A repeated-ring kinetic equation for the velocity-autocorrelation function in a hard-sphere fluid is presented. In this theory contributions from uncorrelated (Enskog) collisions, correlated (single-ring) collisions, and multiple (repeated-ring) collisions are included. A quasihydrodynamic approximation is made to describe the intermediate propagation between correlated collisions. We obtain an expression for the self-diffusion coefficient $D$ which suggests that its density dependence arises from a competition between the shear and density fluctuations of the fluid. Using interpolation formulas to numerically evaluate the coupling of the test-particle motion to the fluid fluctuations we have computed $D$. The variation of $D$ with density is found to be in good qualitative agreement with computer molecular-dynamics simulation. For low to moderate density the coupling of the test-particle motion to the fluid shear (vortex) fluctuations leads to an enhancement of $D$ relative to its Enskog value. At high density the coupling to the fluid-density fluctuations yields a sharp decrease in $D$.

The velocity-autocorrelation function of a tagged particle moving in a classical liquid is expressed in terms of a characteristic oscillator frequency and a frequency-dependent relaxation kernel. The relaxation spectrum is approximated by calculating the interaction of the tagged particle with current excitations of the liquid. The interaction with the longitudinal modes is shown to be responsible for the observed peak structure of the correlation function. The results of the present theory, in particular the values for the diffusion constant, agree well with the molecular-dynamics experiments on argon and rubidium.

The lowest-order corrections to the critical temperature, pressure, and density are given for a system characterized by a Lennard-Jones interparticle potential. Results are found in three and two dimensions using a truncated virial expansion. The results are compared with critical data for the noble gases.

The statistical equilibrium of charged nonrelativistic multiply periodic classical mechanical systems and random classical electromagnetic radiation is considered in the narrow-linewidth approximation using calculations by van Vleck based on action-angle variables. It is found that the nonrelativistic classical mechanical systems and the classical radiation are both in equilibrium when the distribution of the mechanical systems in phase space is given by the Boltzman distribution and the radiation spectrum is given by the Rayleigh-Jeans law. This result was reported by van Vleck in 1924 but the calculations were never published. It is also shown that nonrelativistic periodic systems with no harmonics are in equilibrium in classical zero-point radiation provided the phase-space distribution is given by $e^{-\frac{2J}{h}}$, where $J$ is the action variable for the system and $h$ is Planck's constant. In the case of a nonrelativistic nonlinear dipole oscillator the action-angle description is connected with earlier work, also leading to the Rayleigh-Jeans law, which did not use the narrow-linewidth approximation. It is pointed out that the use of relativity for the mechanical systems will lead to expressions different from those used in this work, and it is conjectured that the appearance of the Rayleigh-Jeans law as an equilibrium spectrum may reflect the nonrelativistic character of the scattering systems.

The adiabatic hypothesis of old quantum theory which formed the basis for Bohr-Sommerfeld quantization assumed that the adiabatic invariants $J_l=\oint \int p_{l}d{q}_l$ of multiply periodic mechanical systems should be quantized with the discrete values $J_l=nh$ or sometimes $J_l=(n+\frac{1}{2})h$, where $h$ is Planck's constant. The hypothesis did not follow from any more general assumption. In the present paper we derive some related results within the theory of classical electrodynamics with classical electromagnetic zero-point radiation. We derive the fact that the adiabatic invariants of a charged nonrelativistic periodic mechanical system with no harmonics remain adiabatic invariants when placed in the classical electromagnetic zero-point radiation spectrum and only in this spectrum. Furthermore the phase-space distribution for the three-dimensional mechanical system in zero-point radiation becomes $P(J_1, J_2, J_3)=\mathrm{const}\times \exp [-2\mathrm{}\frac{(J_1+J_2+J_3)}{h}]$ leading to the average value $〈J_l〉=\frac{1}{2h}$ for each of the adiabatic invariants. Here $h$, which is naturally chosen as Planck's constant, is just the scale factor appearing in the classical zero-point radiation. For each of the systems in equilibrium with zero-point radiation already treated in the literature, we sketch the connection with the unifying formalism of action-angle variables.

