We present here a comparison between measurements made in numerical experiments and approximate solutions to the Boltzmann equation for a dilute gas in a finite geometry and subjected to a temperature gradient. We first briefly recall a moment method that has been developed to solve the stationary state for the Boltzmann equation with boundary conditions independently of any small gradient expansions and we then apply it to the case of a temperature gradient within the so-called four- and six-moment approximations. These solutions are then compared to the results obtained by nonequilibrium molecular dynamics simulations of a dilute gas enclosed between parallel plates held at different temperatures: Local measurements of the moments of the one-particle distribution function, related to the densities, the temperature, and the fluxes, are reported for various Knudsen numbers and for various temperature gradients.

We consider neural networks of the Hopfield type with couplings $J_{\mathrm{ij}}$ which need not be symmetric. From the master equation for microscopic states we derive an evolution equation for the probability density of the macroscopic parameters $q_{\mu }$, which measure the overlap of the instantaneous microscopic state (or image) with one of the built-in patterns. No restrictions are imposed on the choice of the patterns. For three different temperatures this equation is used to illustrate retrieval in the standard Hopfield network and limit-cycle behavior in nonsymmetric models.

We derive the correlation of spatially coupled subsystems in the case in which a dissipative organization is sustained in an open reactive system which operates far from equilibrium. This phenomenon is viewed in the context of phase-ordering dynamics of thermodynamically unstable systems, where the organization is achieved by amplification of nonequilibrium fluctuations (as in spinodal decompositions and in nucleation processes). Thus we show how the correlation of subsystems leads to the contraction of phase space to a center manifold and to the propagation of fluctuations along the center manifold in order to reach the dissipative structure.

In this paper we show that diffusion processes in a ‘‘complex’’ phase space with many local minima can be mapped into a random-walk problem on a tree structure. We then rigorously solve the latter problem for regular trees, under quite general assumptions about the rates. Finally, we extend our results to the case of inhomogeneous trees.

We study the growth of a surface through deterministic local rules. A scaling theory for the generalized deterministic Kardar-Parisi-Zhang equation ${\partial }_t$h=D Δh+λ‖∇h${\Vert }^{\beta }$, with β≥1, is developed. A one-dimensional surface model, which corresponds to β=1, is solved exactly. It can be obtained as a limiting case of ballistic deposition, or as the deterministic limit of the Eden model. We determine the scaling exponents, the correlation functions, and the skewness of the surface. We point out analogies to the Burgers equation (β=2), for which such detailed properties are not known.

We study the elasticity, topological defects, and hydrodynamics of the recently discovered incommensurate smectic ($A_{\mathrm{IC}}$) phase, characterized by two collinear mass density waves of incommensurate spatial frequency. The low-energy long-wavelength excitations of the system can be described by a displacement field u(x) and a ‘‘phason’’ field w(x) associated, respectively, with collective and relative motion of the two constituent density waves. We formulate the elastic free energy in terms of these two variables and find that when w=0, its functional dependence on u is identical to that of a conventional smectic liquid crystal, while when u=0, its functional dependence on w is the same as that for the angle variable in a slightly anisotropic XY model. An arbitrariness in the definition of u and w allows a choice that eliminates all relevant couplings between them in the long-wavelength elastic energy. The topological defects of the system are dislocations with nonzero u and w components. We introduce a two-dimensional Burgers lattice for these dislocations, and compute the interaction between them. This has two parts: one arising from the u field that is short ranged and identical to the interaction between dislocations in an ordinary smectic liquid crystal, and one arising from the w field that is long ranged and identical to the logarithmic interaction between vortices in an XY model. The hydrodynamic modes of the $A_{\mathrm{IC}}$ include first- and second-sound modes whose direction-dependent velocities are identical to those in ordinary smectics. The sound attenuations have a different direction dependence, however. The breakdown of hydrodynamics found in conventional smectic liquid crystals, with three of the five viscosities diverging as 1/ω at small frequencies ω, occurs in these systems as well and is identical in all its details. In addition, there is a diffusive phason mode, not found in ordinary smectic liquid crystals, that leads to anomalously slow mechanical response analogous to that predicted in quasicrystals, but on a far more experimentally accessible time scale.

