Combinatorial Models of Rigidity and Renormalization

We first introduce the percolation problems associated with the graph theoretical concepts of (k,l)-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for (k,l)-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of (k,l)-percolation problems on hierarchical graphs, for k≤l<2k. We introduce and solve by renormalization such a model, which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions, depending on the values of the parameters.     

The Knudsen Ion Layer Problem in the Theory of the Collisionless Sheath

The interaction of a partially ionized continuum gas with a negatively charged electrode is investigated theoretically. It is supposed that l_d ? 1, where l_d is the Debye length, l is the mean-free-path. The self-consistent solution is given for two cases: l_I ? l and l_I ～ l, l_I is the effective ionisation length. The matched asymptotic expansions method is used. The Bohm sheath criterion is shown to be satisfied automatically in the considered cases.     

Self-consistent approach to low-energy resonance. Resonances of πN system

A method is worked out for determining the position of low-energy P, D-resonances of an arbitrary two-particle system with a strong interaction on the assumption that the values of the partial-wave amplitude and its first derivative are known in one points _F =s ₀. This is made possible by the modification of Balázs' N/D method for two unknown parametersF _i ^l using the elastic approximation.Although the otherl-states of the system in the functionN _l ^I are generally contained in the three- or two-pole form mentioned, in spite of the asymptotic behaviour of the amplitudes they cannot be consistent with the approximation used.The explicit solution with a generally given subtraction points ₀ is carried out for the pionnucleon system.     

Billiard Representation for Multidimensional Multi-Scalar Cosmological Model with Exponential Potentials

A multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potentials is considered. The dynamics of the model near the singularity is reduced to a billiard on the (N–1)-dimensional Lobachevsky space H ^N–1, N =ls; n+l. It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotic Kasner-like behavior. For the case of one scale factor (n =1) billiards with finite volumes (e.g. coinciding with that of the Bianchi-IX model) are described and oscillating behaviour of scalar fields near the singularity is obtained.     

Lingering Random Walks in Random Environment on a Strip

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the (log t)² asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the (log t)² asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem.     

Long range diffusion in ultrametric spaces

A model for diffusion in spaces with ultrametric topology is introduced. The parameters of the model are two sets of numbers: the branching numberm _l+1 /m _l describing the hierarchical structure of the metric and the transition ratesr _l between different levels of the hierarchy. This model is expected to be relevant to the spin glas problem. The exact solution of the dynamical problem can be given in the general case. In the special casem _l =m ^l andr _l R ^l anomalous long time behaviour of the autocorrelation function is found which decays with a power law multiplied by a periodically varying amplitude.Dedicated to B. Mühlschlegel on the occasion of his 60th birthdayThis work has been supported by the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting.

The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.     

Modelling capillary filling dynamics using lattice Boltzmann simulations

We investigate the dynamics of capillary filling using two lattice Boltzmann schemes: a liquid-gas model and a binary model. The simulation results are compared to the well-known Washburn's law, which predicts that the filled length of the capillary scales with time as l ∝ t ^1/2. We find that the liquid-gas model does not reproduce Washburn's law due to condensation of the gas phase at the interface, which causes the asymptotic behaviour of the capillary penetration to be faster than t ^1/2. The binary model, on the other hand, captures the correct scaling behaviour when the viscosity ratio between the two phases is sufficiently high.     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

The Density Matrix for Single-mode Light after k-Photon Absorption

In order to continue and generalize the studies of the density matrix of a light field undergoing k-photon absorption, in this paper we put the emphasis on the off-diagonal elements. The solution obtained earlier for the diagonal elements describing the photon statistics can be found as a special case but will not be discussed again. The general solution calculated by recursion shows an asymptotic behaviour if the initial photon number is sufficiently high. Only the initial phase information survives. Illustrating the solution we start with coherent light and a generalized coherent state.     

Inflation within closed friedmann universe models

For the LagrangianL = R ²,the de Sitter space-time is known to be an attractor solution. Here, we classify for closed Friedmann models in which initial conditions lead asymptotically to a de Sitter phase and what the behaviour is for the other solutions. Four types of solutions form together a generic set, and three of them are asymptotically de Sitter; the fourth one has both an initial and final singularity. Furthermore, exactly seven other solutions exist and can be given in closed form. Three of them are known, the other four are new and have a linear asymptotic behaviour of the cosmic scale factor.     

Families of commuting transfer matrices and integrable models with disorder

A family of commuting transfer matrices is shown to be associated to each symmetry transformation of a given Yang-Baxter algebra. This applies in lattices models and field theory.The Yang-Baxter algebra remains unchanged when an arbitrary parameter μ_l is associated to each lattice site. We generate in this way integrable one-dimensional hamiltonians with long-range couplings and disorder given by the <{;μ₁<};. These operators are lattice versions of the non-local charges in sigma models. As a simple example we get a Dzialozhinski-Moriya interaction with an arbitrary coupling per site from the six-vertex model. A similar model with a disordered magnetic field follows too. Their exact solution by an algebraic Bethe ansatz is presented. We derive the excitations spectrum in terms of the density of parameters (μ).As another application, the total spin S² is computed for a XXZ Heisenberg chain (μ_l ≡ 0) as a function of the anisotropy Δ (− ∞ < Δ < + ∞).     

