Aid or competition between two laser modes oscillating on different transitions of a three-level medium are studied following the method of Lamb. An arbitrary amount of Doppler broadening is allowed for. The "electric-quadrupole" term generated by back-to-back transitions of the two modes but having vanishing electric dipole moment is found to have little effect on the oscillation intensities in typically observed circumstances. This fact justifies, in part, the rate-equation approximation commonly used in multilevel analysis. For a bidirectional ring cascade configuration, however, this coherence term can be important because of constructive interference familiar from two-photon, Doppler-free spectroscopy. The ordinary (i.e., hole-burning) cross-saturation terms for cascade transitions produce mutual mode aid, rather than competition as in the competitive cases. A curious consequence of this difference is greater accuracy in the third-order theory of cascade transitions than in that of competitive transitions, as a result of partial cancellation of errors incurred through neglect of higher orders. The treatment is generalized to multilevel media (four or more levels) with numerical illustration and discussion of the method's relationship to laser oscillation involving diatomic molecules.

In the present paper, we present additional calculations of the contributions to the logarithmic term in the virial expansion of the transport coefficients of hard spheres. We consider the limiting cases of Brownian and Lorentzian gases and compare with previously known results.

The existence of an oscillatory mode which can propagate in a nematic liquid crystal under the influence of an electric field is demonstrated. A real dispersion relation can be derived from the continuum-mechanics equations of motion for certain values of material constants relative to the applied electric field and boundary conditions. The velocity of the wave is extremely small, on the order of 10 μm/sec for typical experimental conditions.

The cooperative-problem lattice model for gases, liquids, and solids on a face-centered-cubic lattice is approximately solved. A hard-core interaction is assumed between particles on nearest-, next-nearest-, and on next-but-one nearest-neighbor lattice sites. Particles on lattice sites which are at a greater distance from one another are connected by an arbitrary soft pair interaction whose zero Fourier component may be positive or negative. By considering not only homogeneous particle distributions but also some simple periodical particle distributions on the lattice we can, in the case of a mean attractive soft particle interaction, prove the possibility of the existence of four thermodynamical stable phases: a gas phase, a liquid phase, a solid modification with a face-centered-cubic periodical structure, and a solid modification with a simple cubic periodical structure. Transitions between these phases can take place.

It is shown that if the critical-point exponent $\eta$ vanishes, then phase transitions can exist only in greater than two space dimensions.

Asymptotic expansions for the thermodynamic functions of an ideal Fermi gas are obtained about the point $z=1\mathrm{}$, where $z$ is the fugacity. We do this in three dimensions for the three different types of single-particle energy spectra, namely nonrelativistic, relativistic, and massless. To relate to liquid-${}_{}{}^3{}_{}{}^{}\mathrm{He}$ films, we also study the nonrelativistic spectrum in two dimensions.

The fluctuation-dissipation theorem (FDT) valid both for canonical systems in equilibrium and for purely irreversible stochastic processes with detailed balance is shown to imply that the renormalized values of all coefficients in the field equation ("mass" and "charges") are simply related to static cumulants of the field. As a consequence, the problem of determining the static behavior of the system (which reduces to quadratures when the FDT holds) is completely separated from the dynamic problem; the perturbation theory for the time dependence of the two-point cumulant has, if properly renormalized, the full statics of the system incorporated exactly; the renormalized perturbation theory (RPT) will, in general, be much more useful than the unrenormalized one since in contrast to the latter it does not necessarily require the system to be nearly Gaussian. If the nonlinearity in the field equation involves only a quartic charge matrix, the renormalized charge matrix is $\frac{C^{\mathrm{}\left(4\right)\mathrm{}}}{C^{\mathrm{}\left(2\right)4}}$, where $C^{\mathrm{}\left(2\right)\mathrm{}}$ and $C^{\mathrm{}\left(4\right)\mathrm{}}$ are the static cumulants of order 2 and 4, respectively, whereupon the expansion parameter in the RPT becomes $\frac{C^{\mathrm{}\left(4\right)\mathrm{}}}{C^{\mathrm{}\left(2\right)2}}$; when the static cumulants are known, the smallness of this expansion parameter (which does not necessarily require $C^{\mathrm{}\left(4\right)\mathrm{}}\approx 0$, i.e., near-Gaussian statics) can be checked explicitly. The usefulness and necessity of mass and charge renormalization is demonstrated by a perturbative calculation of the linewidth of the Van der Pol oscillator; although this oscillator is far from Gaussian near and above threshold, the result (below, near, and above threshold) is quite satisfactory already in second and excellent in fourth order; the unrenormalized perturbation expansion, on the other hand, yields nonsense in any finite order except very far below threshold where the oscillator is indeed nearly Gaussian. For many-body systems the static cumulants defining the renormalized mass and charge(s) can, in general, not be evaluated exactly; however, in many cases sufficiently accurate approximate values or experimental information are available to render the RPT practical.

