In this paper we discuss the effects of fluctuations on the phase diagram of an isotropic Ising model with ferromagnetic first-neighbor and antiferromagnetic second- and third-neighbor interactions. We find that, in certain cases, there are substantial deviations from mean-field behavior. Thus the phase transitions between the one-dimensional modulated phases and paramagnetic phase are found to undergo fluctuation-induced first-order phase transitions. We also discuss that point of the phase diagram which, within mean-field theory, was predicted to be an isotropic Lifshitz point. It is argued that the existence of such a point is unlikely, and some alternative scenarios are proposed.

In earlier papers [McKane, Luckock, and Bray, Phys. Rev. A 41, 644 (1990); Bray, McKane, and Newman, ibid. 41, 657 (1990)] the path-integral approach was used to describe the behavior of a particle coupled to weak, exponentially correlated noise and moving in a one-dimensional potential. The method of steepest descents was used to calculate the leading-order exponential contributions to various quantities of physical interest. This analysis is developed further here. By accounting for small fluctuations about the paths of steepest descent, we determine the prefactors that multiply the dominant exponential contributions to these quantities. In particular, we calculate the escape rate for a particle over a potential barrier to second order in the noise correlation time (which is assumed to be short).

Deformation effects of droplet fluctuations on the dynamics in an Ising ferromagnetic state in two dimensions are studied in the case of an order-parameter-nonconserving system by investigating an equation of motion for a domain boundary. Analytic and numerical studies show that the deformation effects on the survival probability of the droplets simply result in changing the time scale or renormalizing a kinetic coefficient.

A binary mixture subject to a large temperature gradient is studied. Solutions of the evolution equations for the time-dependent fluctuation correlations are obtained using mode-coupling theory subject to the assumption that the mixture is not close to a critical point. From these results predictions for the light-scattering spectrum at the Rayleigh and Brillouin lines are made. The temperature gradient is shown to cause a large enhancement of the scattering amplitude at the Rayleigh line, whereas the effect at the Brillouin line is found to be small.

We study the magnitude of the energy fluctuations of a long polymer chain with excluded-volume interactions placed in a media with a weak random potential. It is shown that there always exists a critical length ${\mathit{N}}^{\mathrm{*}}$ of a polymer segment or a critical length scale ${\mathit{L}}^{\mathrm{*}}$ at which the fluctuations of the interaction energy between the polymer segment and the medium are of the order of kT. At larger scales the problem reduces to that of the minimum energy path in a strong random potential. Using an analogy with the directed polymer problem, we suggest that the size of a polymer coil is of order ${\mathit{L}}^{\mathrm{*}}$(N/${\mathit{N}}^{\mathrm{*}}$${)}^{2/3}$ in three dimensions. In the two-dimensional case, the size of the polymer coil is controlled by excluded-volume effects: R∼${\mathit{N}}^{3/4}$. The magnitudes of the energy barriers between the different, neighboring configurations are (N/${\mathit{N}}^{\mathrm{*}}$${)}^{1/3}$kT and (N/${\mathit{N}}^{\mathrm{*}}$${)}^{3/8}$kT for the cases d=3 and 2, respectively. Those energy barriers cause the trapping of a long polymer chain. The coil-to-globe transition in disordered media is also discussed.

The quasistationary distribution and the transition probabilities for systems driven by weak noise are analyzed to logarithmic accuracy. The problem is reduced to the boundary-value problem for the set of ordinary differential equations. The form of the equations is determined by the shape of the power spectrum of noise. They are solved in the limiting cases of small and large times characterizing noise correlations, and also for a high-frequency narrow-band noise. The dependences of the logarithm of the distribution on the parameters and the coordinate of the system are quite specific in the latter case. The logarithms of the transition probability and of the quasistationary distribution in the point separating the ranges of attraction to different stable states differ from each other.

The time-dependent solution of the periodically forced Fokker-Planck equation is determined, following an iteration procedure. The mixing property of the system is verified. The enhancements of the autocorrelation and the signal-to-noise ratio are distinguished and localized, in different noise strength regions. The conditions for the validity of the adiabatic approximation and the way to systematically improve the adiabatic result are clarified.

We use generalized-moment inequalities for obtaining constraints on the charge-correlation function (CCF) for two types of systems in fully symmetric states: parallel rigid wires and charged particles confined to a plane. The former case resembles closely the three-dimensional one: the second (spatial) moment of the CCF is bounded by the Stillinger-Lovett (classical) value and its quantum generalization. The quantum value is attained only for the one-component plasma. In the latter case the relationship between the CCF and the dielectric function tends to the classical one in the long-wavelength limit. Assuming inverse-power decay of the CCF at infinity [D(r)∼${\mathit{r}}^{\mathrm{-}\mathrm{\alpha }}$] leads to an alternative: either α>3 and external charges are not screened, or α=3 and there is (at least partial) screening. A more general statement, valid also for a damped oscillatory behavior of D(r) near infinity, is ${\lim }_{\mathit{r}\to \mathrm{\infty }}$r where ε is the long-wavelength limit of the static dielectric function.

