Quantum monodromy and molecular spectroscopy

Monodromy (or once round) is a classical property of integrable dynamical systems in two or more degrees of freedom, which imposes a characteristic pattern on the quantum mechanical eigenvalue distribution. This article explains the connection by showing how the presence of an isolated critical point of the Hamiltonian leads to a classical action function that is multi-valued with respect to energy and angular momentum. Consequently, by the Bohr correspondence principle between actions and quantum numbers, there can be no uniquely defined global system of quantum numbers. Implications for the interpretation of highly excited molecular spectra are brought out by reference to quasi-linear molecules, which transfer one degree of freedom from rotational to vibrational motion during the excitation process. Emphasis is placed on the simplest examples, while a brief resumé of the wide scope of the quantum monodromy phenomenon is given in the final section.     

Classical simulations of magnetic structures for chromium clusters: Size effects

Classical (Heisenberg) simulations show that the total magnetization of the lowest-energy states of clusters made of antiferromagnetically coupled chromium atoms is planar, rather than collinear, depending on the arrangement of the atoms. Although the model Hamiltonian is not restrictive, many cluster configurations of various numbers of atoms do not use all three directions for the spins. This result confirms the conclusion drawn from the local-spin DFT calculation by Kohl and Bertsch that clusters of N≤13 have non-collinear magnetic moments. The present simulations show non-collinear spin ordering also for bigger clusters, designed to be as spherical as possible following the bcc arrangement, when atoms interact both with the nearest and next-nearest neighbours. Depending on the signs of the coupling constants frustration appears. The advantage of the discrete model, despite the simplicity, is that very large clusters and magnetization at finite temperatures can be studied. This model predicts that clusters with specific numbers of atoms interacting only with the nearest neighbours have collinear spins as in the bulk. We also apply the model to simulate the destruction of the anti-ferromagnetic ordering by thermal fluctuations. This model shows no unique magnetization of mixed Fe _0.33 Cr _0.67, which is consistent with experimental observations.     

Scaling,Renormalization and Statistical Conservation Laws in the Kraichnan Model of Turbulent Advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of ξ, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

On the steady motions of a flat domain wall in a ferromagnet

A new derivation is presented of Walker's exact solution to Gilbert equation, a solution which mimicks the travelling-wave motion of a flat domain wall at 180^°. It is shown that a process during which the working of the applied magnetic field exactly compensates dissipation (the Walker condition) exists both under the constitutive circumstances considered in the standard Gilbert equation and when either the internal free-energy or the dissipation, or both, are generalized by the introduction of higher-gradient terms; but that such a process cannot solve the generalized Gilbert equation. It is also shown that, when dry-friction dissipation is considered and a suitable magnetic field is applied, the associated Gilbert equation has a Walker-type solution mimicking a flat wall, at 90^° this time, which however does not satisfy the Walker condition. Received 16 November 2001     

Recursive weighted treelike networks

We propose a geometric growth model for weighted scale-free networks, which is controlled by two tunable parameters. We derive exactly the main characteristics of the networks, which are partially determined by the parameters. Analytical results indicate that the resulting networks have power-law distributions of degree, strength, weight and betweenness, a scale-free behavior for degree correlations, logarithmic small average path length and diameter with network size. The obtained properties are in agreement with empirical data observed in many real-life networks, which shows that the presented model may provide valuable insight into the real systems.     

Fuzzy overlapping community detection based on local random walk and multidimensional scaling

A fuzzy overlapping community is an important kind of overlapping community in which each node belongs to each community to different extents. It exists in many real networks but how to identify a fuzzy overlapping community is still a challenging task. In this work, the concept of local random walk and a new distance metric are introduced. Based on the new distance measurement, the dissimilarity index between each node of a network is calculated firstly. Then in order to keep the original node distance as much as possible, the network structure is mapped into low-dimensional space by the multidimensional scaling (MDS). Finally, the fuzzy c$c$-means clustering is employed to find fuzzy communities in a network. The experimental results show that the proposed algorithm is effective and efficient to identify the fuzzy overlapping communities in both artificial networks and real-world networks.     

Lie-optics, geometrical phase and nonlinear dynamics of self-focusing and soliton evolution in a plasma

It is useful to state propagation laws for a self-focusing laser beam or a soliton in group-theoretical form to be called Lie-optical form for being able to predict self-focusing dynamics conveniently and amongst other things, the geometrical phase. It is shown that the propagation of the gaussian laser beam is governed by a rotation group in a non-absorbing medium and by the Lorentz group in an absorbing medium if the additional symmetry of paraxial propagation is imposed on the laser beam. This latter symmetry, however, needs care in its implementation because the electromagnetic wave of the laser sees a different refractive index profile than the laboratory observer in this approximation. It is explained how to estimate this non-Taylor paraxial power series approximation. The group theoretical laws so-stated are used to predict the geometrical or Berry phase of the laser beam by a technique developed by one of us elsewhere. The group-theoretical Lie-optic (or ABCD) laws are also useful in predicting the laser behavior in a more complex optical arrangement like in a laser cavity etc. The nonlinear dynamical consequences of these laws for long distance (or time) predictions are also dealt with. Ergodic dynamics of an ensemble of laser beams on the torus during absorptionless self-focusing is discussed in this context. From the point of view of new physics concepts, we introduce a stroboscopic invariant torus and a stroboscopic generating function in classical mechanics that is useful for long-distance predictions of absorptionless self-focusing.     

Wetting of a plane with a narrow solvophobic stripe

ABSTRACT

We present a numerical study of a simple density functional theory model of fluid adsorption occurring on a planar wall decorated with a narrow deep stripe of a weaker adsorbing (relatively solvophobic) material, where wall-fluid and fluid-fluid intermolecular forces are considered to be dispersive. Both the stripe and outer substrate exhibit first-order wetting transitions with the wetting temperature of the stripe lying above that of the outer material. This geometry leads to a rich phase diagram due to the interplay between the pre-wetting transition of the outer substrate and an unbending transition corresponding to the local evaporation of liquid near the stripe. Depending on the width of the stripe, the line of unbending transitions merges with the pre-wetting line inducing a two-dimensional wetting transition occurring across the substrate. In turn, this leads to the continuous pre-drying of the thick pre-wetting film as the pre-wetting line is approached from above. Interestingly we find that the merging of the unbending and pre-wetting lines occurs even for the widest stripes considered. This contrasts markedly with the scenario where the outer material has the higher wetting temperature, for which the merging of the unbending and pre-wetting lines only occurs for very narrow stripes.