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Feb 2013

Volume 54, Issue 2, Articles (02xxxx)

Issue Cover Spotlight Figure

J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan
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back to top Partial Differential Equations

Radiative transfer and diffusion limits for wave field correlations in locally shifted random media

Habib Ammari, Emmanuel Bossy, Josselin Garnier, Wenjia Jing, and Laurent Seppecher

J. Math. Phys. 54, 021501 (2013); http://dx.doi.org/10.1063/1.4790409 (18 pages)

Online Publication Date: 5 February 2013

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The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.
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05.60.-k Transport processes
02.50.-r Probability theory, stochastic processes, and statistics
02.60.Pn Numerical optimization
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

The equivalence of the Chern-Simons-Schrödinger equations and its self-dual system

Hyungjin Huh and Jinmyoung Seok

J. Math. Phys. 54, 021502 (2013); http://dx.doi.org/10.1063/1.4790487 (5 pages)

Online Publication Date: 5 February 2013

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In this paper, we discuss the equivalence of the second order Chern-Simons-Schrödinger equations and its first order self-dual system.
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11.15.Yc Chern-Simons gauge theory
03.65.Ge Solutions of wave equations: bound states

Multidimensional Yamada-Watanabe theorem and its applications to particle systems

Piotr Graczyk and Jacek Małecki

J. Math. Phys. 54, 021503 (2013); http://dx.doi.org/10.1063/1.4790507 (15 pages)

Online Publication Date: 15 February 2013

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We prove a multidimensional version of the Yamada-Watanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix-valued stochastic processes, called a “spectral” matrix Yamada-Watanabe theorem. The multidimensional Yamada-Watanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The β-versions of these particle systems are also considered.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
02.10.Yn Matrix theory
02.30.Gp Special functions
02.30.Hq Ordinary differential equations
02.50.Ey Stochastic processes

New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials

M. Schmuck

J. Math. Phys. 54, 021504 (2013); http://dx.doi.org/10.1063/1.4790656 (21 pages)

Online Publication Date: 20 February 2013

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We consider the Poisson-Nernst-Planck system which is well-accepted for describing dilute electrolytes as well as transport of charged species in homogeneous environments. Here, we study these equations in porous media whose electric permittivities show a strong contrast compared with the electric permittivity of the electrolyte phase. Our main result is the derivation of convenient low-dimensional equations, that is, of effective macroscopic porous media Poisson-Nernst-Planck equations, which reliably describe ionic transport. The contrast in the electric permittivities between liquid and solid phase and the heterogeneity of the porous medium induce strongly oscillating electric potentials (fields). In order to account for this specific physical scenario, we introduce a modified asymptotic multiple-scale expansion which takes advantage of the nonlinearly coupled structure of the ionic transport equations. This allows for a systematic upscaling resulting in a new effective porous medium formulation which shows a new transport term on the macroscale. Solvability of all arising equations is rigorously verified. The emergence of a new transport term indicates promising physical insights into the influence of the microscale material properties on the macroscale. Hence, systematic upscaling strategies provide a source and a prospective tool to capitalize intrinsic scale effects for scientific, engineering, and industrial applications.
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66.30.Dn Theory of diffusion and ionic conduction in solids
77.22.Ch Permittivity (dielectric function)
82.45.Gj Electrolytes

Long-time dynamics for a class of Kirchhoff models with memory

Marcio Antonio Jorge Silva and To Fu Ma

J. Math. Phys. 54, 021505 (2013); http://dx.doi.org/10.1063/1.4792606 (15 pages)

Online Publication Date: 25 February 2013

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This paper is concerned with a class of Kirchhoff models with memory effects utt+αΔ2u−div(|∇u|p−2∇u)− ∫ 0∞μ(s)Δ2u(t−s)ds−Δut+f(u) = h, defined in a bounded domain of mathN. This non-autonomous equation corresponds to a viscoelastic version of Kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations. Under assumptions that the exponent p and the growth of f(u) are up to the critical range, it turns out that the model corresponds to an autonomous dynamical system in a larger phase space, by adding a component which describes the relative displacement history. Then the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.
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05.45.-a Nonlinear dynamics and chaos
05.45.Df Fractals

Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan

J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Online Publication Date: 27 February 2013

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For a two-dimensional steady supersonic Euler flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic flow from the quiescent gas (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic flow in BV. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e., characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.
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47.40.Ki Supersonic and hypersonic flows
02.30.-f Function theory, analysis
47.32.cd Vortex stability and breakdown
47.35.-i Hydrodynamic waves
47.40.Dc General subsonic flows
47.40.Hg Transonic flows

A dyadic model on a tree

David Barbato, Luigi Amedeo Bianchi, Franco Flandoli, and Francesco Morandin

J. Math. Phys. 54, 021507 (2013); http://dx.doi.org/10.1063/1.4792488 (20 pages)

Online Publication Date: 28 February 2013

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We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.
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47.10.ad Navier-Stokes equations
47.27.-i Turbulent flows
02.10.Ox Combinatorics; graph theory
02.60.Lj Ordinary and partial differential equations; boundary value problems
back to top Representation Theory and Algebraic Methods

Generalized Bell states and principal realization of the Yangian Y(mathN)

Ming Liu, Chengming Bai, Mo-Lin Ge, and Naihuan Jing

J. Math. Phys. 54, 021701 (2013); http://dx.doi.org/10.1063/1.4789317 (11 pages)

Online Publication Date: 1 February 2013

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We prove that the action of the Yangian algebra Y(mathN) is better described by the principal generators on the tensor product of the fundamental representation and its dual. The generalized Bell states or maximally entangled states are permuted by the principal generators in a dramatically simple manner on the tensor product. Under the Yangian symmetry the new quantum number J2 is also explicitly computed, which gives an explanation for these maximally entangled states.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.Lx Quantum computation architectures and implementations
02.10.-v Logic, set theory, and algebra

Extended trigonometric Cherednik algebras and nonstationary Schrödinger equations with delta-potentials

J. T. Hartwig and J. V. Stokman

J. Math. Phys. 54, 021702 (2013); http://dx.doi.org/10.1063/1.4790566 (20 pages)

Online Publication Date: 11 February 2013

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We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
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05.30.Jp Boson systems
02.20.Uw Quantum groups
03.65.Fd Algebraic methods
03.65.Ge Solutions of wave equations: bound states

Group-theoretical derivation of Aharonov-Bohm phase shifts

C. R. Hagen

J. Math. Phys. 54, 021703 (2013); http://dx.doi.org/10.1063/1.4792234 (4 pages)

Online Publication Date: 20 February 2013

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The phase shifts of the Aharonov-Bohm effect are generally determined by means of the partial wave decomposition of the underlying Schrödinger equation. It is shown here that they readily emerge from an math(2,1) calculation of the energy levels employing an added harmonic oscillator potential which discretizes the spectrum.
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03.65.Ge Solutions of wave equations: bound states
03.65.Ta Foundations of quantum mechanics; measurement theory
02.20.-a Group theory
03.65.Fd Algebraic methods
back to top Quantum Mechanics

CPT-conserved effective mass Hamiltonians through first and higher order charge operator C in a supersymmetric framework

B. Bagchi, A. Banerjee, and A. Ganguly

J. Math. Phys. 54, 022101 (2013); http://dx.doi.org/10.1063/1.4792472 (9 pages)

Online Publication Date: 25 February 2013

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This paper examines the features of a generalized position-dependent mass Hamiltonian Hm in a supersymmetric framework in which the constraints of pseudo-Hermiticity and CPT are naturally embedded. Different representations of the charge operator are considered that lead to new mass-deformed superpotentials Wm(x) which are inherently PT-symmetric. The qualitative spectral behavior of Hm is studied and several interesting consequences are noted.
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11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
11.30.Pb Supersymmetry
back to top Quantum Information and Computation