Under the conditions existing in electron-beam-irradiated pellet fusion targets, temperature and density effects on the energy loss and multiple scattering are shown to cause significant effective range shortening. A significant drop in the bremsstrahlung yield is also calculated.

In this paper a general formalism for the treatment of relativistic quantum plasmas is given. It is manifestly covariant and rests on the use of a (covariant) relativistic Wigner function. Here it is applied to the particular case where spin effects are neglected (in most astrophysical applications this is a good approximation): a relativistic quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is given. The Vlasov approximation (Hartree approximation) is then considered and dispersion relations are obtained. Limiting cases (relativistic nonquantum high-temperature plasma and relativistic degenerate zero-temperature plasma) obtained previously by other authors are found anew. Finally, the formalism given appears to be much simpler and physically more transparent than many-body techniques used elsewhere.

The smectic-$B$ to smectic-$E$ transition is discussed in terms of an orientational intermolecular potential recently suggested by Meyer. Within the framework of the molecular-field approximation the transition temperature is obtained from the lowest eigenvalue of an Hermitian matrix, and the ordered state below it is described by the corresponding eigenvector. A condition is found for which the experimentally observed herringbone structure is the most stable one.

A second-order virial equation of state for a symmetric system of nucleons ($N=Z$) interacting through a deuteronlike force is established. The dependence of the virial coefficient on temperature and the square-well parameters is discussed. The isotherms are analyzed, and it is shown that the gas-liquid phase transition takes place at a much lower density than that commonly assumed to be the characteristic density of nuclear matter.

Low-temperature limits for the liquid state, one of which is a stability limit which appears to arise from a structural instability in the liquid and results in crystallization, are analyzed from the viewpoint of the liquid's structure. For the case of the hard-sphere fluid, available theories are employed towards the calculation of the structure factor of the fluid and the consequent determination of a stability limit. The results presented herein indicate that, for the hard-sphere fluid, as this stability limit is approached, the dependence of the structure factor on structural and thermodynamic parameters is similar to the corresponding dependence which has been observed for simple liquids and liquid metals.

Using techniques developed by Greenspoon and Pathria, and by Chaba and Pathria, a rigorous asymptotic analysis of the onset of Bose-Einstein condensation in a finite three-dimensional system at constant pressure, under Dirichlet boundary conditions, is carried out. Finite size corrections arising from a modification of the density of states of the system, as well as from the discreteness of the single-particle energy levels, play an important role in the critical region. At the critical temperature $T_c$, the volume of the system changes abruptly from being proportional to the total number of particles $N$ to being proportional to $N^{\frac{3}{4}}$, the number of particles in the condensate rises sharply from $\propto N^{\frac{1}{4}}$ to $\propto N$, and the specific heat $C_P$ turns out to be proportional to $N^{\frac{9}{8}}$. At somewhat lower temperatures, the finite size effects arising from the discreteness of the single-particle energy levels become dominant and the volume of the system, together with $C_P$, becomes $\propto N^{\frac{3}{5}}$.

Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at $r_s=139\mathrm{}$, where $r_s$ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at $r_s=2\mathrm{}$, to a small but finite value (∼1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit ($r_s\gg 1$).

The three pair-correlation functions in a two-component (electrons and ions) plasma are calculated to second order in the plasma parameter $\epsilon$. Expressions for the correlation energy and compressibility are obtained and their dependence on the degree of ionization $Z$ is explicitly shown.

The intensity spectrum of the scattered radiation in two-atom resonance fluorescence is investigated and compared with the one-atom spectrum. For intense fields the spectra are found to be proportional to one another. A conjecture is made for the $N$-atom spectrum.