The BBGKY hierarchy is analyzed to construct a theory of viscosity in liquids. The approach is an elaboration of a previous theory of self-diffusion. The results reduce to Chapman-Enskog theory at low density and represent a generalization of Enskog theory at high density. The whole analysis is carried out within the framework of Navier-Stokes hydrodynamics; quadratic and cubic contributions to the nonequilibrium correlation functions do not affect the coefficient of viscosity.

The phase velocity and wave amplitude of ion waves launched by a small probe are investigated experimentally and theoretically for all directions in a weakly magnetized plasma. Ion waves in a weakly magnetized plasma are found to propagate omnidirectionally (excluding the effect of plasma flows) with respect to the phase velocity and to be launched bidirectionally along the direction of the magnetic field with respect to the amplitude. The excitation mechanism is also investigated in a fluid model in comparison with the results of a nonmagnetized plasma.

The Enskog transport equation, suitably modified to give the proper short-time behavior, is used to study the phase-space density correlation function of a dense fluid of hard spheres. The method of kinetic models is used to obtain numerical solutions for the dynamic structure factor $S(Q, \omega )$; in particular, results are obtained for liquid argon. It is found that for the hard-sphere calculations to give a satisfactory description of the available experimental data produced by computer molecular dynamics studies and neutron inelastic scattering measurements, the factor $g\left(r_0\right)$, the pair distribution at contact, should be replaced by a wavelength-dependent quantity. Moreover, the wavelength dependence determined by fitting the experimental data shows a close correlation with the well-known behavior of the static structure factor $S\left(Q\right)\mathrm{}$. Possible reasons for a nonlocal $g\left(r_0\right)$ are discussed.

In this paper a general perturbation method is developed for a kinetical analysis of the role and the influence of collisions on the nonlinear effects occurring in weakly or partially ionized gaseous media. The nature of the collision operators leads to an expansion of the electron distribution function in a basis of irreducible Cartesian tensors and to the study of the coupled differential system thus established. It is shown that this system, coupled with Maxwell's equations, can be uncoupled only in weakly dissipative media, where multiple time and space scales can be defined and where the distribution function as well as the electric field can be expanded as a function of a suitably chosen small parameter. The procedure is applied to different physical situations characterized by the order of magnitude of the collisional parameters. It is shown that three cases may be distinguished: collisional, weakly collisional, and an intermediate case. In each of these cases, the propagation of a single wave (longitudinal or transverse) is studied and it is shown how these techniques allow one to determine the orders of magnitude of various nonlinear effects (such as collisional heating and generation of harmonics) as well as the time or space scales at which they occur. Finally, there is a brief discussion of other applications and extensions of the method (such as the influence of a plasma inhomogeneity, the role of the Coulomb collisions, the coupling of several waves, and the low-frequency approximation).

The use of generalized Langevin equations in the study of transport in simple classical liquids is extended by the derivation of the second such equation in the hierarchy of coupled phase-space transport equations. This set of generalized Langevin equations is applied to the description of self-diffusion. An approximation is made to simplify the second generalized Langevin equation, and a perturbation solution is presented for the velocity autocorrelation function. The method introduced is related to the Mori continued-fraction representation, but poses a different point of view for analysis.

