The importance of the flexibility of n-alkyl tail chains in real liquid-crystal systems is reviewed. Two new microscopic, molecular statistical-physics theories (a full statistical theory and a simpler but accurate approximate theory) for the intrachain constraints on the n-alkyl tail-chain flexibility are presented and are compared with each other and with an old, more approximate theory for tail-chain flexibility used in earlier papers. The new approximate approach is computationally much faster than the full statistical method and is the first treatment to generate and explain odd-even effects in multiple smectic-A phases and the first treatment to generate and explain odd-even effects in smectic-A and nematic phases without resorting to ad hoc or arbitrarily adjustable fits to experimental data. Phase stabilities and odd-even effects for various thermodynamic and molecular ordering properties are calculated in the smectic-$A_1$, smectic-$A_d$, and nematic liquid-crystal phases and the isotropic liquid phase using the new approximate method. Some predictions and accompanying physical explanations are made for various systems that have not yet been chemically synthesized and/or experimentally studied. The theoretical results in this paper are in good semiquantitative and (in some cases) quantitative agreement with available experimental data and offer some significant improvements—compared with experiment—over the theoretical results of earlier papers, especially with regard to the relative stabilities of the nematic and multiple smectic-A phases. The calculations in this paper also show for the first time that intrachain constraints on the tail-chain flexibility are by far the major factor responsible for odd-even effects in these liquid-crystal systems.

It is shown that the joint distribution of energy eigenvalues for systems with a varying degree of nonintegrability which has been obtained dynamically by T. Yukawa [Phys. Rev. Lett. 54, 1883 (1985)] can also be deduced by putting his equations of motion in the form of stochastic differential equations. We obtain an interpolation formula for the nearest-neighbor-spacing distribution as a smooth one-parameter family of density functions $P_{\lambda }$(S), 0≤λ<∞. This distribution retains a nonanalytic nature near λ→0; when λ=0 it agrees with the Poissonian distribution but whenever λ≠0 it is proportional to S for small S, as predicted by M. Robnik [J. Phys. A 20, L495 (1987)]. A considerable improvement on the agreement between the energy-level histogram in a real system (hydrogen in a magnetic field) and theoretical formulas which have been studied by Wintgen and Friedrich [Phys. Rev. A 35, 1464 (1987)] is demonstrated.

Light scattering in the bend, splay, and bend-twist modes has been studied in two binary liquid-crystal systems exhibiting the nematic–smectic-A–smectic-C multicritical phase diagram. The results on the behavior of the longitudinal correlation length above the nematic–smectic-A transition are found to be in agreement with x-ray scattering data. The tilt susceptibility above the smectic-A–smectic-C transition is consistent with a simple mean-field description. The pretransitional behavior of the bend and splay elastic constants above the nematic–smectic-C transition in a pure material is found to follow the predictions of the Chen-Lubensky model. Near the multicritical point, the temperature dependence of the light-scattering intensity exhibits an anomalous minimum above the nematic–smectic-C transition in the bend and bend-twist modes but not in the splay mode. This behavior is not described by any existing model.

We perform the linear-stability analysis in the boundary-layer model of dendritic solidification using both the direct matrix-diagonalization method and the differential-equation integration method. The fastest needle-crystal steady state is found to be linearly stable for nonzero anisotropy; successively slower needle crystals have successively more unstable modes. This result agrees with those previously obtained on other models.

The effects of noise reduction and anisotropy enhancement on the Eden and screened-growth models have been explored using computer simulations. In the case of the Eden model a transition from an almost circular asymptotic shape to a diamondlike shape is observed with increasing noise reduction or anisotropy enhancement. This shape transition is continuous and the asymptotic cluster shape seems to be reached at quite small cluster sizes. For the screened-growth model a similar transition from a more or less circular shape to a crosslike shape in the cluster envelope is observed. The simulations are consistent with but do not provide strong support for the idea that noise reduction accelerates the approach to asymptotic behavior without compromising the asymptotic geometric scaling properties of the clusters. As was found earlier for the diffusion-limited aggregation model, noise reduction causes the effective exponents describing the cluster geometry to oscillate with increasing cluster mass and this makes an unambiguous interpretation of the results impossible without a great deal of additional work. This oscillatory behavior is a consequence of the formation of an almost discrete hierarchy of sidebranches with large amounts of noise reduction.

