Volume 6, Issue 2, February 1965
Index of content:

Random Walks on Lattices. II
View Description Hide DescriptionFormulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points.
The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a _{1} n + a _{2} n ^{½} + a _{3} + a _{4} n ^{−½} + …. The constants a _{1} − a _{4} have been obtained for walks on a simple cubic lattice when k = 3 and a _{1} and a _{2} are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited.
The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c.
Most of the results in this paper have been derived by the method of Green's functions.

Analytic Continuation of
View Description Hide DescriptionThe positions of the singularities of the partial wave amplitude in the complex l plane are investigated for that case of a relativistic, two‐particle elastic scattering amplitude satisfying the Mandelstam representation. For a finite number of intermediate states it is shown that the convex hull of singularities in the l plane is related to a growth indicator function. For an infinite number of intermediate states it is demonstrated that the imaginary parts of the singularities in the l plane are unbounded from either above or below.

Effects of Finite Sample Size on the Measurement of Transport Quantities
View Description Hide DescriptionA determination is made of the temporal evolution of the momentum autocorrelation function for a noninteracting gas constrained by walls. A frequency‐dependent dielectric constant is calculated from the latter by applying an appropriate Kubo relation. Deviations from the Drude conductivity are obtained which, for high frequency and/ or large systems, are found to be ∼ o(10^{2}/βmω^{2} L ^{2}), where L is the spatial dimension of the assembly and ω the frequency of the applied field.
A similar calculation is performed to determine the effects of boundaries upon measurements of scattering cross sections for slow neutrons. It is found that, for realistic experimental conditions, wall perturbations are unlikely to be of importance.

Correlation Functionals
View Description Hide DescriptionThe correlations between positions of particles in classical equilibrium statistical mechanics are usually expressed by correlation functions. It can be seen that if the correlation functions of a system are known, its configurational entropy per unit volume is already determined (independently of the interaction potential or other thermodynamical parameters). In this paper we consider all sequences of functions of an increasing number of variables (these functions are candidates for correlation functions) for which an entropy per unit volume may be defined. These sequences are called correlation functionals and are investigated in detail. They prove useful to the study of the limit of an infinite system in statistical mechanics(thermodynamic limit). Furthermore they allow the segregation of certain thermodynamic systems into phases to be made evident.

The Structure of Space and the Formalism of Relativistic Quantum Theory. II
View Description Hide DescriptionA purely group theoretical treatment of interactions of relativistic particles is shown to be possible in the connection of a finite geometry of space‐time. The S operators are constructed by means of the reductions of the irreducible unitary representations of the relativity group of space‐time. The dependence of the S‐matrix elements on the momenta is still nontrivial, since there is a nontrivial distribution of observable momenta in a finite geometry. The Feynman graphs are easily evaluated by a simple computational technique involving five rules. The coupling schemes corresponding to nuclear electromagnetic, and weak forces in finite geometry are considered.

Exponential Ensemble for Random Matrices
View Description Hide DescriptionUsing the ideas of information theory, it is pointed out that the Gaussian ensemble for random Hermitian matrices can be characterized as the ``most random'' ensemble of these matrices. Pursuing the same characterization for positive matrices, we are led in a natural way to the definition of the exponential ensemble. Transforming to a representation of eigenvectors and eigenvalues, the joint distribution function for the eigenvalues of positive Hermitian matrices is found for this ensemble. The asymptotic single‐level density formula is derived, using a semiclassical model. It is found that the density is convex from below over most of the domain of eigenvalues. Since this is similar to the exponential dependence expected for nuclear spectra, the density is examined in a region near the level taken to correspond with the lowest nuclear level. It turns out, however, that the density is concave from below near this level, and that a large number of levels are contained in this concave region. Hence the exponential ensemble does not fairly represent nuclear energy levels, at least in this region. Various changes are made in the measure on the matrix ensemble to determine to what extent the level density depends on this measure. It is seen that the level density graph retains a characteristic shape for a wide variety of measures. The relationship of the limiting behavior of the level density for positive matrices to the semicircle law is noted.

Reduced Density Matrices of Quantum Gases. I. Limit of Infinite Volume
View Description Hide DescriptionThe reduced density matrices for quantum gases are studied by Banach space techniques. For suitably restricted interactions, they are shown to be analytic functions of the activity. As the volume of the system becomes infinite, they tend in some sense to well‐defined limits for which the same analyticity properties hold. As a consequence, the virial expansion is shown to be convergent in a neighborhood of the origin.

