Table of contents

We study the spatial correlations of the one-dimensional KPZ surface for the flat initial condition. It is shown that the multi-point joint distribution for the height is given by a Fredholm determinant, with its kernel in the scaling limit explicitly obtained. This may also describe the dynamics of the largest eigenvalue in the GOE Dyson's Brownian motion model. Our analysis is based on a reformulation of the determinantal Green's function for the totally ASEP in terms of a vicious walk problem.

Investigating the long-time asymptotics of the totally asymmetric simple exclusion process, Sasamoto obtains rather indirectly a formula for the Tracy–Widom distribution for the Gaussian orthogonal ensemble. We establish that his novel formula indeed agrees with more standard expressions.

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising-like) models in equilibrium statistical mechanics, improving on the mean-field approximation and the Bethe–Peierls approximation, which can be regarded as the lowest level of the CVM. In recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models. The foundations of the CVM are briefly reviewed, and the relations with similar techniques are discussed. The main properties of the method are considered, with emphasis on its exactness for particular models and on its asymptotic properties. The problem of the minimization of the variational free energy, which arises in the CVM, is also addressed, and recent results about both provably convergent and message-passing algorithms are discussed.

We calculate the mean joint residence time of two Brownian particles in a sphere, for very general initial conditions. In particular, we focus on the dependence of this residence time as a function of the diffusion coefficients of the two particles. Our results can be useful for describing kinetics of bimolecular diffusion controlled reactions activated by catalytic sites.

For stochastic processes leading to condensation, the condensate, once it is formed, performs an ergodic stationary-state motion over the system. We analyse this motion, and especially its characteristic time, for zero-range processes. The characteristic time is found to grow with the system size much faster than the diffusive time scale, but not exponentially fast. This holds both in the mean-field geometry and on finite-dimensional lattices. In the generic situation where the critical mass distribution follows a power law, the characteristic time grows as a power of the system size.

We study consensus formation in interacting systems that evolve by multi-state majority rule and by plurality rule. In an update event, a group of G agents (with G odd), each endowed with an s-state spin variable, is specified. For majority rule, all group members adopt the local majority state; for plurality rule the group adopts the local plurality state. This update is repeated until a final consensus state is generally reached. In the mean field limit, the consensus time for an N-spin system increases as ln N for both majority and plurality rules, with an amplitude that depends on s and G. For finite spatial dimensions, domains undergo diffusive coarsening in majority rule when s or G is small. For larger s and G, opinions spread ballistically from the few groups with an initial local majority. For plurality rule, there is always diffusive domain coarsening towards consensus.

We studied the pure and dilute Baxter–Wu (BW) models using the Wang–Landau (WL) sampling method to calculate the density-of-states (DOS). We first used the exact result for the DOS of the Ising model to test our code. Then we calculated the DOS of the dilute Ising model to obtain a phase diagram, in good agreement with previous studies. We calculated the energy distribution, together with its first, second and fourth moments, to give the specific heat and the energy fourth order cumulant, better known as the Binder parameter, for the pure BW model. For small samples, the energy distribution displayed a doubly peaked shape, and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with L. The energy distribution yielded the expected BW α = 2/3 critical exponent for the specific heat. The Binder parameter minimum appeared to scale with lattice size L with an exponent θ_B equal to the specific heat exponent. Its location (temperature) showed a large correction-to-scaling term θ₁ = 0.248 ± 0.025. For the dilute BW model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected α = 0 was recovered.

We apply a three-dimensional approach to describe a new parametrization of the L-operators for the two-dimensional Bazhanov–Stroganov (BS) integrable spin model related to the chiral Potts model. This parametrization is based on the solution of the associated classical discrete integrable system. Using a three-dimensional vertex satisfying a modified tetrahedron equation, we construct an operator which generalizes the BS quantum intertwining matrix . This operator describes the isospectral deformations of the integrable BS model.

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg–Kohn functional for electronic or nucleonic densities. We give explicit properties of such polynomials, generalizing Laguerre ones. The nature of the dimension 1 subspace completing such sets is described. A numerical example illustrates the use of such polynomials.

It is shown that for a given Hermitian Hamiltonian possessing supersymmetry, there is always a non-Hermitian Jaynes–Cummings-type Hamiltonian (JCTH) admitting entirely real spectra. The parent supersymmetric Hamiltonian and the corresponding non-Hermitian JCTH are simultaneously diagonalizable. The exact eigenstates of these non-Hermitian Hamiltonians are constructed algebraically for certain shape-invariant potentials, including a non-Hermitian version of the standard Jaynes–Cummings (JC) model for which the parent supersymmetric Hamiltonian is the superoscillator. It is also shown that a non-Hermitian version of the several physically motivated generalizations of the JC model admits entirely real spectra. The positive-definite metric operator in the Hilbert space is constructed explicitly along with the introduction of a new inner product structure, so that the eigenstates form a complete set of orthonormal vectors and the time evolution is unitary.