Resonant scattering (redistribution) of weak monochromatic light by atoms in the presence of collisions has been considered by Omont et al., who allowed for the broadening of both upper and lower levels of the transition. Mollow (who also has considered this problem for intense radiation) has shown that their results are applicable only if collisional repopulation of the levels (in contrast to collisional depopulation) is ignored. We here consider in detail the effect of repopulation. It leads, in the weak-field limit, to a true-coherent component ($\delta$-function spectrum) at the driving frequency, as well as a component at this frequency whose width is essentially the inelastic collisional rate of the lower state. An additional redistributed component due to repopulation of the upper level is also obtained.

A recent paper by Luken and Sinanoǧlu (Phys. Rev. A 13 1293, 1976) has criticized some of our work and contains material on excited states and oscillator strengths. We suggest that satisfaction of upper boundedness via the Hylleraas-Undheim-MacDonald theorem is neither a sufficient nor a necessary condition for obtaining reasonably accurate oscillator strengths. Our method for truncated Hamiltonian matrices which chooses the root which minimizes the energy as well as the correlation overlap $〈\frac{X}{X}〉$ in fact yielded upper bounds and did not suffer a "variational collapse." We point out that for excited valence states embedded in Rydberg or continuum series, the $f$ values are very sensitive to (a) choice of basis sets and (b) Relative position of diagonal matrix elements. The Ni ${}_{}{}^4{}_{}{}^{}S_{}^o{}_{}{}^{}\to 2s2p^4 {}_{}{}^4{}_{}{}^{}P$, Bi ${}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}\to 2s2p^2 {}_{}{}^2{}_{}{}^{}S$, Ci $2p^2 {}_{}{}^3{}_{}{}^{}P\to 2s2p^3 {}_{}{}^3{}_{}{}^{}P_{}^o{}_{}{}^{}$ transitions serve as examples.

The concept of variational collapse is based on the form of a variational trial wave function and not solely on its energy. In particular, a variational calculation is said to suffer from variational collapse when a trial wave function intended to represent an excited state takes on the character of one or more states of lower energy as the energy of the trial wave function decreases. In view of this, we repeat our claims that certain calculations published by Nicolaides and Beck suffered from variational collapse, and consequently the results of these calculations do not represent the non-closed-shell many-electron theory of Sinanoǧlu.

Recently Pollock and Alder have come to the surprising conclusion that the first virial coefficient for the macroscopic electric field produced by a permanent dipole as calculated by a continuum argument differs by a factor of 2 from that found directly from a virial calculation or from a molecular dynamics simulation. Here it is shown that the origin of this discrepancy is the result of an incorrect identification of a cavity field with a macroscopic field. When the correct identification is made there is complete agreement among the results of a continuum calculation, the use of a cubic lattice, and the virial expansion. In higher order the results of these models are expected to differ and molecular dynamics simulations should prove useful as a check on the applicability of the various models.

The molecular-orbital self-consistent-field method is used to determine the molecular energy in the presence of a strong magnetic field. It is shown that in the fields of magnitude ${10}^5$ T the formations of molecules are energetically favorable.

The ground-state wave function of a heliumlike atom is assumed to be $\psi =\phi \left(r_3\right)\exp (-\alpha r_{>}-\beta r_{<})$, where $r_{<}, r_{>}=\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{ }\max (r_1, r_2)$, $r_1$ and $r_2$ being the distances of the electrons from the nucleus. An equation which determines the electron-electron correlation function $\phi \left(r_3\right)$ is derived and numerically solved. The parameters $\alpha$ and $\beta$ are determined by minimizing the energy of the system. This is an extension of the Feshbach-Rubinow-type approximation which has recently been applied to the same problem. The positronium negative ion ($\mathrm{ee}{e}^{+}$) is also examined by the same method. It is found for the binding energies that the improvement over the Feshbach-Rubinow approximation is substantial for the $\mathrm{ee}{e}^{+}$ system, but less significant for heliumlike atoms. It is claimed that the proposed method is particularly useful in calculating quantities which are sensitive to the short-range correlation between the electrons.