A detailed study of the statistical properties of first-passage times in a one-dimensional random environment is presented. It is found that the spectrum of relaxation times of the master equation describing the hopping dynamics in such a system possesses a gap, which separates the leading relaxation time from the next one by an amount that is exponentially large in the square root of the size of the system. The leading relaxation time for a given (typical) system equals τ, the mean first-passage time for that realization. One consequence of the existence of this gap is that for times that are larger than a fraction of the mean first-passage time τ, the probability of the first-passage time to have a value t is approximated, to a high degree of accuracy, by (1/τ)exp(-t/τ). The appearance of the above-mentioned gap is, surprisingly, related to the existence of a dominant single effective trap (or ‘‘potential well’’), the next leading trap being of no importance for large systems. A detailed study of the properties of the mean first-passage time and its fluctuations among the members of the ensemble of realizations is presented (the probability distribution of τ is shown to possess anomalous long-time behavior). The realization-averaged mean first-passage time is shown to be dominated by rare realizations and is contrasted with typical values of that quantity. The mean first-passage-time distribution is shown to be related to a new characterization of random walks that we coin the extent. The latter quantity is investigated in detail.

An extended mean-field approximation for statistical operators is derived within the context of information theory. The ensuing scheme is applied to (many) boson mean-field approaches within a completely general statistical framework. Numerical examples are given within the context of the finite-temperature anharmonic oscillator problem. Fluctuations are also discussed.

By means of the kneading theory, the Lorenz-like map is studied based on admissibility conditions. A criterion for the existence of chaos is derived. The Farey transformation and a subzone of the nonchaotic region are discussed.

We have investigated with molecular dynamics the transport properties and velocity autocorrelation functions (VAF) for liqud ${\mathrm{SiO}}_2$ up to pressures of 20 GPa and at temperatures of 4000 K using 252–1371 particles. A systematic study of the sensitivity of the self-diffusion coefficient to the number of particles in the system showed that around 800 are required for convergence. At low pressures the VAF reveals that motions of silicon and oxygen atoms are decoupled over the 50–75-fs range, across the second negative peak in the VAF of oxygen. As the pressure increases, the decoupling is shifted into the 25–50-fs range, over the first positive peak in the VAF of oxygen. In the spectral domain, the major feature on increasing pressure is the enhanced contribution of frequencies in the 800–1000-${\mathrm{cm}}^{\mathrm{-}1}$ region. This is consistent with high-pressure infrared spectroscopic measurements on silica glass. Analysis of the distribution of coordination numbers revealed that both the fivefold- and sixfold-coordinated silicon become prevalent between 7 and 20 GPa. This accords well with recent findings from NMR studies of silicate glasses. We propose that the importance of fivefold-coordinated states in determining the self-diffusion coefficient needs to be reassessed.

Recently defined distinct diffusion coefficients and distinct time-correlation functions (TCF’s) related to them have been calculated by molecular dynamics in the ordinary NVEp ensemble, where N is the number of particles, V the total volume of the system, E the total energy, and p the total momentum. Lennard-Jones mixtures of simple structure were considered. For the ideal system, the theoretical predictions for the distinct TCF’s were found to be in complete agreement with the molecular-dynamics results. For nonideal systems, the distinct functions are very suitable for indicating phase separation or the inverse phenomena, particularly when the corresponding self-TCF is added. The results show furthermore that the proposed separation of the mass current in self- and distinct terms is completely equivalent to that given in earlier work using one-particle and collective diffusive processes.

The supercritical Hopf bifurcation to oblique traveling rolls in nematic liquid crystals leads to a complex spatiotemporal state. We present an experimental study of the bifurcation scenario in which a secondary bifurcation to a coherent steady state occurs at higher values of the control parameter. Temporal resonant modulation of the driving force can stabilize the four amplitude modes appearing after the first bifurcation, and their superposition can lead to a standing bimodal pattern.

Confined states in phase dynamics–the coexistence of states with different wavelengths–are further studied. In particular, a stable and accurate numerical code is developed to study the dynamic equation of motion for two sets of boundary conditions–a pinned phase and an unpinned phase. With a pinned phase, stable confined states are found to exist for a wide range of parameter values. With an unpinned phase, the localized states are at best neutrally stable. Also, analytic solutions are found for the stationary confined states.

