Table of contents

For two-dimensional lattice equations, one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e. that it is 'consistent around a cube' (CAC). As a consequence of CAC one can construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted a search based on this principle and certain additional assumptions. One of those assumptions was the 'tetrahedron property', which is satisfied by most known equations. We present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property. Its Lax pair is also presented, and some basic properties are discussed.

The fluctuation theorem (FT) has been studied as a far from equilibrium theorem, which relates the symmetry of entropy production. To investigate the application of this theorem, especially to biological physics, we consider the FT for a tilted rachet system. Under natural assumptions, the FT for steady state is derived.

We review recent theoretical progress on the statistical mechanics of error correcting codes, focusing on low-density parity-check (LDPC) codes in general, and on Gallager and MacKay–Neal codes in particular. By exploiting the relation between LDPC codes and Ising spin systems with multi-spin interactions, one can carry out a statistical mechanics based analysis that determines the practical and theoretical limitations of various code constructions, corresponding to dynamical and thermodynamical transitions, respectively, as well as the behaviour of error-exponents averaged over the corresponding code ensemble as a function of channel noise. We also contrast the results obtained using methods of statistical mechanics with those derived in the information theory literature, and show how these methods can be generalized to include other channel types and related communication problems.

Using the generating functional analysis an exact recursion relation is derived for the time evolution of the effective local field of the fully connected Little–Hopfield model. It is shown that, by leaving out the feedback correlations arising from earlier times in this effective dynamics, one precisely finds the recursion relations usually employed in the signal-to-noise approach. The consequences of this approximation as well as the physics behind it are discussed. In particular, it is pointed out why it is hard to notice the effects, especially for model parameters corresponding to retrieval. Numerical simulations confirm these findings. The signal-to-noise analysis is then extended to include all correlations, making it a full theory for dynamics at the level of the generating functional analysis. The results are applied to the frequently employed extremely diluted (a)symmetric architectures and to sequence processing networks.

We investigate the Becker–Döring model of nucleation with three generalizations; an input of monomer, an input of inhibitor and finally, we allow the monomers to form two morphologies of cluster. We assume size-independent aggregation and fragmentation rates. Initially we consider the problem of constant monomer input and determine the steady-state solution approached in the large-time limit, and the manner in which it is approached. Secondly, in addition to a constant input of monomer we allow a constant input of inhibitor, which prevents clusters growing any larger and this removes them from the kinetics of the process; the inhibitor is consumed in the action of poisoning a cluster. We determine a critical ratio of poison to the monomer input below which the cluster concentrations tend to a non-zero steady-state solution and the poison concentration tends to a finite value. Above the critical input ratio, the concentrations of all cluster sizes tend to zero and the poison concentration grows without limit. In both cases the solution in the large-time limit is determined. Finally we consider a model where monomers form two morphologies, but the inhibitor only acts on one morphology. Four cases are identified, depending on the relative poison to monomer input rates and the relative thermodynamic stability. In each case we determine the final cluster distribution and poison concentration. We find that poisoning the less stable cluster type can have a significant impact on the structure of the more stable cluster distribution; a counter-intuitive result. All results are shown to agree with numerical simulation.

The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a 'reconstruction' formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.

Nienhuis' truncated O(n) model gives rise to a model of self-avoiding loops on the hexagonal lattice, each loop having a fugacity of n. We study such loops subjected to a particular kind of staggered field w, which for n → ∞ has the geometrical effect of breaking the three-phase coexistence, linked to the three-colourability of the lattice faces. We show that at T = 0, for w > 1 the model flows to the ferromagnetic Potts model with q = n² states, with an associated fragmentation of the target space of the Coulomb gas. For T > 0, there is a competition between T and w which gives rise to multicritical versions of the dense and dilute loop universality classes. Via an exact mapping, and numerical results, we establish that the latter two critical branches coincide with those found earlier in the O(n) model on the triangular lattice. Using transfer matrix studies, we have found the renormalization group flows in the full phase diagram in the (T, w) plane, with fixed n. Superposing three copies of such hexagonal-lattice loop models with staggered fields produces a variety of one- or three-species fully-packed loop models on the triangular lattice with certain geometrical constraints, possessing integer central charges 0 ⩽ c ⩽ 6. In particular, we show that Benjamini and Schramm's RGB loops have fractal dimension D_f = 3/2.

We obtain exact results for the effective diffusion constant of a two-dimensional Langevin tracer particle in the force field generated by charged point scatterers with quenched positions. We show that if the point scatterers have a screened Coulomb (Yukawa) potential and are uniformly and independently distributed then the effective diffusion constant obeys the Volgel–Fulcher–Tammann law where it vanishes. Exact results are also obtained for pure Coulomb scatterers frozen in an equilibrium configuration of the same temperature as that of the tracer.