A comparison of recent melting data for ${\mathrm{N}}_2$, C${\mathrm{H}}_4$, and Ar suggests that there probably is greater local orientational order in the dense fluid phases of ${\mathrm{N}}_2$ and C${\mathrm{H}}_4$ than in their respective solids near melting. The possible nature of this local orientational order is discussed.

We analyze stimulated scattering of a right-circularly-polarized electromagnetic pump wave, of frequency ${\omega }_0$, from a cold relativistic magnetized electron beam. The pump wave is chosen to satisfy the dispersion relation associated with the magnetized beam. The scattered waves consist of collective plasma oscillations as well as right- and left-polarized electromagnetic waves, traveling parallel and antiparallel to the beam. The frequency of the forward-scattered electromagnetic wave is Doppler shifted up twice and is of order ${{\gamma }_{0z}}^2{\omega }_0$, where ${\gamma }_{0z}={(1-\frac{v_0^2}{c^2})}^{-\frac{1}{2}}$ and $v_0$ is the beam's speed. Enhanced stimulated scattering results if the frequency of the pump, in the beam frame, is approximately equal to the electron's cyclotron frequency. Enhanced growth rates for the scattering process are obtained for the scattered radiation. This theory describes a mechanism which is a plausible explanation of a recent experiment in which ∼ 100 kW of submillimeter radiation was observed. In view of the large up-conversion in frequency and enhanced growth rates of the scattered radiation, a new high-frequency generative device based on this mechanism seems possible.

In this paper, the results of various computer experiments performed on the two-particle, unequal-mass Toda lattice are discussed. These experiments indicate that this system exhibits a transition from near-integrable to macroscopically observable stochastic behavior at a critical value of the total energy $E_c$ which depends on the mass ratio of the two particles ($\frac{m_2}{m_1}$). This transition appears to occur for every mass ratio in the open interval $0<\mathrm{}\frac{m_2}{m_1}<1\mathrm{}$. Details of trajectory behavior at selected energies and mass ratios are exposed through presentation of a number of Poincaré surfaces of section. The transition discussed herein will likely prove to be significant for both mathematics and physics.

The empirically scaled Brinkman-Kramers expression for charge exchange is found to account for the enhancement in target $K$ x-ray production for fully stripped projectiles over that for two-electron projectiles even for collision partners as heavy as Cl on Kr.

We derive analytical solution for a signal field amplified by stimulated Rayleigh and Brillouin effects. The presence of light absorption is taken into account. The solution describes the transient behavior at all frequency differences between the pump and the signal waves. The validity is demonstrated for the simple case of no absorption, and the particular time behavior at $\omega =0\mathrm{}$ is stressed.

Explicit calculations demonstrate that cascade repopulation effects are a possible explanation of the $t^{-1.5\mathrm{}}$ dependence in the decay of Ly-$\alpha$ radiation at long times after excitation observed recently by Braithwaite, Matthews, and Moore. Good agreement with all available experimental data is obtained with reasonable population models; excitation proportional to $n^{-m}$ with $m=2.2-4\mathrm{}$ is indicated.

A six-dimensional integral in the coordinate space of two particles 1 and 2, where the integrand contains Coulomb functions for both the particles, the term $\frac{1}{|{\overrightarrow{\mathrm{r}}}_1-{\overrightarrow{\mathrm{r}}}_2|}$, and an exponential term, is reduced to a double integral in real space. The resulting integral can finally be integrated numerically with a high degree of accuracy. This technique has been verified by its application to the Nordsieck integral for bremsstrahlung with different sets of parameters and compared with the exact method of Nordsieck. The error estimate is always found to be less than 0.01%.

We present electromagnetic self-energy shifts for the $2S$ states of hydrogenic ions for $Z=10, 20, 30, 40, 50, 60, \mathrm{and} 70$. The results are computed nonrelativistically and with retardation according to a recent prescription of ours.