The reflectance spectra from chiral smectic-C (Sm-${\mathit{C}}^{\mathrm{*}}$) liquid-crystalline structure distorted by a magnetic field were calculated and analyzed near their Lifshitz point where coexistence of the three phases Sm-A–Sm-C–Sm-${\mathit{C}}^{\mathrm{*}}$ occurs. Using the 4×4 matrix formulation of Maxwell’s equations we found that an infinite series of reflection peaks that correspond to even harmonics of the helicoidal pitch appear in the case of normal incidence due to the modulation of the tilt angle. Additional infinite odd sequences of peaks appear only at oblique incidence. The peaks’ heights decay strongly with the order and the harmonic. For those that exist in the undistorted structure, they decay also with the perturbation exhibiting critical behavior. For those that appear due to the perturbation, they exhibit a maximum at a universal value of the field at H/${\mathit{H}}_{\mathit{L}}$=1/(2${)}^{1/4}$. We give full explanations to the polarization characteristics by calculating the form factor for scattering that provides selection rules for the scattering events. Analytic expressions were obtained within the framework of the kinematical theory of scattering. It is shown that the theory is valid in the range very close to the transition, suggesting a way of investigating this type of phase transition by reflection spectra techniques.

We report constant-pressure Monte Carlo simulations of a system of hard ellipses with aspect ratios k=2, 4, and 6. The results show three different phases: isotropic, nematic, and solid, except for k=2, where no nematic phase is formed. In all cases the fluid-solid transition appears to be first order. However, the isotropic-nematic transition looks first order for k=4 and continuous, via disclination unbinding, for k=6. Thus a tricritical point for some k between these two values is predicted. The simulation results are also compared with a previous theoretical calculation. Satisfactory agreement is found for the equation of state, but the theoretical prediction for the location of the isotropic-nematic transition disagrees with the simulation results. Finally, we observed that the solid phase near melting exhibits anomalously large fluctuations.

A simple model of oil, water, and amphiphile, in which the third is favored energetically to sit between the other two, is studied in one and two dimensions by transfer-matrix methods, Monte Carlo simulations, and the Müller-Hartman-Zittartz approximation. Phase diagrams, correlation functions, scattering intensities, and the surface tension of the oil-water interface have been calculated. The results obtained here confirm many of the conclusions drawn from the analysis of the model in three dimensions via mean-field theory. In particular, we find that in the microemulsion, the water-water correlation function shows damped oscillatory behavior at large distances, whereas the amphiphile-amphiphile correlation function decays exponentially. With increasing amphiphile concentration, the phase diagram of the two-dimensional system, like that of the three-dimensional one, shows the commonly observed progression from three-phase coexistence through a microemulsion to two-phase coexistence with a lamellar phase. In contrast to the behavior in three dimensions, the microemulsion in two dimensions is stabilized even in the limit of very low temperatures. The comparison of results obtained here with those of mean-field theory elucidates the role that fluctuations play in this system.

A classical model consisting of two chains coupled in a ladder configuration with interactions between all adjacent atoms is considered. It is shown that the ground states can be obtained from a model with a single chain of pseudoparticles in an external substrate potential and with new effective interactions between adjacent pseudoparticles. When the original interactions are either everywhere convex or Lennard-Jones like, the ground-state behavior can be understood by considering a simpler analytic model with Lennard-Jones interactions between adjacent atoms. In the ground-state phase diagram, not only are the expected pinched phases found but dimerized phases and an infinite series of superdegenerate points accumulating at a multiphase point, the end point of an accumulation line, are seen. The variation of the linear density is discussed in terms of measurable quantities. Finally, the effects of other forms of the interparticle potentials are considered and the relevance to one-dimensional liquid crystals and other physical systems is discussed.

Dynamic light scattering (DLS) is used to study the diffusional behavior of polystyrene latex spheres incorporated in polyacrylamide gels. It is shown that the gel systems exhibit nonergodic features, implying that the time-averaged intensity correlation function (ICF), the quantity obtained from a single DLS experiment, is not equal to the ensemble-averaged ICF. It is demonstrated that the theory of Pusey and van Megen [P. N. Pusey and W. van Megen, Physica A 157, 705 (1989)] on DLS by nonergodic media can be used to extract dynamic structure factors from single DLS experiments. It appears from this analysis that large values of the diffusion coefficient of the tracer particles, which would have been obtained in a classical analysis, do not imply rapid particle motions but result simply from an incorrect analysis of the data. The initial decay of the dynamic structure factor gives a short-time diffusion coefficient of the particles that hardly depends on the degree of cross-linking. We also show that a proper analysis of the data results in a dynamic structure factor that develops a nondecaying component. This component is a measure for the fraction of frozen-in fluctuations resulting from constrained diffusion of the tracer particles. From the nondecaying component in the dynamic structure factor, a distribution of root-mean-square displacements of the particles is calculated that appears to be an exponential distribution of which the decay length decreases with increasing degree of cross-linking. Although the scattering from the systems with probes is much higher than that of the matrix, we cannot neglect the gel scattering completely. We present theoretical expressions allowing corrections for gel scattering.