A class of exact master equations descriptive of a Markovian process is obtained, starting from the Perron-Frobenius equation for a chaotic dynamical system. The conditions that must be satisfied by the initial probability density for the validity of the master equation are derived. The approach employs projection operator techniques and provides one with a dynamical prescription for carrying out coarse-graining in a systematic manner.

We present a computationally efficient scheme of modeling the phase-ordering dynamics of thermodynamically unstable phases. The scheme utilizes space-time discrete dynamical systems, viz., cell dynamical systems (CDS). Our proposal is tantamount to proposing new Ansa$iuml—tze for the kinetic-level description of the dynamics. Our present exposition consists of two parts: part I (this paper) deals mainly with methodology and part II [S. Puri and Y. Oono, Phys. Rev. A (to be published)] gives detailed demonstrations. In this paper we provide a detailed exposition of model construction, structural stability of constructed models (i.e., insensitivity to details), stability of the scheme, etc. We also consider the relationship between the CDS modeling and the conventional description in terms of partial differential equations. This leads to a new discretization scheme for semilinear parabolic equations and suggests the necessity of a branch of applied mathematics which could be called ‘‘qualitative numerical analysis.’’

The transitions between the 2S and 2P states of the muonic helium atom (αμ or ${}_{}{}^3{}_{}{}^{}\mathit{\mu }$) in collisions with hydrogen atoms are examined using the rectilinear-trajectory coupled-equation method. The 2S level shift is included semiempirically and the target atom is approximated using the frozen ground-state charge density. Partial decoupling of the coupled equations is achieved, and some simple approximations for the 2S-to-2P transition probability are derived and compared with the exact solution of the coupled equations. The cross section obtained is fit by ${\sigma }_{2S\mathrm{-}2P}$≃5.5×${10}^{\mathrm{-}3}$/ $v^{1.8}$$a_0^2$ for v≳1, with the velocity v in atomic units, and muon-catalyzed fusion parameters are reevaluated using this expression.

Using molecular-dynamics (MD) data on a binary-alloy model, we have computed the self (incoherent) -part of the density autocorrelation functions of both species in the supercooled liquid and near the glass transition, over an extensive range of wave numbers. Standard theoretical models of liquid-state theory fail to reproduce the data, while the Chudley-Elliott jump diffusion model yields reasonable results in the glass range. With a suitable scaling of the time axis, the data for different temperatures can be brought onto a single master curve, which is well fitted by a Kohlrausch (‘‘stretched-exponential’’) function with a wave-number-dependent exponent.

The properties of the power spectra of a broad class of smooth d-dimensional (d≥2) dynamical systems exhibiting a reverse sequence of pairwise mergings of chaotic bands is discussed. In particular, the width of the power pairwise spectral feature associated with the pairwise merging of chaotic bands scales as ‖p-$p_m$${\Vert }^{\gamma }$, where p is a system parameter, the mth merging occurs at p=$p_m$, and γ is the critical exponent. Numerical experiments are performed which verify theoretical predictions for γ. For example, near the accumulation of period doublings the result for the critical exponent reduces to γ${=(1/2+2\mathrm{}}^{\mathrm{-}m}$η/ln(1/J), where η is a universal number, J is the Jacobian determinant, and $2^m$ is the number of bands (before merging). (In the trivial case of smooth one-dimensional maps J→0 and the exponent is (1/2.) .AE

For a classical monatomic fluid, gas and liquid phases are differentiated on the basis of their statistical-mechanical entropy expressions. In the customary picture of a gas, each particle is first allowed to move independently over the entire volume; interactions are then introduced, and the configurational entropy becomes 1 minus the interaction corrections. In a liquid, the configurational entropy is assumed to be dominated by correlations, and is expressed as $S^{\mathrm{}\left(2\right)\mathrm{}}$ plus higher-order correlation corrections, where the two-particle term $S^{\mathrm{}\left(2\right)\mathrm{}}$ is of order -2. For liquid argon at 85 K on the saturation curve, $S^{\mathrm{}\left(2\right)\mathrm{}}$ is evaluated from the measured radial distribution function, and the entropy is accurately given by liquid statistical theory, with the higher-order correlation corrections being approximately 13% of the magnitude of $S^{\mathrm{}\left(2\right)\mathrm{}}$. A qualitative gas-liquid phase diagram is drawn for a classical monatomic fluid.