We study the Pöschl–Teller equation in complex domain and deduce infinite families of TQ and Bethe ansatz equations, classified by four integers. In all these models the form of T is very simple, while Q can be explicitly written in terms of the Heun function. At particular values there is an interesting interpretation in terms of finite lattice spin- XXZ quantum chain with (for free–free boundary conditions), or (for periodic boundary conditions). This result generalizes the findings of Fridkin, Stroganov and Zagier. We also discuss the continuous (field theory) limit of these systems in view of the so-called ODE/IM correspondence.

In this paper, the stochastic randomization is introduced in two different multi-value cellular automata (CA) models in order to model the bicycle flow. It is shown that with the randomization effect considered, the multiple states in the deterministic multi-value CA models disappear and the unique flow-density relations (fundamental diagrams) exist. The fundamental diagrams, the spacetime plots of the two models, are studied in detail. It is found that the transition from free flow to congested flow is smooth in one model while it is of second order in the other model. The comparison of the results of the two models indicates that in the bicycle flow, the priority of the movement should be given to slow bicycles in order to reach a larger maximum flow rate.

We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of Montanari and Ricci-Tersenghi (2003 Eur. Phys. J. B 33 339) to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clause density α or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for α < α_m, with α_m ≈ 4.153. On the other hand, the SAT–UNSAT phase transition seems to be correctly described within 1RSB.

Critical slowing down associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block-circulant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present algorithm per iteration is O(rslog s) +  delops, where the stiffness matrix A is partitioned into r × r blocks such that each block is an s × s matrix, and delops represents the operational count associated with solving a block-diagonal matrix with r × r dense matrix blocks. This algorithm using the block-circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient algorithm, and alleviates the critical slowing down that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.

Totally asymmetric simple exclusion processes (TASEP) with particles which occupy more than one lattice site and with a local inhomogeneity far away from the boundaries are investigated. These non-equilibrium processes are relevant for the understanding of many biological and chemical phenomena. The steady-state phase diagrams, currents and bulk densities are calculated using a simple approximate theory and extensive Monte Carlo computer simulations. It is found that the phase diagram for TASEP with a local inhomogeneity is qualitatively similar to homogeneous models, although the phase boundaries are significantly shifted. The complex dynamics is discussed in terms of domain-wall theory for driven lattice systems.

In this work, we present a vertex–face correspondence between an elliptic R-operator and Boltzmann weights related to the Lie superalgebra sl(m∣n).

It has been argued that for a finite two-dimensional classical Coulomb system of characteristic size R, in its conducting phase, as R → ∞ the total free energy (times the inverse temperature β) admits an expansion of the form: , where χ is the Euler characteristic of the manifold where the system exists. The first two terms represent the bulk free energy and the surface free energy, respectively. The coefficients A and B are non-universal but the coefficient of ln R is universal: it does not depend on the detail of the microscopic constitution of the system (particle densities, temperature, etc). By doing the usual Legendre transform this universal finite-size correction is also present in the grand potential. The explicit form of the expansion has been checked for some exactly solvable models for a special value of the coulombic coupling. In this paper we present a method for obtaining these finite-size corrections in the Debye–Hückel regime. It is based on the sine-Gordon field theory to find an expression for the grand canonical partition function in terms of the spectrum of the Laplace operator. As an example we find explicitly the grand potential expansion for a Coulomb system confined in a disc and in an annulus with ideal conductor walls.

The plane-wave decomposition method, a widely used means of numerically finding eigenstates of the Helmholtz equation in billiard systems is described as a variant of the mathematically well-established boundary integral method (BIM). A new unified framework encompassing the two methods is discussed. Furthermore, a third numerical method, which we call the gauge freedom method is derived from the BIM equations. This opens the way to further improvements in eigenstate search techniques.

The stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) in stochastic coupled dynamical systems (ordinary differential equations) is considered. A general approach is presented, based on the master stability function, Gershgörin disc theory and the extreme value theory in statistics, to yield constraints on the distribution of coupling to ensure the stability of synchronized dynamics. Three types of different behaviour: global-stable, exponential-stable and power-stable, are found, depending on the nature of the distribution of the interactions between units. Systems with specific coupling schemes are used as examples to illustrate our general method.

In a previous contribution (Stöckmann H J 2002 J. Phys. A: Math. Gen.35 5165), the density of states was calculated for a billiard with randomly distributed delta-like scatterers, doubly averaged over the positions of the impurities and the billiard shape. This result is now extended to the k-point correlation function. Using supersymmetric methods, we show that the correlations in the bulk are always identical to those of the Gaussian unitary ensemble (GUE) of random matrices. In passing from the band centre to the tail states, the density of states is depleted considerably and the two-point correlation function shows a gradual change from the GUE behaviour to that found for completely uncorrelated eigenvalues. This can be viewed as similar to a mobility edge.