We study the influence of entanglement on the relation between the statistical entropy of an open quantum system and the heat exchanged with a low temperature environment. A model of quantum Brownian motion of the Caldeira–Leggett type—for which a violation of the Clausius inequality has been stated by Th M Nieuwenhuizen and A E Allahverdyan (2002 Phys. Rev. E 66 036102)—is re-examined and the results of the cited work are put into perspective. In order to address the problem from an information theoretical viewpoint a model of two coupled Brownian oscillators is formulated that can also be viewed as a continuum version of a two-qubit system. The influence of an additional internal coupling parameter on heat and entropy changes is described and the findings are compared to the case of a single Brownian particle.

In molecular quantum similarity measure between two molecules, using the molecular electron density, the major task involves the accurate numerical evaluation of overlap between the two electron densities in the integral measure. By expanding the electron densities in terms of atomic orbitals, using the linear combination of atomic orbitals approach, a large number of overlap-like quantum similarity integrals over the basis functions in the two molecules will be required accurately for the calculation of a meaningful quantum similarity measure. Improvement of the computational methods of these integrals would be indispensable to a further development in computational studies of large molecular systems. Analytic expressions were obtained for these overlap-like quantum similarity integrals over Slater-type functions, in terms of the usual two-centre overlap integrals over B functions. Different approaches were developed for the numerical evaluation of these two-centre overlap integrals over B functions, which can be used for the numerical evaluation of the two-centre overlap-like quantum similarity integrals over Slater-type functions. In this work we present our approach which is based on the use of nonlinear transformations for improving convergence of highly oscillatory integrals. In the case of more complicated multicentre integrals, this approach is shown to be able to produce remarkably good results. We also present different representations, which were obtained by Steinborn group, and which can be used for a fast and accurate computation of the two-centre overlap-like quantum similarity integrals over Slater-type functions.

An amplitude-phase formula for the S matrix using two Milne solutions and the regular Schrödinger solution is derived. The formula is particularly useful in the analysis of Regge poles located far out in the complex ℓ-plane, particularly for discontinuous scattering potentials. Numerical applications for an attractive square-well potential and an inverse-power potential ∼r⁻⁴ are presented.

By using the concept of concurrence, the entanglement of periodic anisotropic XY chains in a transverse field is studied numerically. It is found that the derivatives ∂_λC(1) of nearest-neighbour concurrence diverge at quantum critical points. By proper scaling, we found that all the derivatives ∂_λC(1) for periodic XY chains in the vicinity of quantum critical points have the same behaviours as that of a uniform chain.

In this paper, the dynamics of a relativistic particle of spin 1/2, interacting with an external electromagnetic field in noncommutative space, is studied in the path integral framework. By adopting the Fradkin–Gitman formulation, the exact Green's function in noncommutative space (NCGF) for the quadratic case of a constant electromagnetic field is computed, and it is shown that its form is similar to its counterpart given in commutative space. In addition, it is deduced that the effect of noncommutativity has the same effect as an additional constant field depending on a noncommutative θ matrix.

We consider the massive Klein–Gordon field on the half line with and without a Robin boundary potential. The field is coupled at the boundary to a harmonic oscillator. We solve the system classically and observe the existence of classical boundary bound states in some regions of the parameter space. The system is then quantized, the quantum reflection matrix and reflection cross section are calculated. Resonances and Ramsauer–Townsend effects are observed in the cross section. The pole structure of the reflection matrix is discussed.

The Dirac equation for an electron in the central Coulomb field of a point-like and almost point-like nucleus with the charge greater than 137 is considered. This singular problem, to which the fall-down onto the centre is inherent, is addressed using a new approach, based on a special concept of the singular centre, capable of producing results independent of the nucleus size. To this end, the Dirac equation is presented as a generalized eigenvalue boundary problem of a self-adjoint operator. The eigenfunctions make complete sets, orthogonal with a singular measure, and describe particles, asymptotically free and delta-function-normalizable both at infinity and near the singular centre r = 0. The barrier transmission coefficient for these particles responsible for the effects of electron absorption and spontaneous electron–positron pair production is found analytically as a function of electron energy and charge of the nucleus. The singular threshold behaviour of the corresponding amplitudes substitutes for the resonance behaviour, typical of the conventional theory that depends on the cut-off procedure at the edge of a finite-size nucleus.