) is a Gaussian random function or stirring force with f(t)f(t) = (t – t). We show how to obtain approximately the coefficients of the expansion of the equal-time Green's functions as power series in (1/R)ⁿ, whereR is the internal Reynolds number ()^1/2/, by using a new expansion for the path integral representation of the generating functional for the correlation functions. Exploiting the fact that the action for the randomly driven system is related to that of a quantum mechanical anharmonic oscillator with Hamiltonianp ²/2 +m ² x ²/2 +vx ⁴ +x ⁶/2, we evaluate the path integral on a lattice by assuming that thex ⁶ term dominates the action. This gives an expansion of the lattice theory Green's functions as power series in 1/(a)^1/3, wherea is the lattice spacing. Using Padé approximants to extrapolate toa = 0, we obtain the desired large-Reynolds-number expansion of the two-point function.Supported financially by the National Science Foundation and the U.S. Department of Energy.     

A new nodal expansion for classical plasmas

In calculating the equation of state for plasmas we find that diagrammatic expansions for the free energy become unwieldy at high density. At best, many terms must be retained in order to obtain meaningful results. We present a new expansion technique which can be applied to plasmas in the interiors of Jupiter and white dwarf stars. In such cases the older techniques are unsatisfactory because of the size of the ion coupling parameter. Our work yields expansions for which this parameter is supplanted by ion correlation functions, which can be supplied by external computations. In this paper we assume a two-species plasma of classical particles, thereby focusing on combinatorial techniques. The final result is a new nodal expansion in terms of ion correlation functions and an electron coupling parameter.     

Phase space cell expansion and borel summability for the Euclidean φ 3 4 theory

The stability of the free energy is proved for complex values of the coupling constant by the way of a convergent expansion. As a consequence, one obtains the Borel summability of the perturbation series.On leave of absence from the Centre de Physique Théorique, Ecole Polytechnique, Palaiseau, FranceSupported in part by the National Science Foundation under Grant PHY 75-21212     

Decay of correlations in classical lattice models at high temperature

In classical statistical mechanical lattice models with many body potentials of finite or infinite range and arbitrary spin it is shown that the truncated pair correlation function decays in the same weighted summability sense as the potential, at high temperature.Research partially supported by the National Science Foundation under Grant MCS 78-00680     

Incremental expansion of the classical partition functions

The incremental expansion of the canonical partition function of a classical many-particle system is obtained. The incremental expansion coefficients are calculated for the case of a pair interaction potential. The results are used to derive the incremental expansion of the system free energy.     

Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems

We present and discuss some physical hypotheses on the decrease of truncated correlation functions and we show that they imply the analyticity of the thermodynamic limits of the pressure and of all correlation functions with respect to the reciprocal temperature and the magnetic fieldh (or the chemical potential ) at all (real) points (₀,h ₀) (or (₀, ₀)) where they are supposed to hold. A decrease close to our hypotheses is derived in certain particular situations at the end.     

Low-fugacity asymptotic expansion for classical lattice dipole gases

We consider a classical dipole gas in the grand canonical ensemble. We prove that in dimensions greater than or equal to three, and for all temperatures, the free energy and the charges-dipoles correlation functions have an expansion in powers ofz, the fugacity of the system, which is asymptotic to all orders. We also give some information about the decay of correlations.on leave from Institut de Physique Théorique Université de Louvian, Belgium. Supported by N.S.F. grant No. PHY-15920.     

Equilibrium states for classical systems

It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ?, and for the spin correlation functions σ, are essentially equivalent for all ?, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.     

The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials

We study the classical field limit as $\text{h}\text{?}\text{→ 0}$ of nonrelativistic many-boson systems with two-body interaction in the neighborhood of a fixed ?-independent solution of the classical evolution equation (namely, the Hartree equation). The unitary group describing the evolution of the quantum system, after multiplication from both sides by suitable ?-dependent Weyl operators which somehow subtract out the classical motion, has an expansion in power series of $\text{h}\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, the zeroth-order term of which describes the quantum fluctuations of the system around the classical motion. We prove that for bounded two-body interaction potentials and small time intervals, this series is strongly Borel summable; namely, it yields a power series which is Borel summable in vector norm when applied to fixed ?-independent vectors taken from a suitable dense set.