Dimensions, lengths, and separability in finite-dimensional quantum systems

Lin Chen and Dragomir Ž. Ðoković

J. Math. Phys. 54, 022201 (2013); http://dx.doi.org/10.1063/1.4790405 (13 pages)

Online Publication Date: 6 February 2013

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Many important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set S of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in several low-dimensional systems. By using dimension arguments, we show that there exist separable states which are not convex combinations of d or less pure product states. For instance, such states exist in bipartite M⊗N systems when (M − 2)(N − 2) > 1. This solves an open problem proposed by DiVincenzo, Terhal and Thapliyal about 12 years ago. We prove that there exist a separable state ρ and a pure product state, whose mixture has smaller length than that of ρ. We show that any real ρ∈S, which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2⊗N system, the number r of product states can be taken to be r = rank ρ. We also show that the general multipartite separability problem can be reduced to the case of real states. Regarding the separability problem, we propose two conjectures describing S as a semialgebraic set, which may eventually lead to an analytic solution in some low-dimensional systems such as 2⊗4, 3⊗3, and 2⊗2⊗2.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.10.Ab Logic and set theory

Bipartite entanglement, spherical actions, and geometry of local unitary orbits

Alan Huckleberry, Marek Kuś, and Adam Sawicki

J. Math. Phys. 54, 022202 (2013); http://dx.doi.org/10.1063/1.4791681 (19 pages)

Online Publication Date: 21 February 2013

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We use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map onto coadjoint orbits of local transformations (unitary transformations which do not change entanglement). Thus, the geometry of coadjoint orbits provides a partial classification of different entanglement classes. To achieve the full classification, a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions, and distinguishable particles putting a special emphasis on the generality of the approach allowing one to consider all three cases in completely parallel manners.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
05.30.Fk Fermion systems and electron gas
05.30.Jp Boson systems
02.30.Uu Integral transforms
back to top Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes

A. Kempf, A. Chatwin-Davies, and R. T. W. Martin

J. Math. Phys. 54, 022301 (2013); http://dx.doi.org/10.1063/1.4790482 (22 pages)

Online Publication Date: 13 February 2013

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While a natural ultraviolet cutoff, presumably at the Planck length, is widely assumed to exist in nature, it is nontrivial to implement a minimum length scale covariantly. This is because the presence of a fixed minimum length needs to be reconciled with the ability of Lorentz transformations to contract lengths. In this paper, we implement a fully covariant Planck scale cutoff by cutting off the spectrum of the d’Alembertian. In this scenario, consistent with Lorentz contractions, wavelengths that are arbitrarily smaller than the Planck length continue to exist. However, the dynamics of modes of wavelengths that are significantly smaller than the Planck length possess a very small bandwidth. This has the effect of freezing the dynamics of such modes. While both wavelengths and bandwidths are frame dependent, Lorentz contraction and time dilation conspire to make the freezing of modes of trans-Planckian wavelengths covariant. In particular, we show that this ultraviolet cutoff can be implemented covariantly also in curved spacetimes. We focus on Friedmann Robertson Walker spacetimes and their much-discussed trans-Planckian question: The physical wavelength of each comoving mode was smaller than the Planck scale at sufficiently early times. What was the mode's dynamics then? Here, we show that in the presence of the covariant UV cutoff, the dynamical bandwidth of a comoving mode is essentially zero up until its physical wavelength starts exceeding the Planck length. In particular, we show that under general assumptions, the number of dynamical degrees of freedom of each comoving mode all the way up to some arbitrary finite time is actually finite. Our results also open the way to calculating the impact of this natural UV cutoff on inflationary predictions for the cosmic microwave background.
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04.20.Jb Exact solutions
98.80.Jk Mathematical and relativistic aspects of cosmology
04.20.Gz Spacetime topology, causal structure, spinor structure

