We consider Lévy flights characterized by the step index $f$ in a quenched isotropic short-range random force field to one loop order. By means of a dynamic renormalization group analysis, we find that the dynamic exponent $z$ for $f<\mathrm{}2\mathrm{}$ locks onto $f$, independent of dimension and independent of the presence of weak quenched disorder. The critical dimension for $f<\mathrm{}2\mathrm{}$ is given by $d_c=2f-2\mathrm{}$. For $d<\mathrm{}{d}_c$ the disorder is relevant, corresponding to a nontrivial fixed point for the force correlation function. We also discuss the behavior of the subleading diffusive term.

We present a general nonperturbative method for constructing an integrable Hamiltonian that is close to a given nonintegrable one. The existing invariant tori of the original Hamiltonian are accurately approximated by the constructed one. We illustrate the method by applying it to apparently near-integrable gravitational potentials for which no underlying integrable Hamiltonians are known.

A general framework is presented for the discussion of Brownian motion in crystals with randomly distributed topological defects. In a two-dimensional lattice with disclinations one finds a nonuniversal subdiffusional behavior if screening in the disclination ensemble is taken into account. Without screening, a Sinai-type diffusion is expected. In a three-dimensional random array of parallel screw dislocations, a Brownian particle shows anisotropic normal diffusion. However, the process no longer is Gaussian and displays long-time tails in the fourth-order cumulants.

We study a stochastic lattice gas of particles undergoing asymmetric diffusion in two dimensions. Transitions between a low-density uniform phase and high-density nonuniform phases characterized by localized or extended structure are found. We develop a mean-field theory which relates coarse-grained parameters to microscopic ones. Detailed predictions for finite-size ($L$) scaling and density profiles agree excellently with simulations. Unusual large-$L$ behavior of the transition point parallel to that of self-organized sandpile models is found.

A proposal to demonstrate the nonlocality of a single photon is described. This is accomplished without using inequalities, and in a way that brings out a very curious feature of quantum mechanics. Unlike the first proposal to investigate the nonlocality of a single photon due to Tan, Walls, and Collett, this proposal does not require supplementary assumptions, and therefore rules out the whole class of local hidden variable interpretations.

Late-stage coarsening in a 2D binary mixture has been investigated via direct imaging of "bubble" domains in an amphiphilic monolayer at an air-water interface. In a regime with growth exponent $n\simeq 0.28$ for the mean domain radius, the radius distribution exhibits dynamic scaling, closely approaching a Gaussian shape. The local structure of disordered bubble patterns ensures short-range screening of topological charge. Pattern statistics correspond to equilibrated random Voronoi lattices and indicate a prominent role of entropy maximization. Pattern stabilization against bubble coalescence is attributed to electrostatic interactions.

The equation of state of the hydrogen plasma is calculated by the restricted path integral Monte Carlo method. We have investigated the plasma from the classical weak coupling regime to the quantum strongly coupled regime. Good agreement is found with the existing theories for low electronic degeneracy. Inception of molecular formation is observed at low densities and temperatures.

Generalized statistical distributions for identical particles are introduced for the case where filling a single-particle quantum state by particles depends on filling states of different momenta. The system of one-dimensional bosons with a two-body potential that can be solved by means of the thermodynamic Bethe ansatz is shown to be equivalent thermodynamically to a system of free particles obeying statistical distributions of the above class. The quantum statistics arising in this way are completely determined by the two-particle scattering phases of the corresponding interacting systems. An equation determining the statistical distributions for these statistics is derived.

We present the exact solution to a one-dimensional multicomponent quantum lattice model interacting by an exchange operator which falls off as the inverse sinh square of the distance. This interaction contains a variable range as a parameter and can thus interpolate between the known solutions for the nearest-neighbor chain and the inverse-square chain. The energy, susceptibility, charge stiffness, and the dispersion relations for low-lying excitations are explicitly calculated for the absolute ground state, as a function of both the range of the interaction and the number of species of fermions.

The conjecture is verified that the optimum, energy minimizing, magnetic flux for a half-filled band of electrons hopping on a planar, bipartite graph is $\pi$ per square plaquette. We require only that the graph has periodicity in one direction and the result includes the hexagonal lattice (with flux 0 per hexagon) as a special case. The theorem goes beyond previous conjectures in several ways: (1) It does not assume, a priori, that all plaquettes have the same flux (as in Hofstadter's model). (2) A Hubbard-type on-site interaction of any sign, as well as certain longer range interactions, can be included. (3) The conclusion holds for positive temperature as well as the ground state. (4) The results hold in $D\ge 2$ dimensions if there is periodicity in $D-1\mathrm{}$ directions (e.g., the cubic lattice has the lowest energy if there is flux $\pi$ in each square face).