Reduced Density Matrices of Quantum Gases. II. Cluster Property
View Description Hide DescriptionThe reduced density matrices of quantum gases are studied by means of a Wiener integral representation described in a previous paper. They are shown to satisfy a cluster property in the form of an absolute integrability condition of the natural quantum analogues of the Ursell functions, considered as functions of the differences of their arguments. Use is made of the natural transposition to the quantum case of the algebraic formalism introduced by Ruelle in the classical case. By‐products are two results on the signs of the coefficients of the Mayer expansion, in the case of Maxwell‐Boltzmann and Fermi‐Dirac statistics, respectively.

Two‐Photon Ionization of Atomic Hydrogen. II
View Description Hide DescriptionThis paper contains a generalization of results described in previous work. A general formula is given for the total cross section for two‐photon ionization of a hydrogenic state. An implicit method for evaluating the second‐order radial matrix elements that occur in this expression is described in detail. Numerical results were obtained for the states with principal quantum numbers n = 1 through 5. It is concluded that, when both one‐photon and two‐photon ionization are energetically possible, the effect of the latter may be expressed as an intensity‐dependent correction to the Gaunt factor which may therefore be written as G = g _{1} + ζIλ^{3} g _{2}. Here g _{1} is the usual Gaunt factor, I is the intensity of the light in W cm^{−2}, λ is the wavelength in cm, g _{2} is a dimensionless factor of order unity, and ζ is a constant given by ζ = 0.1504 W^{−1} cm^{−1}. Graphs of g _{2} are given as a function of electron energy.
These results do not include the effect of three‐photon processes, which can also contribute to the first‐order intensity‐dependent correction to the Gaunt factor as a result of interference between the first‐ and third‐order amplitudes.

Use of Analytic Signals in Third‐ and Fourth‐Order Coherence Functions
View Description Hide DescriptionIn this paper complex forms of the third‐ and fourth‐order coherence functions are defined using analytic signals. The relations between the complex forms and the real forms are given in detail. It is shown how to use the complex forms in the solution of radiation problems. The quasimonochromatic approximation greatly simplifies the calculations and this simplification is discussed for radiation problems in terms of the complex forms. The paper concludes with a discussion of quasimonochromatic radiation from a modulated incoherent source representing, perhaps, a self‐luminous turbulent region.

Derivation of Low‐Temperature Expansions for the Ising Model of a Ferromagnet and an Antiferromagnet
View Description Hide DescriptionLow‐temperature expansions for the free energy of the Ising model of a ferromagnet and an antiferromagnet are derived for the more usual two‐ and three‐dimensional lattices. The underlying enumerative problem is studied and a new method described that makes it possible to obtain more terms than available previously without undue labor.

Special Class of Feynman Integrals in Two‐Dimensional Space‐Time
View Description Hide DescriptionContributions of Feynman diagrams consisting of a single loop with an arbitrary number of vertices are explicitly evaluated in two‐dimensional space‐time. The result can be written as a sum of logarithms multiplied by algebraic expressions. Each logarithm is characteristic of a simple diagram of one loop with two external lines, while the coefficients can be obtained from rules analogous to the rules of residue calculus.

Exact Solution of a Many‐Fermion System and Its Associated Boson Field
View Description Hide DescriptionLuttinger's exactly soluble model of a one‐dimensional many‐fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [ρ(p), ρ(−p′)], which always vanish for any finite number of particles, no longer vanish in the field‐theoretic limit of a filled Dirac sea. In fact the operators ρ(p) define a boson field which is ipso facto associated with the Fermi‐Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter n̄_{k} , and find: ∂n̄_{k} /∂k_{ kF } = ∞ (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.

Adiabatic Particle Orbits in Discontinuous Fields
View Description Hide DescriptionThe invariance properties of the action integral J = ∮ p dq are studied for the motion of charged particles in one‐dimensional electromagnetic fields. Attention is concentrated on those situations where the field gradients become large. Whereas in the case of smooth fields and finite gradients the asymptotic theory of the one‐dimensional oscillator as developed by Gardner and by Lenard applies, the presence of large gradients, requires a special treatment. The present considerations are restricted to cases where the strong field variation is confined to a small region such that the transition can be approximated by a discontinuity. It is shown that for time intervals of order 1/ε, J is an adiabatic invariant of at least order ε^{½}, ε measuring the time variation of the fields. As an example, the motion of high‐speed particles through a plane discontinuity is shown to be adiabatic in that sense. Consequently, the initial and the final magnetic moments differ only slightly.

The Dirac Equation and the Unitary Representations of the Inhomogeneous Lorentz Group
View Description Hide DescriptionThe Dirac equation in its quantized form is discussed in order to deduce the structure of the unitary representation of the Poincaré group in Fock space. There is a short account of discrete symmetries in Fock space and a simple method of calculating spin matrix elements is outlined.