We describe the structure factors of a fairly general family of self-similar deterministic arrays (fractals). These are constructed recursively by an inflation method which imitates real physical processes where large clusters are grown from smaller entities. An algebraic approach is used to describe the behavior of the structure factors in the limit of arbitrarily large objects. These structure factors exhibit several scaling properties which reflect the self-similarity of the direct space arrays.

We investigate two dynamical systems in which nonlinearity and disorder are in competition. In both systems, the disorder is introduced by randomly modulating the square frequency. From computer simulation, it becomes evident that the introduction of a nonlinearity drastically modifies the divergent behavior in time of the random harmonic oscillator and gives rise to a new confining structure in the phase space. Near certain key energies, the evolution can be approximated by deterministic equations of motion in which the randomness appears in terms of average values.

Time-domain spectroscopy is used to study the complex permittivity of two types of liquid crystals: heptylcyanobiphenyl (7CB) and octylcyanobiphenyl (8CB). The variation of the dominant relaxation frequency values of both 7CB and 8CB as a function of temperature are very similar, showing no influence of the smectic structure in 8CB. The experimental relaxation measurements in the nematic phase agree qualitatively with the existing theory.

A direct derivation for the first-passage time statistics of processes driven by white Poisson noise is given. This derivation illustrates the difficulty of boundary conditions for a Markovian process driven by a non-Gaussian white noise. In order to gain information on the first-passage time distribution of non-Markovian processes, we carry out numerical simulations for a free process driven by colored Poisson noise with rectangular and exponential pulses. We study the case of stationary and nonstationary noise. Gaussian limits, given by a non-Markovian noise and the Ornstein-Uhlenbeck noise, are also analyzed.

To study the stationary probability distribution of master equations in the large-system-size limit, we introduce a kind of master Hamiltonian which is similar to the Fokker-Planck Hamiltonian suggested by Graham et al. in the study of Fokker-Planck equations in the weak-noise limit. The general procedure to associate the potential of the master equation with a certain separatrix of the master Hamiltonian system is described. With some solvable models we show how the smoothness of the potential of the master equation is related to the integrability of the corresponding master Hamiltonian system.

Formal solution of the diffusion equation (∂/∂t)V=(∂/∂x)[D(x)(∂/∂x)V] is obtained as a functional of D(x). The solution is expressed by means of certain operators, which are somewhat similar to the time-ordering operator of quantum mechanics. This formalism provides us with a good perspective of the problem and serves as a basis for approximation methods. In particular, the equation is exactly solvable when D(x) is a discrete (stepwise) function. Coarse graining of the diffusion coefficient is also discussed. It is shown that the reciprocal of the coarse-grained diffusion coefficient is equal to the arithmetical average of 1/D(x). Finally, a stochastic equation obtained by adding a decay term and noise terms to the diffusion equation is considered. Spatial correlation function is calculated exactly for the case where D(x) is a step function.

The scaling structure of the growth probability distribution in the surface layer of the generalized diffusion-limited aggregation model (η model) is derived by making use of the real-space renormalization-group method. A conductance of the surface layer is defined and renormalized as the growth-bond conductance. The renormalization-group transformation equation is derived for the growth-bond conductance. The equation has a nontrivial solution which is a stable fixed point. The growth probability assigned to each growth bond is represented by a random multiplicative process of the cell’s growth probabilities evaluated at the fixed point. A hierarchy of generalized dimensions D(q) is calculated and the α-f spectrum is found for diffusion-limited aggregation. The dependence of the α-f spectra is found on the parameter η describing the different dielectric breakdown models.

A cubic map lattice, which consists of an array of cubic discrete-time maps coupled through nearest-neighbor diffusion interactions, is studied in a parameter region where the isolated maps possess bistable steady states. Within this parameter region, the spatio-temporal evolution of this deterministic dynamical system exhibits the phenomena of phase separation and nucleation. When the two coexisting states have the same stability, the domain growth is like that for continuous systems with a nonconserved order parameter, and follows Allen-Cahn scaling on long distance and time scales. When the states have different stabilities, growth typically occurs by a nucleation process. In addition to exploring these phenomena, which have their analogs in continuous systems, features of the inhomogeneous states peculiar to the discrete model are also investigated.

