Using a generalized version of the Volterra-Wiener functional technique, we study a model for a many-mode laser action in which the coupling among modes takes place through the intensities. It is shown that the coupled Langevin equations for the field amplitudes can be solved avoiding the infinities at the mode thresholds. In particular, mean photon numbers, variances, and correlation and cross-correlation functions of the modes can be computed. In order actually to carry out the computations, it is necessary to evaluate the solution of a set of coupled deterministic first-order differential equations and to diagonalize a matrix whose dimension is given by the number of modes envisaged. As an application, the two-mode case is treated in detail and, besides a derivation of the stationary properties, in part already known, the transient photon statistics, during the buildup of the two modes, is thoroughly studied. The advantage of this approach is due to its simplicity and to the possibility of dealing with a large number of interacting modes.

The effects of thermal fluctuations on the convective instability are considered. It is shown that the Langevin equations for hydrodynamic fluctuations are equivalent, near the instability, to a model for the crystallization of a fluid in equilibrium. Unlike the usual models, however, the free energy of the present system does not possess terms cubic in the order parameter, and therefore the system undergoes a second-order transition in mean-field theory. The effects of fluctuations on such a model were recently discussed by Brazovskii, who found a first-order transition in three dimensions. A similar argument also leads to a discontinuous transition for the convective model, which behaves two dimensionally for sufficiently large lateral dimensions. The magnitude of the jump is unobservably small, however, because of the weakness of the thermal fluctuations being considered. The relation of the present analysis to the work of Graham and Pleiner is discussed.

The Fokker-Planck equation governing the probability density $P(\overrightarrow{\omega }, t; {\overrightarrow{\omega }}_0)$ of the angular velocity of a body undergoing rotational Brownian motion is solved without approximation for the case of a symmetric top. Correlation functions of the components of angular velocity in a principal body coordinate system. $〈{\omega }_i\mathrm{}\left(t\right)\mathrm{}{\omega }_l\mathrm{}\left(0\right)\mathrm{}〉$, are calculated without approximation for a symmetric top by use of $P(\overrightarrow{\omega }, t; {\overrightarrow{\omega }}_0)$, and also by use of rotational Langevin equations. For a body of general shape, Fourier transforms, and related correlation times, are calculated from rotational Langevin equations, the results being correct to second order in the differences of the principal moments of inertia of the body.

We calculate the smectic order-parameter contribution to the nematic elastic energy near a second-order nematic-smectic-$A$ phase transition and show that the elastic behavior is nonlinear for large strains. We show that the Fredericksz transition in a magnetic field becomes first order to a highly strained state within 0.1°C of the phase transition under typical experimental conditions.

The Schrödinger equation for a system of $N$ mutually gravitating particles is solved in one dimension. Solutions are constructed out of a product of Airy functions. The eigenvalues are shown to be the zeros of the derivative of the Airy function. The total energy of the system is found to be proportional to $N^{\frac{5}{3}}$ for a boson system.

The dispersion curve for sound propagating in a layer of fluid undergoing thermally driven convection is shown to exhibit an "energy gap" phenomenon associated with the spatially periodic structure of the convective flow. Numerical estimates for the phase velocity, group velocity, and attenuation anomalies at the gap are given for a typical laboratory experiment.

The ${}_{}{}^3{}_{}{}^{}P\to {}_{}{}^1{}_{}{}^{}D$ electron-impact excitation cross section within the ground configuration of both neutral carbon and silicon is calculated in the distorted-wave approximation of the two-state Hartree-Fock coupled equations. An essential element of the present treatment is that we do not assume orthogonality to core orbitals in deriving equations for the scattering orbitals. A local adiabatic polarization potential is also added to the distorted-wave equations. Both elements are necessary in getting good agreement with close-coupling results for carbon to low impacting energies. The agreement is sufficiently good that our predictions for silicon should be accurate to within a factor of 2.

The effect of the transition zone on the surface tension of Ar near the triple point is reexamined. The surface tension is calculated from the Kirkwood-Buff formula for the cases of a linear density profile and also a cubic profile and the formulas differ from those previously obtained by Fitts. The realistic Barker-Fisher-Watts potential is used and the radial distribution function in the bulk liquid is taken from experiment. Several approximate representations of the distribution function in the transition zone are used. The results indicate that the surface tension is a slowly varying function of the thickness of the transition zone, and that Fowler's step-function profile does not introduce substantial error. When this value is combined with the three-body nonadditivity contribution, the total tension is significantly smaller than the experimental value.