A recently synthesized and characterized homologous series of liquid-crystal compounds (n4COOBC) is shown by high-resolution ac microcalorimetry experiments (on the n=3,6 homologs) to most likely exhibit smectic-A to smectic-B (hexatic) (Sm$B_H$) phase transitions exclusively. This is in partial contrast to earlier tentative assignments, based on low-resolution calorimetry, x-ray, and microscopy studies, which had crystal smectic-B (Sm$B_x$) for n=4–8 and Sm$B_H$ for n=3 and 9. The SmA-Sm$B_H$ thermal anomalies reported here exhibit strong fluctuation contributions but are clearly first order. Experimental results are compared with the extensively characterized n-alkyl-4’-alkoxy-biphenyl-4-carboxylates (nmOBC’s) and discussed in the context of Goodby’s empirical molecular structure rules for the existence of the Sm$B_H$, and in the context of a generic phase diagram (SmA, Sm$B_x$, Sm$B_H$) recently predicted by Aharony et al. The latter illustrates the physical content of our earlier suggestion that short-range hexagonal positional order is responsible for ubiquitous tricriticality of the SmA-Sm$B_H$ phase transition.

A recursive relation to calculate the pair-correlation function of the q-state (ferromagnetic) Potts model on a Cayley tree with a homogeneous magnetic field applied to its surface spins is presented. The results are reduced to the calculation of the pair-correlation function of the model defined on a finite chain.

The influence of the pressure thermostat on the equilibrium properties in the Nosé-Hoover (NH) method is evaluated. Three tests were made to choose the isobaric relaxation time ${\tau }_p^{\mathrm{*}}$. For the present system (N=108 particles, ρ${\sigma }^3$=0.75, and kT/ε=0.9) the lowest value for which ${\tau }_p^{\mathrm{*}}$ is Gaussian was ∼0.1, which is twice the mean collision time τ. For ${\tau }_p^{\mathrm{*}}$>0.1 the NH method obeys the general Langevin equation giving the same Mori coefficients, and for ${\tau }_p^{\mathrm{*}}$<0.1 the method breaks down.

The simplicity of the state dependence at relatively high temperatures of the many-body potential contribution to the pressure and energy has been pointed out previously [J. Ram and P. A. Egelstaff, J. Phys. Chem. Liq. 14, 29 (1984); A. Teitsima and P. A. Egelstaff, Phys. Rev. A 21, 367 (1980)]. In this paper, we investigate how far these many-body potential terms may be represented by simple models in the case of krypton on the 423-, 273-, 190-, and 150-K isotherms, and xenon on the 170-, 210-, and 270-K isotherms. At the higher temperatures the best agreement is found for the mean-field type of theory, and some consequences are pointed out. On the lower isotherms a state point is found where the many-body energy vanishes, and large departures from mean-field behavior are observed. This is attributed to the influence of short-ranged many-body forces.

We study the propagation of an elliptically polarized light beam normally incident onto an arbitrarily oriented liquid crystal in steady-state conditions. The Euler-Lagrange equations for the molecular director and the equations describing the evolution of the beam polarization in the birefringent medium are derived from a unique variational principle, which is proved to be consistent with the geometric-optics approximation. The Hamiltonian formulation of the theory is studied in detail. The conservation of total angular momentum and total free energy in the process is derived from Noether’s theorem, and the theory of the adiabatic invariants is used to obtain a new proof of Mauguin’s theorem of crystal optics. The general analytical solution of the propagation problem is presented for the important case of pure twisted structures. It is proved that two particular solutions exist (called Mauguin’s solutions) obeying Mauguin’s theorem rigorously. Only these solutions may exhibit the occurrence of the optical Fréedericksz transition. In general, multiple optical thresholds are found. An analytical formula to obtain the thresholds is also derived.