Quasipolynomial (or QP) mappings constitute a wide generalization of the well-known Lotka–Volterra mappings, of importance in different fields such as population dynamics, physics, chemistry or economy. In addition, QP mappings are a natural discrete-time analogue of the continuous QP systems, which have been extensively used in different pure and applied domains. After presenting the basic definitions and properties of QP mappings in a previous paper [1], the purpose of this work is to focus on their characterization by considering the existence of symplectic QP mappings. In what follows such QP symplectic maps are completely characterized. Moreover, use of the QP formalism can be made in order to demonstrate that all QP symplectic mappings have an analytical solution that is explicitly and generally constructed. Examples are given.

Oscillatory instabilities in Hamiltonian anharmonic lattices are known to appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions of multibreather type. Here, we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear: the three-site discrete nonlinear Schrödinger model with periodic boundary conditions. The stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes. We numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types. Close to the lower-amplitude threshold stable two-frequency quasi-periodic solutions exist surrounding the unstable stationary solution, and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small. By contrast, close to the higher-amplitude threshold all two-frequency quasi-periodic solutions are detached from the unstable stationary solution, and the resulting dynamics is of 'population-inversion' type involving also the originally unexcited site.

We show that adaptation and control of the target system can be achieved by linking the modification of its dynamical properties to the estimated difference of the distribution of the dynamical characteristics of the control and target signals. Subsequently the target system, which has initially different dynamical properties adjusts its dynamics via changes of its control parameters and synchronizes with the control one. The differences in the evolving probability distributions are evaluated through entropy estimation, causing the adaptation to be based solely on the statistical properties of the control and target signals without explicit knowledge of the underlying equations of the system.

Let H be any -symmetric Schrödinger operator of the type −ℏ²Δ + (x²₁ + ⋅⋅⋅ + x²_d) + igW(x₁, ..., x_d) on , where W is any odd homogeneous polynomial and . It is proved that is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of H, i.e., the eigenvalues of . Moreover we explicitly construct the canonical expansion of H and determine the singular values μ_j of H through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues λ_j of H by Weyl's inequalities.

In the present paper the non-Noether symmetries of the Toda model, nonlinear Schrödinger equation and Korteweg–de Vries equations (KdV and mKdV) are discussed. It appears that these symmetries yield the complete sets of conservation laws in involution and lead to the bi-Hamiltonian realizations of the above-mentioned models.

We calculate the long time and distance asymptotics of the one-particle correlation functions in the model of impenetrable spin 1/2 fermions in 1 + 1 dimensions. We consider the spin disordered zero temperature regime, which occurs when the limit T → 0 is taken at a positive chemical potential. The asymptotic expressions are found from the asymptotic solution of the matrix Riemann–Hilbert problem related to the determinant representation of the correlation functions.

The Kramer–Neugebauer limit of their solution of the Ernst equation to give the Tomimatsu–Sato class of solutions is obtained. This gives insight into Nakamura's conjecture.

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the h-deformed plane and the quantum group GL_p,q(2) are recovered in this way. Geometric structures such as metrics and compatible linear connections are introduced.

The k-point correlation functions of the Gaussian random matrix ensembles are certain determinants of functions which depend on only two arguments. They are referred to as kernels, since they are the building blocks of all correlations. We show that the kernels are obtained, for arbitrary level number, directly from supermatrix models for one-point functions. More precisely, the generating functions of the one-point functions are equivalent to the kernels. This is surprising, because it implies that already the one-point generating function holds essential information about the k-point correlations. This also establishes a link to the averaged ratios of spectral determinants, i.e. of characteristic polynomials.

The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard dp ∧ dq structure on . In this paper, we describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations. The multiplication in this algebra generates an associative ∗ product of functions on the phase space. This ∗ product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane. A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer. Zero and constant curvature cases are considered as examples. The quantization with both static and time-dependent electromagnetic fields is obtained. The expansion of the ∗ product by the deformation parameter ℏ, written in the covariant form, is compared with the known deformation quantization formulae.

We consider the random sequence x_n = x_n−1 + γx_q, with γ > 0, where q = 0, 1, ..., n − 1 is chosen randomly from a probability distribution P_n(q). When all q are chosen with equal probability, i.e. P_n(q) = 1/n, we obtain an exact solution for the mean ⟨x_n⟩ and the divergence of the second moment ⟨x²_n⟩ as functions of n and γ. For γ = 1 we examine the divergence of the mean value of x_n, as a function of n, for the random sequences generated by power law and exponential P_n(q) and for the non-random sequence P_n(q) = δ_q,β(n−1).

We consider time-independent Schrödinger equations in one dimension with both periodic and Stark potentials. By means of an iterative procedure we obtain a formal power series for the Wannier–Stark ladders. In the case of strongly singular periodic potentials we prove that such a formal power series is of Gevrey type.