Zwanzig and Bixon have introduced a hydrodynamic model for self-diffusion, in which a single tagged particle moves through a viscoelastic continuum. After a small error in their calculation is corrected, it is shown that the agreement between theory and molecular-dynamics calculations of the velocity autocorrelation function is improved. We apply a similar type of model to the vibrational relaxation (dephasing and population relaxation) of a diatomic molecule in a monatomic fluid. The quantitative agreement with expectations from experiment is poor, suggesting a breakdown of the approach at the high frequencies involved in vibrational motion; however, the qualitative predictions of the model may be useful in correlating experimental data on vibrational Raman line shapes. Futhermore, the results imply that vibrational relaxation experiments in liquids will provide nonhydrodynamic information on the poorly understood high-frequency viscoelastic properties of liquids.

In a previous paper a noniterative procedure was developed to calculate the radial-distribution function for a classical electron gas. Here we extend this method to a one-component system of charged particles whose interparticle potentials have a short-range as well as a long-range contribution. Numerical results are presented for charged hard spheres; comparison is made with results generated by an integral-equation technique.

A theory of spin-lattice relaxation due to layer sliding in smectic-$B$ liquid crystals is developed. For the layer sliding three, two-dimensional, diffusion-type models of motion are used. One is of continuous nature appropriate for weak longitudinal correlations. The other two are of jump nature and are more appropriate when longitudinal correlations are strong. Results for relaxation dispersion and anisotropy are presented graphically. The theoretical results compare quite well with the experimental dispersion of $T_1$ data of TBBA in the smectic-$B$ phase. However, the agreement is not conclusive. A sliding correlation time which would satisfy the experiment is about ${10}^{-5\mathrm{}}$ sec.

The nonlinear kinetics of an intense, incoherent, high-frequency (${\omega }_t\gg {\omega }_p$), isotropic photon spectrum of wide spectral width ($\delta \omega \gg {\Delta }_{\mathrm{Doppler}}\omega \gg {\omega }_p$) propagating through a rarefied isotropic plasma is examined when induced Raman and Compton effects are both included. In the presence of weak plasma turbulence, $\frac{W}{n_e\mathrm{kT}}\gtrsim \left(\frac{{\omega }_t}{{\omega }_p}\right)\left(\frac{{k_t}^3}{n_e}\right)$, Kompaneet's differential equation must be replaced by a new nonlinear kinetic equation of the Burger type which accounts for the linear Raman diffusion and for the nonlinear induced Compton scattering.

For an excitation by a small probe in a magnetized plasma, we observed ray velocities (phase velocities in the direction of group velocities) of ion waves near the ion cyclotron frequency. The propagation of ion waves from the exciter was found to be restricted inside a fixed cone angle which corresponds to a low-frequency resonance cone.

The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\to 0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O\left(r^{-2\mathrm{}}\right)$ as $r\to \infty$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.

We have investigated elastic deformations by external magnetic fields in flat samples of smectic $C$ with fixed boundary conditions. In the calculations the internal parameters (tilt angle, density, interlayer distance) are assumed to be fixed, distortions of the smectic layers are neglected, and only reorientations of the director about the normal to the smectic layers are allowed. The problem is solved exactly assuming a one-dimensional variation of the order parameter. Stability conditions and explicit expressions for the orientation of the director as a function of position are derived for general orientations of the magnetic field. Solutions of the variational problem can be classified according to the maximum deviation angle, $\Phi$, of the director. In general there are several separated allowed regions of $\Phi$. When Fréedericksz transitions occur, they are usually discontinuous first-order transitions. The properties of the solutions are discussed and some special examples are considered in detail. Transitions are investigated both as a function of the magnitude and of the orientation of the magnetic field. Expressions for the free energy are also derived.

This paper employs a hybrid Mayer-Salpeter cluster expansion to reinterpret a previously developed collective-coordinate approach to the radial distribution functions for a classical one-component electron gas.

Assertions that stimulated emission may cause photon correlations (or, properly, photocount correlations) are discussed. A particular association of terms in the density matrix rate equation for first order, linear, interactions is urged, following the literature of maser amplifiers. With this association of terms, a consistent picture results in which stimulated emission amplifies without altering the statistical character of an incident field. Spontaneous emission generates excess correlations, through its interference with the incident field. Objections to Webb's thermodynamic argument for an association between stimulated emission and the occurrence of photon correlations are also discussed.

It is explicitly shown how the Scully-Lamb strong-signal master equation can be deduced from the single-mode laser model of Haken and co-workers.

Absolute cross sections for electron transfer into the $3p$ and $3d$ states of hydrogen have been measured for 2.2-to 8.2-keV proton impact on He, Ar, Kr, ${\mathrm{O}}_2$, ${\mathrm{H}}_2$, and ${\mathrm{N}}_2$. An apparent maximum occurs in the $3d$ cross section for both Ar and ${\mathrm{H}}_2$. Balmer-alpha cross sections are synthesized from these results and previously reported $3s$ cross sections.

We consider a simple relativistic correction to the plane-wave Born-approximation (PWBA) model, in the refined form of Basbas, Brandt, and Laubert. This correction improves the agreement between the PWBA model and experimental data.