Refinement of a formula for decay after weak coherent excitation of a sphere

A new formula has been given recently by A.A. Svidzinsky and M.O. Scully to describe the temporal evolution of the excitation function in a large sphere satisfying the Markov condition after excitation by a single photon. This formula is based on a physically reasonable Ansatz from which differential equations are inferred for the undetermined radial functions in the Ansatz. The solution to these differential equations leads to the formula for β. Numerical calculations from this formula yield a value ∼10% for the maximum probability of occupancy of secondary excited states.In this paper, we refine the formula of Svidzinsky and Scully by allowing the radial functions in the Ansatz to depend on the angular index l of the spherical Bessel functions. By using the Debye formula for the asymptotic behavior of j_l(u) for large l as well as u, we obtain differential equations in each angular sector, similar to theirs but with a dependence on l. The solution to these equations yields our improved formula, from which we calculate 17.1% for the maximum probability of secondary excited states.     

The physical interpretation of a painlevé transcendent spacetime

We construct a nondiagonalizable solution of the stationary axially symmetric vacuum Einstein equations involving Painlevé transcendents III(V). From the asymptotic behaviour of this solution we identify the corresponding Newtonian potential as that of a modulated line mass at=0, and we identify our transcendental solution as corresponding to a very special differentially rotating modulated line source.     

The apparition of a new reaction at lower temperature in equiatomic CuAu alloy

Recently a lot of interest has been devoted to the study of order–disorder transitions in different materials. Although the equiatomic CuAu alloy represents a classical model of this type of transformation, it still receives considerable attention because many questions about phase transitions are still raised according to its equilibrium diagram. In this context, the present paper carries a new result observed in CuAu alloy consisting of a new anomaly observed on differential scanning calorimetry (DSC) curves at low temperature and new peaks in X-ray diffraction (XRD) spectra. For that, we will try to give an explanation of the origin of this new reaction, with the help of other techniques as electrical resistivity and microhardness measurements. The kinetic behaviour of this new reaction has been also studied by anisothermal analysis during DSC tests to estimate the kinetics parameters as activation energy E _act and Avrami exponent n.     

Inflation and conformal invariance: the perspective from radial quantization

According to the dS/CFT correspondence, correlators of fields generated during a primordial de Sitter phase are constrained by three‐dimensional conformal invariance. Using the properties of radially quantized conformal field theories and the operator‐state correspondence, we glean information on some points. The Higuchi bound on the masses of spin‐s states in de Sitter is a direct consequence of reflection positivity in radially quantized CFT₃ and the fact that scaling dimensions of operators are energies of states. The partial massless states appearing in de Sitter correspond from the boundary CFT₃ perspective to boundary states with highest weight for the conformal group. Finally, we discuss the inflationary consistency relations and the role of asymptotic symmetries which transform asymptotic vacua to new physically inequivalent vacua by generating long perturbation modes. We show that on the CFT₃ side, asymptotic symmetries have a nice quantum mechanics interpretation. For instance, acting with the asymptotic dilation symmetry corresponds to evolving states forward (or backward) in “time” and the charge generating the asymptotic symmetry transformation is the Hamiltonian itself.     

Quintessential Inflation with Dissipative Fluid

We have investigated cosmological models with a self-interacting scalar field and a dissipative matter fluid as the sources of matter. Different variables are expressed in terms of a generating function. Exact solutions are obtained for one particular choice of the generating function. The potential corresponding to this generating function is a standard tree-level potential arising in the perturbative regime in quantum field theory. With suitable choice of parameters, the scale factor in our model exhibits both decelerating behaviour in the early time as well as an accelerating phase at late times. For certain choices of the parameter the solution also exhibits an attractor nature towards an asymptotic de-Sitter universe.     

Regge Models and Dispersion Relations for Partial Waves

The asymptotic behaviour of partial wave amplitudes is calculated supposing various Regge models for the total scattering amplitude A(s, t, u). The high energy partial wave behaviour obtained is combined with the validity of partial wave dispersion relations. It is shown that consistency of these assumptions can only be achieved by demanding. 1) a definite asymptotic behaviour of the discontinuity of the left hand cut of partial wave amplitudes. 2) the validity of partial wave sum rules of similar kind as the well-known finite energy sum rules for the total amplitude. All steps of the derivation shall first be demonstrated for elastic scattering of identical scalar particles. Then within the helicity formalism the results are generalized for particles with arbitrary spin and different masses. Finally the question is studied whether the sum rules can be employed to determine unknown CDD-pole parameters in an N/D approach for the I = J = 1/2 state in πN scattering. It is shown that the sum rules of highest order are able to do that.     

Asymptotic behaviour and general properties of harmonic generation susceptibilities

We use the Kubo response function formalism to derive the asymptotic behaviour of the harmonic generation susceptibilities to all orders n. The results show a stringent correspondence with the ones previously obtained from the classical anharmonic oscillator model. They are characterized by a dependence and a coefficient proportional to the trace of the (n+1)th derivative of the potential energy on the equilibrium density matrix. Using the above results we derive new Kramers-Kr?nig relations and sum rules for all orders of harmonics susceptibilities. Received 17 April 2000     

Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation II: The Low Density Regime

In this paper we analyse the asymptotic dynamics of a system of N identical quantum particles in a low-density regime. Our approach follows the strategy introduced by the authors in a previous work,⁽²⁾ to treat the simpler weak coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula, in the spirit of the paper by Lanford.⁽³²⁾ For short times and small interaction potential, we rigorously prove that a subseries of the complete perturbative series, converges to the solution of the nonlinear Boltzmann equation that is physically relevant in the context. An important point is that we completely identify the cross-section entering the limiting Boltzmann equation, as being the Born series expansion of quantum scattering.As in ref. 2, our convergence result is only partial, in that we merely characterize the asymptotic behaviour of a subseries of the complete original perturbative expansion. We only have plausibility arguments in the direction of proving that the terms we neglect, when going from the original series to its associated subseries, are indeed vanishing in the limit.The present study holds in any dimension d ≥ 3.