For any Ising model on a planar lattice without bond crossings, we prove that the anisotropic interface tension is given by the lattice Green’s function of a free random-walk problem defined on the dual lattice. This fact, derived via Vdovichenko’s combinatorial method, reveals a general relation between the bulk free energy and the interface tension for two-dimensional Ising models. As an important consequence, we show that the equilibrium crystal shape corresponds to imaginary zeros of the partition function. We also discuss generalizations to non-Ising and non-solvable systems.

A new theorem of inverse formula is introduced for a kind of infinite series. Thus some new results for important inverse problems in physics are presented in this paper. These are the inverse problems for obtaining the phonon density of states, the inverse blackbody radiation problem for remote sensing, and the solution for inverse Ewald summation. Of more importance, it shows the possibility of the application of number theory to physical problems.

It is shown that one can convert a chaotic attractor to any one of a large number of possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. The method utilizes delay coordinate embedding, and so is applicable to experimental situations in which a priori analytical knowledge of the system dynamics is not available. Important issues include the length of the chaotic transient preceding the periodic motion, and the effect of noise. These are illustrated with a numerical example.

Quadratic moments of a particle distribution being transported through a linear Hamiltonian system are considered. A complete set of kinematic invariants made out of these moments are constructed leading to the discovery of new invariants.

We examine the finite-size-scaling amplitudes of the free energy in incommensurate phases on a torus with periodic and twisted boundary conditions. We show that these amplitudes are equivalent to those of the six-vertex model with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while the electric defect lines are generated by competition between the domain-wall separation and the finite system size. We calculate the amplitudes exactly for the free-fermion model and the spin-(1/2 XXZ chain, and conjectire the form of these amplitudes for a more general model. Numerical calculations employing the Bethe ansatz confirm our conjecture.

We consider the effect of disorder on coherent tunneling of electrons through a double-barrier structure. When the disorder potential is treated in leading order we obtain a result with transparent physical consequences. The effect of interface roughness on resonant tunneling is shown to depend qualitatively on the location of the rough interface. Our results may help to explain the often observed asymmetry in the current-voltage characteristics of double-barrier structures.

Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.

I give a second-quantized example of infinite statistics (in which any representation of the symmetric group can occur) for identical particles and discuss the properties of the example.

The second virial coefficient ${\mathrm{B}}_2$ for a system of spin-one-half flux-carrying fermions is computed. It is found that there arise significant differences from known results for the corresponding spinless system. Apart from the existence of cusps and a restricted periodicity of ${\mathrm{B}}_2$ when viewed as a function of the flux parameter α, one obtains here the remarkable result that there exist discontinuities in ${\mathrm{B}}_2$ at all even, nonzero values of α.

A simple derivation of the ideals in the Temperley Lieb algebras is presented together with a convenient basis. Using this basis it is shown how to derive the eigenvalue equations for operators defined on the algebra and a relation to the Bethe-Ansatz equations is noticed. The irreducible content of the spin-(1/2 XXZ model is also discussed.

The scattering of relativistic spin-one-half particles in an Aharonov-Bohm potential is considered. It is shown that earlier approaches to this problem have neglected a crucial delta function contribution to the potential. By formulating the problem with a source of finite radius which is then allowed to go to zero, it is established that this is the delta function alone that causes solutions that are singular at the origin to become relevant. The changes in the amplitude which arise from the inclusion of spin are seen to modify the cross section for the case of polarized beams. Finally, the calculated Aharonov-Bohm amplitude is shown to describe the scattering of particles with arbitrary spin in the c=∞ limit.

A periodically driven relaxation-oscillator circuit is used to study experimentally the multifractal structure of the quasiperiodic parameter set on the critical line. The experimental data provide evidence of a phase transition in the thermodynamic free energy q(τ). The experimental thermodynamics compares well with that of the sine circle map and provides strong evidence for universality in the parameter scaling at the borderline of chaos.

It is shown that for any physical system there is an infinite number of measurements which are related to one of the undecidable problems. The undecidability of the physical systems indicates that there is an infinite number of incomputable correlation functions for each physical system. These results show that there is an inherent and irreducible limitation on the knowledge of the nature of physical systems. This uncertainty principle is caused by the undecidability principle.

We present a semiclassical theory for the two-point correlation function of the eigenphases of the S matrix for chaotic scattering. It is expressed as a sum of contributions from unstable periodic orbits of the classical scatttering mapping. Backed by numerical results and for correlation ranges $r^{\mathrm{*}}$(∼1/ħ) we obtain a universal function which is consistent with the result for Dyson’s circular ensemble. This result adds to the conjecture that universal fluctuations governed by random matrix theory are the quantum manifestation of classical chaos.

We sutdy a stochastic interacting particle system which displays a nonequilibrium transition in its relaxation dynamics in the infinite volume limit. The transition is destroyed by restriction to a finite volume, but its remnants remain until a crossover time $T_c$(L,ε), where L is the system size and ε is the control parameter measuring the distance from the bulk transition. We find that the crossover time $T_c$(L,ε) diverges where ε→0 in a fixed volume. Thus this finite volume system displays arbitrarily long time scales near the transition.

Intrinsic molecular fluctuations are associated with macrovariables whose time evolution is described by macrovariable equations. When the macrovariable equations describe chaotic trajectories, the covariance matrix for the molecular fluctuations diverges rapidly. This implies that the macrovariable equations are not stable and cannot be justified from an underlying molecluar description.

The properties of energy transport are numerically investigated with the use of a one-dimensional cellular-automaton model called 26R. The validity of Fourier’s law and the Green-Kubo formula for thermal conductivity κ is demonstrated for this model in the limit of large systems. Nonlinear correction to Fourier’s law and the recovery of left-right symmetry are also discussed.

Using fully quantal methods, we investigate various manifestations of the geometric phase in bound and free diatom systems. Adiabaticity and degeneracy are not invoked in the discussion.

The coupled-cluster method is widely applicable to quantum systems of interacting particles or fields. Such concepts as supercoherent states, generalized order parameters, and exact mappings onto corresponding multilocal classical field theories arise naturally within it, particularly in its extended version. Its holomorphic representation is applied here to the anharmonic oscillator, in order to demonstrate several key mathematical features.

We discuss a new momentum-conserving lattice-gas model in which particles are allowed to exchange momentum between distant sites. The interactions may be tuned so that a first-order transition occurs between a dense and a light phase. An equation of state may be predicted with the assumption that the lattice is in a factorized state just after the particles have propagated. This method accurately predicts the pressure of the stable and unstable states.