We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems in three dimensions with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential. A new algorithm is discussed which incorporates ideas from control theory and constrained nonequilibrium molecular dynamics. Equilibrium and nonequilibrium steady states are examined. The latter are generated by the application of an external field ${\mathrm{F}}_e$ through which equal numbers of particles are accelerated in opposite directions, and by thermostatting the system using Nosé-Hoover or Gauss mechanics. In equilibrium ${(\mathrm{F}}_e$=0) the Lyapunov spectra are symmetrical and may be understood in terms of a simple Debye model for vibrational modes in solids. For nonequilibrium steady states ${(\mathrm{F}}_e$≠0) the Lyapunov spectra are not symmetrical and indicate a collapse of the phase-space density onto an attracting fractal subspace with an associated loss in dimensionality proportional to the square of the applied field. Because of this attractor’s vanishing volume in phase space and the instability of the corresponding repellor it is not possible to observe trajectories violating the second law of thermodynamics in spite of the time-reversal invariance of the equations of motion. Thus Nosé-Hoover mechanics, of which Gauss’s isokinetic mechanics is a special case, resolves the reversibility paradox first stated by J. Loschmidt [Sitzungsber. kais. Akad. Wiss. Wien 2. Abt. 73, 128 (1876)] for nonequilibrium steady-state systems.

Correlation energy differences have been evaluated using an empirical polarization potential for the two-, three-, and four-electron atom series. In contrast to local-density approximations the Z dependency of the correlation energies has been well reproduced. Moreover, the quantitative agreement with experimental values is surprisingly good. Deviations are smaller than 0.01 hartree.

A constrained least-squares fit procedure wherein the integral F[ρ(r)-${\rho }_0$${\mathrm{}\left(\mathrm{r}\right)\mathrm{}]}^2$ dr is minimized [${\rho }_0$(r) is the reference near–Hartree-Fock (NHF) electron density distribution and ρ(r) is the refined one obtained from a single Slater determinant] has been developed. The constraints applied are the exact theoretical 〈$p^2$〉 and 〈$r^2$〉 expectation values. These expectation values are expected to tailor the electron density around the nuclear and tail regions, respectively. The procedure has been applied to lithium and beryllium atoms as test cases. Nearly all the 〈$r^n$〉 and 〈$p^n$〉 (for n=-2, -1, 1, 3, 4, and 5) expectation values have been improved with use of this procedure. The sacrifice in the electronic energy, in comparison to the corresponding NHF one, is about 0.02% in both cases.

A new method is described for solving the time-independent one-dimensional finite-range-potential scattering problem. The method utilizes positivity properties of the scattering wave function, implemented through the classical theory of moments. It is shown to provide exact narrow bounds to the scattering amplitudes. The method is particularly suited to treat the strong-interaction regimes, where perturbative approximations fail.

A very sensitive test of the electron-atom ionization threshold law is suggested: for spin-aligned heavy negative ions it consists of measuring the polarization asymmetry $A_{\mathrm{PA}}$ coming from double detachment by left- versus right-circularly polarized light. The respective yields are worked out for the ${\mathrm{Te}}^{\mathrm{-}}$[(5p${)}^5$ ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{3\mathrm{r}/2}$] ion. The Coulomb-dipole theory predicts $A_{\mathrm{PA}}$ to be the ratio of two oscillating functions in sharp contrast to any power law (specifically the Wannier) for which the ratio is expected to be a smooth function of energy.

Electron-stripping (or ionization) cross sections of charged ions by energetic proton impact are calculated. We use the binary-encounter-approximation model to describe collisions between an incident proton and electrons in a target ion and the Roothaan-Hartree-Fock method to obtain the electron-velocity distributions of the target ion. Results calculated for Ne ions are discussed and compared with experiments and other calculations in some cases.

We present a discussion which should be helpful in clarifying some key steps in the formulation of a multichannel extension of Schwinger’s variational principle for electron-molecule collisions [K. Takatsuka and V. McKoy, Phys. Rev. A 30, 1734 (1984)].

The photoionization cross section for ${\mathrm{Al}}^{+}$ has been calculated within the framework of the relativistic random-phase approximation including all 1s, 2s, 2p, and 3s channels. Except for the first few eV above threshold, it is found that the simple Hartree-Slater central-field result agrees with this cross section of 10% or better, indicating that multichannel effects drop off rapidly with stage of ionization.

Recent experiment on ${\mathrm{K}}^{+}$ is compared with measurements on neutral K to test the suggestion that inner-shell photoionization cross sections are largely unaffected by the removal of outer-shell electrons.