A Maxwell's demon type "information engine" that extracts work from a bath is constructed from a microscopic Hamiltonian for the whole system including a subsystem, a thermal bath, and a nonequilibrium bath of phonons or photons that represents an information source or sink. The kinetics of the engine is calculated self-consistently from the state of the nonequilibrium bath, and the relation of this kinetics to the underlying microscopic thermodynamics is established.

Generation and prediction of time series are analyzed for the case of a bit generator: a perceptron where in each time step the input units are shifted one bit to the right with the state of the leftmost input unit set equal to the output unit in the previous time step. The long-time dynamical behavior of the bit generator consists of cycles whose typical period scales polynomially with the size of the network and whose spatial structure is periodic with a typical finite wavelength. The generalization error on a cycle is zero for a finite training set, and global dynamical behaviors can also be learned in a finite time. Hence, a projection of a rule can be learned in a finite time.

We show that the particles in the Calogero-Sutherland model obey fractional exclusion statistics as defined by Haldane. We construct anyon number densities and derive the energy distribution function. We show that the partition function factorizes in the form characteristic of an ideal gas. The virial expansion is exactly computable and interestingly it is only the second virial coefficient that encodes the statistics information.

A particular class of random walks with a spin factor on a three-dimensional cubic lattice is studied. This three-dimensional random walk model is a simple generalization of random walk for the two-dimensional Ising model. All critical diffusion constants and associated critical exponents are calculated. Continuum field theories such as Klein-Gordon, Dirac, and massive Chern-Simons theories are constructed near several critical points.

A generalization of periodic boundary conditions to incorporate fluctuations with wavelength larger than the system size is discussed. These boundary conditions are applied to Monte Carlo calculations on the two-dimensional $\mathrm{XY}$ model. The finite-size effect on the vortex-vortex correlation function, and thereby the dielectric function, is shown to be virtually eliminated. These Monte Carlo data are shown to be in excellent agreement with the Kosterlitz renormalization group equations, and the critical temperature is found to be $T_c=0.8922\mathrm{}\left(2\right)\mathrm{}$.

We show that the dynamical fixed point of Ising-like models, characterized by a single scalar, nonconserved ordering field, is stable near four dimensions with respect to all dynamic perturbations, including those of a nonequilibrium nature.

We have obtained a class of solitary wave solutions to novel exactly integrable nonlinear wave equations. Conservation laws can be identified and velocities of propagation predicted. We propose to test our predictions in the optical domain with two-color experiments.

We show that a saturable amplifier can beat the "3 dB" quantum noise figure of linear amplifiers, approaching the ideal "0 dB" Heisenberg limit. Noise suppression is due to nonlinear saturation effects which cut off the high-intensity tail of the photon distribution. A Monte Carlo simulation of an actual semiconductor laser amplifier leads to a minimum noise figure of 0.51 dB, with a still sizable gain of 10.8 dB. The best performance corresponds to an output signal photon number which equals 0.7 times the saturation number.

We propose a new experiment employing two independent sources of spin correlated photon pairs. Two photons from different unpolarized sources each pass through a polarizer to a detector. Although their trajectories never mix or cross, they exhibit fourth-order-interference-like correlations when the other two photons interfere on a beam splitter even when the latter two do not pass any polarizers at all. A wave packet calculation shows that the experiment permits a very discriminatory test of hidden variable theories.

We simplify the derivation of the Holevo upper bound on the maximum information extractable from a quantum communication channel. This simplification leads to upper and lower bounds for binary channels, both of which depend explicitly on the message ensemble.

I show that neither local nonuniform classical acceleration nor spatial separation of the quantized spin components plays an essential role in a number of neutron interferometer experiments designed to detect phases ${\phi }_{\pm }\mathrm{}(B,E)\mathrm{}$ akin to the Aharonov-Bohm phases. These neutral particle phases, which simply manifest quantum interference in spin space, can also be detected by single beams, as exemplified by a recent breakthrough experiment by Sangster et al.

