We derive a new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler’s equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

It is shown that it is possible to demonstrate nonlocality for two particles without using inequalities for all entangled states except maximally entangled states such as the singlet state. The eigenvectors corresponding to the measurements that must be performed to do this are exhibited and found to have a particularly simple relationship to the entangled state.

We consider a Brownian particle in a periodic potential under heavy damping. The second law forbids it from displaying any net drift speed, even if the symmetry of the potential is broken. But if the particle is subject to an external force having time correlations, detailed balance is lost and the particle can exhibit a nonzero net drift speed. Thus, broken symmetry and time correlations are sufficient ingredients for transport.

We discuss a new mechanism generating long range temporal correlations in dynamical systems coupled to a source of white noise. The external noise induces dynamical events uncorrelated on a logarithmic time scale and produces a fluctuating output with ‘‘1/f’’ power spectrum. This behavior requires a complex phase space with many traps, which can arise due to strong cooperative effects. As a demonstration, we numerically analyze a system of many coupled degrees of freedom, which is externally driven and subject to a white noise perturbation. We find both Poisson statistics for events in log(time) and the 1/f power spectrum.

If a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m≤n is conjectured to be ${\mathit{S}}_{\mathit{m},\mathit{n}}$= ${\mathit{S}}_{\mathit{k}=\mathit{n}+1\mathrm{}}^{\mathit{mn}}$ 1/k-m-1/2n and is shown to be ≃lnm-m/2n for 1≪m≤n. Thus there is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.

An approximate wave number formula is derived for one-dimensional wave transmission in linear systems that are uniform on a large scale. It is a low-frequency approximation that coincides at zero frequency with equivalent medium theory. The formula is evaluated for various lattice media models for which exact calculations can also be made.

A model of a two-dimensional autonomous system subject to external noise is investigated. Without noise the system has a stable limit cycle in a certain region of control parameter. Various noise-induced effects have been found numerically, such as a noise-induced frequency shift in the presence of the deterministic limit cycle, and noise-induced coherent oscillations in the absence of the deterministic limit cycle. An interesting result is that the stochastic resonance phenomenon appears in a system without an external signal and when the asymptotic state of the deterministic system is stationary.

We analyze the ensemble of surfaces surrounding critical clusters at T=${\mathit{T}}_{\mathit{c}}$ in the 3D Ising model. We find that ${\mathit{N}}_{\mathit{g}}$(A), the number of surfaces of genus g and area A, behaves as ${\mathit{A}}^{\mathit{x}\left(\mathit{g}\right)}$${\mathit{e}}^{\mathrm{-}\mathrm{\mu }\mathit{A}}$. We show that μ is constant and x(g) is approximately linear; the sum ${\mathit{tsum}}_{\mathit{g}}$ ${\mathrm{N}}_{\mathit{g}}$(A) scales as a power of A. The cluster volume is proportional to its surface area. We discuss similar reuslts for the ordinary spin clusters of the 3D Ising model and for 3D bond percolation.

We consider a topological transition which is an abrupt variation from π to zero of Berry’s phase, in the case when it is a phase difference of interfering waves. It manifests itself in a steplike current–magnetic field and current–gate voltage characteristics predicted for in-plane magnetoresistance of rings in noncentrosymmetric materials. Transition points occur at external magnetic fields equal to momentum-dependent effective magnetic fields for different tunneling channels of a quasi-one-dimensional ring. Similar effects due to the angular anisotropy of electron g factor are considered.

Bogomolny’s formula for energy-smoothed scars is applied to a nonspecific, nonscalable Hamiltonian, a two-dimensional anharmonic oscillator. The semiclassical theory reproduces well the exact quantal results over a large spatial and energy range.

The ground-state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere. Possible connections with the physics of black holes are discussed.

We consider an exactly solvable random matrix model related to the random transfer matrix model for disordered conductors. In the conventional random matrix models the spacing distribution of nearest neighbor eigenvalues, when expressed in units of average spacing, has a universal behavior known generally as the Wigner distribution. In contrast, our model has a single parameter, as a function of which the spacing distribution crosses over from a Wigner to a distribution which is increasingly more Poisson-like, a feature common to a wide variety of physical systems including disorder and chaos.

An essential signature of the topological nature of all Aharonov-Bohm type phases is that they are nondispersive, i.e., independent of the velocity (wavelength) of the interfering particles. This implies that an Aharonov-Bohm phase shift can greatly exceed the usual limit given by the coherence length of the interfering beams. We report the results of a polarized neutron experiment demonstrating this property for a spin-rotation analog of the scalar Aharonov-Bohm effect.

A general theory of coherent behavior (‘‘locking’’) in a globally coupled ensemble of pulse-emitting units is presented. Each unit is modeled as a dynamic threshold device with arbitrary excitability function and noise. The interaction is described by a general linear-response kernel that includes a transmission delay. In the bulk limit, the dynamics is solved exactly. Two types of solutions are studied, viz., coherent states with synchronous activity of all units and incoherent stationary states, and their stability is analyzed in the low-noise limit.

We present a new Monte Carlo sampling algorithm, with which one can obtain any desired distribution of the sampling in one Monte Carlo simulation. The free energy and the entropy of a system can thus be obtained from a simple exercise of this algorithm. The main idea is to sample directly the entropy of a system at infinite temperature. Importance sampling is shown to be a particular case of the new algorithm. The algorithm is tested against the exact partition function of the L=4 simple cubic Ising model. A comparison with the multicanonical ensemble for the L=12, q=10 Potts model shows that the new algorithm is more general and more efficient.

The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of δ-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the δ-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.

We investigate the properties of the Kuramoto-Sivashinsky equation in two spatial dimensions. We show by an explicit, numerical, coarse-graining procedure that its long-wavelength properties are described by a stochastic, partial differential equation of the Kardar-Parisi-Zhang type. From the computed parameters in our effective, stochastic equation we argue that the length and time scales over which the correlation functions cross over from linear diffusive to those of the full nonlinear equation are very large. The behavior of the three-dimensional equation is also discussed.

Although the master equation describing long-range random walks on an energetically disordered lattice is governed by a nonsymmetric transition matrix W, it may be mapped through a similarity transform onto an imaginary-time Schrödinger equation governed by a Hermitian (Hamiltonian) operator ${\mathit{H}}_0$ having a nondegenerate ground state. Under this mapping the diffusion constant D can be expressed in terms of the exact ground state energy of operators that are infinitesimally perturbed from ${\mathit{H}}_0$.

In this paper, we investigate a family of one-dimensional multicomponent quantum many-body systems. The interaction is an exchange interaction based on the familiar family of integrable systems which includes the inverse square potential. We show these systems to be integrable, and exploit this integrability to completely determine the spectrum including degeneracy, and thus the thermodynamics. The perioidic inverse square case is worked out explicitly. Next, we show that in the limit of strong interaction of the ‘‘spin’’ degrees of freedom decouple. Taking this limit for our example, we obtain a complete solution to a lattice system introduced recently by Shastry and by Haldane; our solution reproduces the numerical results. Finally, we emphasize the simple explanation for the high multiplicities found in this model.

The dynamics of bistable stochastic systems driven by oscillating fields is analyzed by numerically solving the Fokker-Planck equation. The long time response is oscillatory in time and out of phase with respect to the field. The phase shift as a function of the noise D has a peak for weak fields and low frequencies if intrawell fluctuations are considered, in agreement with linear response theory. The amplitude has a peak, but for a different value of D, even if intrawell fluctuations are neglected. We also analyze the behavior for strong fields so that linear response theory becomes questionable.