Spectral action for a one-parameter family of Dirac operators on SU(2) and SU(3)

Alan Lai and Kevin Teh

J. Math. Phys. 54, 022302 (2013); http://dx.doi.org/10.1063/1.4790484 (21 pages)

Online Publication Date: 13 February 2013

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The one-parameter family of Dirac operators containing the Levi-Civita, cubic, and the trivial Dirac operators on a compact Lie group is analyzed. The spectra for the one-parameter family of Dirac Laplacians on SU(2) and SU(3) are computed by considering a diagonally embedded Casimir operator. Then the asymptotic expansions of the spectral actions for SU(2) and SU(3) are computed, using the Poisson summation formula and the two-dimensional Euler-Maclaurin formula, respectively. The inflation potential and slow-roll parameters for the corresponding pure gravity inflationary theory are generated, using the full asymptotic expansion of the spectral action on SU(2).
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11.30.Ly Other internal and higher symmetries
98.80.Bp Origin and formation of the Universe
98.80.Jk Mathematical and relativistic aspects of cosmology
04.50.-h Higher-dimensional gravity and other theories of gravity

Conformal field theories with infinitely many conservation laws

Ivan Todorov

J. Math. Phys. 54, 022303 (2013); http://dx.doi.org/10.1063/1.4790408 (14 pages)

Online Publication Date: 19 February 2013

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Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Unitary positive energy representations of scalar bilocal fields,” Commun. Math. Phys. 271, 223–246 (2007)10.1007/s00220-006-0182-2; e-print arXiv:math-ph/0604069v3; B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Infinite dimensional Lie algebras in 4D conformal quantum field theory,” J. Phys. A Math Theor. 41, 194002 (2008)10.1088/1751-8113/41/19/194002; e-print arXiv:0711.0627v2 [hep-th]]. Recently, conformal field theories “with higher spin symmetry” were considered for D = 3 by Maldacena and Zhiboedov [“Constraining conformal field theories with higher spin symmetry,” e-print arXiv:1112.1016 [hep-th]] where a similar result was obtained (exploiting earlier study of CFT correlators). We suggest that the proper generalization of the notion of a 2D chiral algebra to arbitrary (even or odd) dimension is precisely a conformal field theory (CFT) with an infinite series of conserved currents. We recast and complement (part of) the argument of Maldacena and Zhiboedov into the framework of our earlier work. We extend to D = 4 the auxiliary Weyl-spinor formalism developed by Giombi et al. [“A note on CFT correlators in three dimensions,” e-print arXiv:1104.4317v3 [hep-th]] for D = 3. The free field construction only follows for D > 3 under additional assumptions about the operator product algebra. The problem of whether a rational CFT in 4D Minkowski space is necessarily trivial remains open.
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11.15.-q Gauge field theories
11.25.Hf Conformal field theory, algebraic structures
11.30.Ly Other internal and higher symmetries
11.30.Rd Chiral symmetries
02.10.Ud Linear algebra
02.20.-a Group theory
02.30.Lt Sequences, series, and summability
04.62.+v Quantum fields in curved spacetime

Conformally invariant formalism for the electromagnetic field with currents in Robertson-Walker spaces

E. Huguet and J. Renaud

J. Math. Phys. 54, 022304 (2013); http://dx.doi.org/10.1063/1.4791688 (18 pages)

Online Publication Date: 22 February 2013

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We show that the Laplace-Beltrami equation □6a = j in (math6,η), η ≔ diag(+ − − − − +), leads under very moderate assumptions to both the Maxwell equations and the conformal Eastwood-Singer gauge condition on conformally flat spaces including the spaces with a Robertson-Walker metric. This result is obtained through a geometric formalism which gives, as byproduct, simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space.
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03.50.De Classical electromagnetism, Maxwell equations
02.30.-f Function theory, analysis
02.40.-k Geometry, differential geometry, and topology