We study experimentally the patterns behind a moving cholesteric–smectic-A interface. There are three interface types, each of which corresponds to one of the three basic topologically possible configurations.

The saddle-point variational (SPV) method is applied to the n=2 excited states of the Dirac equation for a Coulomb potential. The variational energies agree well with the exact energy for quite large coupling strength (λ=0.4 and 0.8). Fine-structure differences are also well reproduced. An extended SPV method gives better results than explicit orthogonalization of the variational states.

It is shown that if in the projection-operator method an expansion in powers of the interaction Hamiltonian is in view, the usage of the interaction picture is more advantageous than the Schrödinger picture. This is also demonstrated in an example.

Scully and Lamb’s quantum theory of the laser is extended to deal with nonresonant operation of a single-mode laser in a homogeneously broadened medium consisting of three-level atoms. In the case of weak-field intensity, the master equation and operation characters such as the threshold condition, the average photon number, and the half-width of the photon distribution are obtained and discussed.

Dynamical implications of the polaron theory of electrons solvated in molten salts [G. Malescio and M. Parrinello, Phys. Rev. A 35, 897 (1987)] are discussed as they pertain to the electronic absorption spectra. Theoretical results are compared with numerical data and experimental evidence.

From an analysis of cluster growth on the Cayley tree exact results are obtained for the average number of particles on different levels and the mean level number at a general branching ratio. The gyration radius is derived with the recurrence-relation method. The case of weakly biased diffusion is discussed under approximations.

We point out the existence of self-similarity as a function of dimension for the system of Schrödinger equations or of the matrix elements of the tight-binding Hamiltonian in site representation on a discrete hypercubic lattice of finite size. We show how this property of self-similarity can generate an efficient algorithm for the explicit calculation of eigenvalues and densities of states in any dimension.

We study the critical behavior of a Frenkel-Kontorova model, extended to include a second harmonic of the external potential. For certain parameter values of the model the transition from the locked to the sliding incommensurate phase shows a scaling anomaly for the correlation length, with the critical exponent varying with the potential. We conjecture that this behavior indicates a crossover to a new universality class.

By using the Born approximation, calculations are made for photon angular distributions of the radiative electron capture to the $L$ shell and for linear polarization of photons induced by the radiative electron capture to the $K$ shell for relativistic incident velocities. The angular distributions have minimum value in the forward directions and maximum values in the backward directions relative to 90°. This behavior becomes significant with an increase of velocity. The polarization correlation of the emitted photon represents the crossover feature in the case of relativistically high energy impacts.

We calculate the counting (multiplicity) distribution $P_N=〈N\left|\rho \right|N〉$ where $\rho$ is (i) the density matrix corresponding to a (pure) squeezed state and (ii) the density matrix of a superposition of squeezed states with thermal light. In the first case, we find that $P_N$ is an oscillating function of $N$. In the second case, this behavior, characteristic for squeezed states, disappears even for small amounts of noise.

Experimental study in gaseous helium at low temperature (4 K) of thermal convection up to a Rayleigh number $R={10}^{11}$. Three regimes are observed, a chaotic state up to $R=2.5\mathrm{}\times {10}^5$, a soft-turbulence state up to $R=4\mathrm{}\times {10}^7$, and then a hard-turbulence state. We associate those three regimes to the boundary-layer formation and dynamics.

In our model, a voltage is applied across an insulating lipid membrane embedded in a conducting aqueous medium and containing a pore. It is shown to produce along the pore boundary negative contributions to both the edge energy and the lateral tension of the membrane. One effect promotes opening and the other closure of the pore. Surprisingly, we find that voltage may inhibit pore opening and stabilize pores of a certain size.