The gauging of two-dimensional bosonic sigma models on world-sheets with defects

We extend our analysis of the gauging of rigid symmetries in bosonic two-dimensional sigma models with Wess-Zumino terms in the action to the case of world-sheets with defects. A structure that permits a non-anomalous coupling of such sigma models to world-sheet gauge fields of arbitrary topology is analysed, together with obstructions to its existence, and the classification of its inequivalent choices.Comment: 94 pages, 1 figur

Evolution of collision numbers for a chaotic gas dynamics

We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time-integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger number of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.Comment: 4 pages, published versio

THEORY OF INTEGRAL INDIVIDUALITY BY V. S. MERLIN: HISTORY AND NOWADAYS

The study is devoted to overview and analysis of V. S. Merlin’ theory of integral individuality.The aim of the study is to reveal a system’s background of the theory of integral individuality; to designate its current issues and to put new tasks of its further advancement.Methodology and research methods. Problematic and comparative analyses are used. A systematization of the main assumptions of the theory by V. S. Merlin shows that it is based on a general systemic approach and current ideas about integration.Results. It is demonstrated that the system-based approach provides a multi-focus perspective to view the integral individuality. Mostly, the following system ideas are embedded in V. S. Merlin’s theory. They are the concepts of structural levels, teleology, and polymorphism. With respect to the theory of integral individuality, a human is shown as a big system. It consists of a hierarchical set not included in each other, but relatively autonomous operative multilevel subsystems. They link one to another in a polymorphic multi-valued (many-tomany) way. The main features of integral individuality are seen as the hierarchical arrangement and levels, integration and differentiation, teleology and causality, flexibility of polymorphic links (between levels) and rigidity of causal links (within levels). In spite of its maturity, this theory can be put in a further progress. This perspective has been elaborated based on three key ideas – multi-quality, commonality, and isomerism.Scientific novelty. The routes of the phenomenon of integral individuality are uncovered. Its main properties are described: a systemic version of integrity, hierarchy, and polymorphism. Some topical problems are highlighted within the theory of integral individuality. Next tasks can be set to further develop the theory of integral individuality. They focus on shift from the systemic viewpoint to a multi-systemic outline, to combine integrity and commonality, to provide an isomerism coming from the polymorphic framework.Practical significance. The materials and thesis stressed in this article can be useful for researchers studying holistic conceptions of human. The theory of integral individuality can guide investigations designated to test Merlin’s assumptions under various conditions

Drift of particles in self-similar systems and its Liouvillian interpretation

We study the dynamics of classical particles in different classes of spatially extended self-similar systems, consisting of (i) a self-similar Lorentz billiard channel, (ii) a self-similar graph, and (iii) a master equation. In all three systems the particles typically drift at constant velocity and spread ballistically. These transport properties are analyzed in terms of the spectral properties of the operator evolving the probability densities. For systems (i) and (ii), we explain the drift from the properties of the Pollicott-Ruelle resonance spectrum and corresponding eigenvectorsComment: To appear in Phys. Rev.

A convenient criterion under which Z_2-graded operators are Hamiltonian

We formulate a simple and convenient criterion under which skew-adjoint Z_2-graded total differential operators are Hamiltonian, provided that their images are closed under commutation in the Lie algebras of evolutionary vector fields on the infinite jet spaces for vector bundles over smooth manifolds.Comment: J.Phys.Conf.Ser.: Mathematical and Physical Aspects of Symmetry. Proc. 28th Int. colloq. on group-theoretical methods in Physics (July 26-30, 2010; Newcastle-upon-Tyne, UK), 6 pages (in press

Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases I: Equilibrium Systems

We compute the Lyapunov spectrum and the Kolmogorov-Sinai entropy for a moving particle placed in a dilute, random array of hard disk or hard sphere scatterers - i.e. the dilute Lorentz gas model. This is carried out in two ways: First we use simple kinetic theory arguments to compute the Lyapunov spectrum for both two and three dimensional systems. In order to provide a method that can easily be generalized to non-uniform systems we then use a method based upon extensions of the Lorentz-Boltzmann (LB) equation to include variables that characterize the chaotic behavior of the system. The extended LB equations depend upon the number of dimensions and on whether one is computing positive or negative Lyapunov exponents. In the latter case the extended LB equation is closely related to an "anti-Lorentz-Boltzmann equation" where the collision operator has the opposite sign from the ordinary LB equation. Finally we compare our results with computer simulations of Dellago and Posch and find very good agreement.Comment: 48 pages, 3 ps fig

Why nonlocal recursion operators produce local symmetries: new results and applications

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all known today recursion operators and are much easier to verify than those found in earlier work. We also give explicit formulas for the nonlocal parts of higher recursion operators, Poisson and symplectic structures of integrable systems in (1+1) dimensions. Using these two results we prove, under some natural assumptions, the Maltsev--Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.Comment: 10 pages, LaTeX 2e, final versio

Crystallization of the ordered vortex phase in high temperature superconductors

The Landau-Khalatnikov time-dependent equation is applied to describe the crystallization process of the ordered vortex lattice in high temperature superconductors after a sudden application of a magnetic field. Dynamic coexistence of a stable ordered phase and an unstable disordered phase, with a sharp interface between them, is demonstrated. The transformation to the equilibrium ordered state proceeds by movement of this interface from the sample center toward its edge. The theoretical analysis dictates specific conditions for the creation of a propagating interface, and provides the time scale for this process.Comment: 8 pages and 3 figures; to be published in Phys. Rev. B (Rapid Communications section

Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients

The foundations of the chaotic scattering theory for transport and reaction-rate coefficients for classical many-body systems are considered here in some detail. The thermodynamic formalism of Sinai, Bowen, and Ruelle is employed to obtain an expression for the escape-rate for a phase space trajectory to leave a finite open region of phase space for the first time. This expression relates the escape rate to the difference between the sum of the positive Lyapunov exponents and the K-S entropy for the fractal set of trajectories which are trapped forever in the open region. This result is well known for systems of a few degrees of freedom and is here extended to systems of many degrees of freedom. The formalism is applied to smooth hyperbolic systems, to cellular-automata lattice gases, and to hard sphere sytems. In the latter case, the goemetric constructions of Sinai {\it et al} for billiard systems are used to describe the relevant chaotic scattering phenomena. Some applications of this formalism to non-hyperbolic systems are also discussed.Comment: 35 pages, compressed file, follow directions in header for ps file. Figures are available on request from [email protected]

Lyapunov Exponents from Kinetic Theory for a Dilute, Field-driven Lorentz Gas

Positive and negative Lyapunov exponents for a dilute, random, two-dimensional Lorentz gas in an applied field, $\vec{E}$, in a steady state at constant energy are computed to order $E^{2}$. The results are: $\lambda_{\pm}=\lambda_{\pm}^{0}-a_{\pm}(qE/mv)^{2}t_{0}$ where $\lambda_{\pm}^{0}$ are the exponents for the field-free Lorentz gas, $a_{+}=11/48, a_{-}=7/48$, $t_{0}$ is the mean free time between collisions, $q$ is the charge, $m$ the mass and $v$ is the speed of the particle. The calculation is based on an extended Boltzmann equation in which a radius of curvature, characterizing the separation of two nearby trajectories, is one of the variables in the distribution function. The analytical results are in excellent agreement with computer simulations. These simulations provide additional evidence for logarithmic terms in the density expansion of the diffusion coefficient.Comment: 7 pages, revtex, 3 postscript figure