Every irreducible finite-dimensional representation of the quantized enveloping algebra can be extended to the quantum affine algebra via the evaluation homomorphism. We give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.

A bilinear variable separation approach is used to construct some special solutions for a differential-difference Toda equation. The semi-discrete form of the continuous formula which describes some types of special solutions for many (2 + 1)-dimensional continuous systems is found for a suitable quantity of the differential-difference Toda equation. Thus abundant semi-discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions.

Variational extensions of the classical mechanics of the particles are considered, on the basis of a re-formulation of Hamilton's principle aimed at a more general approach to the analysis of evolution phenomena. Some generalized forms of stochastic and, consequently, quantum mechanics are then obtained.

We consider transmission and reflection of narrow Gaussian wave packets by delta potentials in the cases of constant and specific (inverse linear) time-dependent strength. Both transmitted and reflected packets exhibit some 'squeezing' in the momentum probability distributions. Several different definitions of the transmission time are introduced and compared.

By virtue of the entangled state representation |ξ⟩⟩, we solve the dynamics of a generalized parametric amplifier whose Hamiltonian is composed of two forced quantum oscillators plus a parametric down-conversion interaction in the resonant case. The solutions and state vectors of the Schrödinger equation are derived, of which the simplest solution is a squeezed coherent state. The method of characteristics is employed.

Teleportation is viewed as a quantum channel. We present an explicit expression for the general teleportation channel in the Kraus decomposition form. We then analyse optimal teleportation procedures for a noisy entangled resource, Bell measurement by the sender and arbitrary operations by the receiver. Our general result allows us to derive the corresponding quantum channel and fidelity, thereby enabling us to formulate the fidelity-based optimization problem and to conclude that this is a problem of semidefinite programming. We offer an alternative viewpoint on optimal teleportation, namely one can perform corrective operations at the receiver's side instead of first manipulating entanglement, and we give an optimal teleportation strategy.

Exceptional points associated with non-Hermitian operators, i.e. operators being non-Hermitian for real parameter values, are investigated. The specific characteristics of the eigenfunctions at the exceptional point are worked out. Within the domain of real parameters the exceptional points are the points where eigenvalues switch from real to complex values. These and other results are exemplified by a classical problem leading to exceptional points of a non-Hermitian matrix.

A three state Hamiltonian with a degenerate ground state is shown to exhibit the property of generating quantum gates for fixed values of the energy parameters just by using the ability of this quantum system to evolve in time yielding suitable sets of quantum probabilities. A method is presented, based on the conventional rules of quantum mechanics, for obtaining several gates without changing the system configuration, i.e. just by adjusting the times in which the interaction is switched on and off. The mathematical rules for generating such a set of gates are completely defined and described in this work. The corresponding physical implementation should involve fast Rabi pulses and very precise switches of the kind proposed in quantum geometric computation.

We study gauge theories based on Abelian p-forms on real compact hyperbolic manifolds. An explicit formula for the trace anomaly corresponding to skew-symmetric tensor fields is obtained, by using zeta-function regularization and the trace tensor kernel formula. Explicit exact and numerical values of the anomaly for p-forms of order up to p = 4 in spaces of dimension up to n = 10 are then calculated.

We consider arbitrary U(1) charged matter non-minimally coupled to the self-dual field in d = 2 + 1. The coupling includes a linear and a rather general quadratic term in the self-dual field. By using both Lagragian gauge embedding and master action approaches we derive the dual Maxwell Chern–Simons-type model and show the classical equivalence between the two theories. At the quantum level the master action approach in general requires the addition of an awkward extra term to the Maxwell Chern–Simons-type theory. Only in the case of a linear coupling in the self-dual field can the extra term be dropped and we are able to establish the quantum equivalence of gauge invariant correlation functions in both theories.

As generalizations of the fermion seniority model, four multi-mode Hamiltonians are considered to investigate some of the consequences of the pairing of parafermions of order 2. Two- and four-particle states are explicitly constructed for H_A ≡ −GA†A with and the distinct H_C ≡ −GC†C with , and for the time-reversal invariant H₍₋₎ ≡ −G(A† − C†)(A − C) and H₍₊₎ ≡ −G(A† + C†)(A + C), which has no analogue in the fermion case. The spectra and degeneracies are compared with those of the usual fermion seniority model.

We examine, in detail, the possibility of applying the Darboux transformation to non-Hermitian Hamiltonians. In particular we propose a simple method of constructing exactly solvable symmetric potentials by applying Darboux transformation to higher states of an exactly solvable symmetric potential. It is shown that the resulting Hamiltonian and the original one are pseudo supersymmetric partners. We also discuss application of the Darboux transformation to Hamiltonians with spontaneously broken symmetry.

A gauge-independent formulation of the theory developed in J. Phys. A: Math. Gen.36 8341 (2003) is proposed.