We formally establish the relationship between spatial fractal behavior and long-range temporal correlations for a broad range of self-organized (and not self-organized) critical phenomena including directed percolation, interface depinning, and a simple evolution model. The recurrent activity at any particular site forms a fractal in time, with a power spectrum $S\left(f\right)\mathrm{}\sim \frac{1}{f^{\tilde{d}}}$. The exponent $\tilde{d}=\frac{\mathrm{}(D-d)\mathrm{}}{z}$, where $d$ is the spatial dimension, $D$ is the avalanche dimension, and $z$ is the usual dynamical exponent. Theoretical results agree with numerical simulations.

It is shown that the corrected form of physical supplementary conditions in singular Lagrangian systems is that physical observables have weakly vanishing Poisson brackets with the elements of the minimum-evolution closed Poisson-bracket subalgebra of the first-class constraints, as well as with all the second-class constraints. A simple Cawley's example with the above features is studied. The origin of the gauge conditions is discussed and the corrected form and number of the gauge conditions in some more general singular Lagrangian systems are given. Our result differs from the accustomed conclusion which is that the number of gauge conditions or gauge freedoms is always equal to the number of first-class constraints. These results can provide a method for the quantization of some singular systems.

We study the low-energy effective action for the theory of a one-component real scalar field in three Euclidean dimensions (3D), in the symmetric phase, concentrating on its static part—effective potential $V_{\mathrm{eff}}\left(\varphi \right)$. We compute it from the probability distributions of the average magnetization in the 3D Ising model in an external field, obtained by Monte Carlo computation. We find that the ${\varphi }^6$ term in $V_{\mathrm{eff}}$ is important in 3D, and compute the values of the universal four-point and six-point couplings.

The contractor renormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is introduced. The method combines contraction and variational techniques with the real-space renormalization group approach. It applies to lattice systems of infinite extent and is ideal for studying phase structure and critical phenomena. The CORE approximation is systematically improvable and can treat systems with dynamical fermions. The method is tested using the (1+1)-dimensional Ising model.

Relaxing the distinguished ordering underlying the derivation of soliton supporting equations leads to new equations endowed with nonlinear dispersion crucial for the formation and coexistence of compactons, solitons with a compact support, and conventional solitons. Vibrations of the anharmonic mass-spring chain lead to a new Boussinesq equation admitting compactons and compact breathers. The model equation $u_t+{\left[\frac{\delta u+3\mathrm{}\gamma u^2}{2+u^{1-\omega }{\left(u^{\omega }{u}_x\right)}_x}\right]}_x+\nu u_{\mathrm{txx}}=0(\omega ,\nu ,\delta ,\gamma \mathrm{const})$ admits compactons and for $2\omega =\nu \gamma =1\mathrm{}$ has a bi-Hamiltonian structure. The infinite sequence of commuting flows generates an integrable, compacton's supporting variant of the Harry Dym equation.

We have obtained the complete finite-size structure of energy-density correlation functions at all temperatures defined for two-dimensional Ising models on cylindrical lattices, extending the original results of Hecht. This reveals novel structure in terms of winding numbers and confirms predictions from conformal invariance theory to leading order.

The Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a non-Abelian Berry vector potential. We ask what object produces the non-Abelian potential in parameter space, in the same sense that the Abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole (centered at the point where the level becomes degenerate with another) as shown by Berry. We find that it is a meron, an object in four Euclidean dimensions with instanton number ½, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.

The viscous damping of fracton vibrations in 2D percolation clusters is computed. It is found that the anomalous damping is due to the local irregular distribution of the vibration amplitudes. There exists a localization of the damping within the usual localization of fractions. This effect is strongly enhanced by nonlinear dissipation. This is the first evidence of the damping power of fractal structures.

One-dimensional model of nonrelativistic particles with inverse-square interaction potential known as the Calogero-Sutherland model is shown to possess fractional statistics. Using the theory of Jack polynomials the exact dynamical density-density correlation function and the hole propagator part of one-particle Green's function at any rational interaction coupling constant are obtained and used to show clear evidences of the fractional exclusion statistics in the sense of Haldane's "generalized Pauli exclusion principle." This model is also endowed with the corresponding natural exchange statistics.

A new class of random quantum-dynamical systems in continuous space is introduced. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time independent, and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is $\delta$ correlated in time and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both averaged states and observables are obtained. If the deterministic part is that of a particle subject to a constant magnetic field, spatial variance of the averaged state grows diffusively for long times independent of the initial state.