We introduce a class of stochastic process, the truncated Lévy flight (TLF), in which the arbitrarily large steps of a Lévy flight are eliminated. We find that the convergence of the sum of $n$ independent TLFs to a Gaussian process can require a remarkably large value of $n$—typically $n\approx {10}^4$ in contrast to $n\approx 10$ for common distributions. We find a well-defined crossover between a Lévy and a Gaussian regime, and that the crossover carries information about the relevant parameters of the underlying stochastic process.

The most general form for radial potentials with the same phase shifts as a given real potential but arbitrarily different bound spectra is derived with a sequence of supersymmetric factorizations. The wave functions of the phase-equivalent potentials are expressed analytically in terms of the wave functions of the original potential.

By tracking the Brownian trajectory of a sphere of 1 μm radius in a suspension of 35 nm radius beads, we probe the depth of the entropic potential experienced by a large sphere close to the cell wall. This potential is caused by the pressure exerted by the fluid of small beads when they cannot fit in the gap between the large bead and the wall. We find that the potential depth is of order $\mathrm{kT}$. Its dependence on the volume fraction of small spheres is significantly overstimated by a theory which treats the small beads as an ideal gas.

In arrays of globally coupled Josephson junctions, antiphase states have high multiplicity in phase space. For an array of $N$ oscillators, there are ($N-1\mathrm{}$)! distinct antiphase states. In this Letter we show, using the modern ideas of controlling a saddle on a basin boundary, how to switch from one antiphase state to another. The cases of three and four Josephson junctions are explicitly considered. The method we present is directly applicable to real experiments of globally coupled oscillators.

We present a generalized circle theorem which includes the Lee-Yang theorem for symmetric transitions as a special case. It is found that zeros of the partition function can be written in terms of discontinuities in the derivatives of the free energy. For asymmetric transitions, the locus of the zeros is tangent to the unit circle if the partition functions of the two phases are added up with unequal prefactors. This conclusion is substantiated by explicit calculation of zeros of the partition function for the Blume-Capel model near and at the triple line at low temperatures.

We present the results of numerical investigations of Ising models undergoing continuous ordering which indicate, for systems with large but finite interaction range $R$, that the time interval during which the linear theory of Cahn, Hilliard, and Cook fits the simulation data depends strongly on the length scale of observation. We associate the initial linear theory deviation at small length scales with the appearance of isolated structures or domains which form after a quench into the unstable region of thermodynamic space. These domains cannot be described by a linear theory; this implies, in contrast to earlier results, that the linear theory "breakdown" cannot be obtained by investigation of the linear itself for inconsistencies.

An explicit formula for the boundary magnetization of a two-dimensional Ising model with a strip of inhomogeneous interactions is obtained by means of a transfer matrix mean-field method introduced by Lipowski and Suzuki. There is clear numerical evidence that the formula is exact. By taking the limit where the width of the strip approaches infinity and the interactions have well-defined bulk limits, I arrive at the boundary magnetization for a model which includes the Hilhorst-van Leeuwen model. The rich critical behavior of the latter magnetization is thereby rederived with little effort.

We derive the nested Bethe ansatz solution of the fully packed $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behavior. In the $n=0\mathrm{}$ limit we obtain the exact compact exponents $\gamma =1\mathrm{}$ and $\nu =\frac{1}{2}$ for Hamiltonian walks, along with the exact value ${\kappa }^2=3\mathrm{}\sqrt{\frac{3}{4}}$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.

We propose three physical tests to measure correlations in random numbers used in Monte Carlo simulations. The first test uses autocorrelation times of certain physical quantities when the Ising model is simulated with the Wolff algorithm. The second test is based on random walks and the third on blocks of $n$ successive numbers. We apply the tests to show that recent errors in high precision Ising simulations using generalized feedback shift register algorithms are due to short range correlations in random number sequences.