The cluster expansion in statistical mechanics

The Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by finite range potentials. The HamiltonianH ₀+V need be stable in the extended sense thatH ₀+4V+BN0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. We define a class of interacting boson and fermion particle theories with a matter-like potential, 1/r suitably truncated at large distance. This system would collapse in the absence of the exclusion principle—the potential is unstable—but the Hamiltonian is stable. This provides an example of a system for which our method proves existence of the infinite volume limit, that is not covered by the classic work of Ginibre, which requires stable potentials.One key ingredient is a type of Holder inequality for the expectation values of spatially smeared Euclidean densities, a special interpolation theorem. We also obtain a result on the absolute value of the fermion measure, it equals the boson measure.This work was supported in part by NSF Grant MPS 75-10751Michigan Junior Fellow     

Statistical mechanics of a one-dimensional lattice gas

We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be aK-system.     

Poisson measure representation and cluster expansion in classical statistical mechanics

A new representation for distribution functions of the grand canonical ensemble by the Poisson measure functional integral is obtained. Due to the ultralocal nature of the measure, the construction of the cluster expansion is very simple. For the convergence of the cluster expansion, the requirement of exponential decay of the interaction potential is not necessary.Research partially supported by Ukrainian State Committee of Science and Technology     

Derivation of the density functional theory from the cluster expansion

The density functional theory is derived from a cluster expansion by truncating the higher-order correlations in one and only one term in the kinetic energy. The formulation allows self-consistent calculation of the exchange correlation effect without imposing additional assumptions to generalize the local density approximation. The pair correlation is described as a two-body collision of bound-state electrons, and modifies the electron- electron interaction energy as well as the kinetic energy. The theory admits excited states, and has no self-interaction energy.     

Statistical mechanics of an ideal gas of non-Abelian anyons

We study the thermodynamical properties of an ideal gas of non-Abelian Chern–Simons particles and we compute the second virial coefficient, considering the effect of general soft-core boundary conditions for the two-body wavefunction at zero distance. The behaviour of the second virial coefficient is studied as a function of the Chern–Simons coupling, the isospin quantum number and the hard-core parameters. Expressions for the main thermodynamical quantities at the lower order of the virial expansion are also obtained: we find that at this order the relation between the internal energy and the pressure is the same found (exactly) for 2D Bose and Fermi ideal gases. A discussion of the comparison of obtained findings with available results in literature for systems of hard-core non-Abelian Chern–Simons particles is also supplied.     

Statistical mechanics of a one-dimensional lattice gas with exponential-polynomial interactions

Some properties of the transfer-matrix for a one-dimensional classical lattice-gas with exponential-polynomial pair interactions are studied using Hilbert space techniques.     

Statistical mechanics of a confinement electron gas in a uniform electromagnetic field

In this paper we study an ideal electron gas in the presence of a uniform electromagnetic field and confinement by a three-dimensional harmonic potential. We find the partition function of this system and in the sequel we examine the Boltzmann statistics and Fermi-Dirac statistics applying the grand canonical ensemble method.     

Statistical mechanics of a classical one-component coulomb gas using collective coordinates

We develop a new expansion for the logarithm of the canonical partition function ln Q for the classical one-component Coulomb gas, using collective coordinates. Our initial use of collective coordinates is similar to that of Iuknovskii 1958, and our expansion resembles that of Abe (1959). Our result for the lowest-order correction to the Debye-Huckel theory is the same as these earlier results, while our next order correction is different. From our expansion for ln Q we obtain an expansion for the grand function $\text{Ω = F ? μN = ?pV}$. The ultimate purpose of this work is to develop a new mathematical technique for obtaining thermodynamic properties of an ionized gas from quantum statistical mechanics.     

Universal expansion of the powers of a derivation

We give the expansion of the powers of the Lie operator = ⁱ _i in any dimension, where is either a smooth function or a formal power series over an infinite set of commutatives indeterminates. We describe an algorithm for computer treatment and we give, as an example, a table for the orders 1 to 6.     

A new isotherm equation for multilayer adsorption on heterogeneous surfaces yielding the Dubinin-Radushkevich isotherm in the submonolayer region

The Cerofolini procedure for obtaining multilayer adsorption isotherms on heterogeneous surfaces is generalized, to take into account the full form of the BET equation as the local adsorption isotherm.     

Adsorption of gas mixtures on heterogeneous surfaces: The integral representation for a monolayer total adsorption isotherm

The authors extended the integral equation for a one-component adsorption isotherm on to n-component adsorption from gaseous mixtures. It appears that the n-component adsorption isotherm on heterogeneous surfaces can be presented as a multiple integral. Two methods were used for evaluation of the surface heterogeneity with regard to a gas mixture. The first method is based on the n-dimensional gaussian energy distribution. The second one concerns the extension of our method for studying adsorption systems of high surface heterogeneity. Two-component adsorption systems have been analyzed in details.     

Statistical mechanics of the isothermal lane-emden equation

For classical point particles in a box with potential energy H^(N)=N ^–1(1/2) _ij=1 ^N V(x _i,x _j) we investigate the canonical ensemble for largeN. We prove that asN the correlation functions are determined by the global minima of a certain free energy functional. Locally the distribution of particles is given by a superposition of Poisson fields. We study the particular case =[–L, L] andV(x, y)=}- cos(x–y),L}>0, }>0.References     

Statistical mechanics of dissipative transport in crystals

We present a microscopic derivation of dissipative (nonideal) hydrodynamic-like equations for crystals. The local density changes and the lattice displacement (distortion) are treated as independent. In this way we take into account all the eight slow modes predicted by the phenomenological theory, including the socalled vacancy diffusion mode. However, microscopically derived hydrodynamic-like equations are simpler than those postulated in the phenomenological theory. We also derive the exact formal expressions for the solid-state transport coefficients in terms of the Green-Kubo integrals.This paper is dedicated to Prof. Matthieu Ernst on the occasion of his 60th birthday. His advice and encouragement were invaluable in early stages of the research described here.     

Quantum shot noise: Expansions in powers of the pulse density

Quantum shot noise consists of individual pulses which contribute time-dependent (operator) potentials toward a total potentialV(t). The averaged quantity T exp _t0 ^t dtV(t) in general can no longer be calculated explicitly, in contrast to the classical case, and expansions are of interest. Noncommutative cumulant expansions are not directly applicable if the correlation functions ofV(t) have singularities, as happens in applications. It is shown here that these expansions, when applied to quantum shot noise, can be partially summed to give expansions in powers of the pulse density. Three types of such expansions are established explicitly, and for two of them the derivation is direct. For one of them the first-order approximation is closely connected to the so-called unified theory of spectral-line broadening.