It is shown that the use of the self-interaction-corrected Ξa method leads to the values of the Sternheimer quadrupole antishielding factor ${\gamma }_{\infty }$ for ${\mathrm{F}}^{\mathrm{-}}$, ${\mathrm{Cl}}^{\mathrm{-}}$, and ${\mathrm{Br}}^{\mathrm{-}}$ as -25, -65, and -146, respectively, which provide a significant improvement over the corresponding values of -41, -82, and -195 derived from the Xα wave functions. The Hartree-Fock estimates of ${\gamma }_{\infty }$ as given by -22, -55, and -155 for ${\mathrm{F}}^{\mathrm{-}}$, ${\mathrm{Cl}}^{\mathrm{-}}$, and ${\mathrm{Br}}^{\mathrm{-}}$, respectively, are therefore approximated quite well by means of the Ξa wave functions.

The ionization process for slow collisions of protons with ground-state ${\mathrm{He}}^{+}$ions is studied. A simple model is formulated based on analytic properties of quasimolecular adiabatic energy surfaces and the adiabatic theory of inelastic transitions. Cross sections are calculated for center-of-mass collision energies ranging from 8–68 keV. These results are compared with other theoretical and experimental data.

We present differential cross sections for the excitation of atomic hydrogen to the 2s and 2p states, for incident positron (and electron) energies ranging from 100 to 400 eV, obtained by using the unitarized eikonal-Born-series method.

We consider, in the context of nonrelativistic quantum electrodynamics, the transformation from the minimal-coupling Hamiltonian to one where the interaction is quadratic in the electromagnetic vector potential. The new effective Hamiltonian is used to obtain the energy shifts in an atom caused by its interaction with the electromagnetic field. This demonstration is very compact since all the terms contributing to the shift arise from the single quadratic-coupling Hamiltonian. A brief comparison with other Hamiltonians having alternative couplings is made.

Laboratory and particle simulation experiments are used to show that moderate-power microwaves (laser light) [I${\lambda }^2$/${\mathit{T}}_{\mathit{e}}$(W/${\mathrm{cm}}^2$)(μ${\mathrm{m}}^2$/keV)≲5×${10}^{14}$ for ${\mathit{ZT}}_{\mathit{e}}$/${\mathit{T}}_{{}_{}{}^{\mathrm{u}}{}_{}{}^{}}$=10] strongly heat hot electrons due to parametric instabilities in an underdense plasma, when the microwaves are incident on a large-scale-length plasma. The anomalous collision frequency is as high as ${\nu }^{\mathrm{*}}$/${\omega }_0$∼4×${10}^{\mathrm{-}2}$. The hot electrons are strongly heated due to the large size of the instability region.

The isothermal elastic constants of the perfect fcc hard-sphere crystal are computed from our approximate density-functional theory and found to be in qualitative agreement with the computer-simulation results over the full density region ranging from marginal stability of the solid up to the close-packing density.

Within the sustained regimes of spatio-temporal intermittency occurring in coupled map lattices and directed percolation with two space dimensions, fluctuating clusters are shown to be well described by ordinary percolation concepts and critical exponents.

After discussing a possible form of the transformation from the wave-function of the Lax equations of the nonlinear Schrödinger (NLS) equation relating to the (n-1)-soliton solution to that relating to the n-soliton solution, an explicit solution of it is found for general n. This is then used to obtain soliton solutions of the NLS equation and solutions of the Lax equations. The usual Bäcklund transformation for the NLS equation is deduced simply. The Zakharov-Shabat equations of the inverse scattering method are also deduced simply, but without the restriction of ${\lambda }_j$ in the upper half-plane. It is shown that the solutions of the NLS equation are regular when no two of ${\lambda }_1$,λ${\mathrm{¯}}_1$,...,${\lambda }_{\mathit{N}}$,λ${\mathrm{¯}}_{\mathit{N}}$ are coincident.

A detailed numerical analysis of the direct correlation function of classical fluids is presented. Yukawa-type tails are assumed, as well as hard-core potentials for the interatomic interactions. The linear approximation relating the tail of the direct correlation function to the corresponding core is discussed and appears to be satisfactory for a variety of conditions.