Higher dimensional abelian Chern-Simons theories and their link invariants

L. Gallot, E. Pilon, and F. Thuillier

J. Math. Phys. 54, 022305 (2013); http://dx.doi.org/10.1063/1.4791677 (27 pages)

Online Publication Date: 25 February 2013

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The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons action, non trivial only in dimensions 4l + 3, whose parameter k is quantized. The generalized Wilson (2l + 1)-loops are observables of the theory and their charges are quantized. The Chern-Simons action is then used to compute invariants for links of (2l + 1)-loops, first on closed (4l + 3)-manifolds through a novel geometric computation, then on math4l+3 through an unconventional field theoretic computation.
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11.15.Yc Chern-Simons gauge theory
02.40.-k Geometry, differential geometry, and topology

Seiberg-Witten equations and non-commutative spectral curves in Liouville theory

Leonid Chekhov, Bertrand Eynard, and Sylvain Ribault

J. Math. Phys. 54, 022306 (2013); http://dx.doi.org/10.1063/1.4792241 (21 pages)

Online Publication Date: 25 February 2013

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We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the energy-momentum tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov, and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.
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11.10.Nx Noncommutative field theory
11.30.Rd Chiral symmetries
02.40.-k Geometry, differential geometry, and topology
11.25.Hf Conformal field theory, algebraic structures
02.30.Rz Integral equations

Wick rotation for quantum field theories on degenerate Moyal space(-time)

Harald Grosse, Gandalf Lechner, Thomas Ludwig, and Rainer Verch

J. Math. Phys. 54, 022307 (2013); http://dx.doi.org/10.1063/1.4790886 (21 pages)

Online Publication Date: 28 February 2013

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In this paper the connection between quantum field theories on flat noncommutative space(-times) in Euclidean and Lorentzian signature is studied for the case that time is still commutative. By making use of the algebraic framework of quantum field theory and an analytic continuation of the symmetry groups which are compatible with the structure of Moyal space, a general correspondence between field theories on Euclidean space satisfying a time zero condition and quantum field theories on Moyal Minkowski space is presented (“Wick rotation”). It is then shown that field theories transferred to Moyal space(-time) by Rieffel deformation and warped convolution fit into this framework, and that the processes of Wick rotation and deformation commute.
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11.10.Nx Noncommutative field theory
02.10.-v Logic, set theory, and algebra
back to top General Relativity and Gravitation

Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group

E. Minguzzi

J. Math. Phys. 54, 022501 (2013); http://dx.doi.org/10.1063/1.4789950 (38 pages)

Online Publication Date: 15 February 2013

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This work is devoted to the relativistic generalization of Chasles' theorem, namely, to the proof that every proper orthochronous isometry of Minkowski spacetime, which sends some point to its chronological future, is generated through the frame displacement of an observer which moves with constant acceleration and constant angular velocity. The acceleration and angular velocity can be chosen either aligned or perpendicular, and in the latter case the angular velocity can be chosen equal or smaller than the acceleration. We start reviewing the classical Euler's and Chasles' theorems both in the Lie algebra and group versions. We recall the relativistic generalization of Euler's theorem and observe that every (infinitesimal) transformation can be recovered from information of algebraic and geometric type, the former being identified with the conjugacy class and the latter with some additional geometric ingredients (the screw axis in the usual non-relativistic version). Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in detail. We prove its exponentiality and identify a causal semigroup and the corresponding Lie cone. Through the identification of new Ad-invariants we classify the conjugacy classes, and show that those which admit a causal representative have special physical significance. These results imply a classification of the inequivalent Killing vector fields of Minkowski spacetime which we express through simple representatives. Finally, we arrive at the mentioned generalization of Chasles' theorem.
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03.30.+p Special relativity
02.20.Sv Lie algebras of Lie groups
02.40.Dr Euclidean and projective geometries

Invariant classification of vacuum pp-waves

R. Milson, D. McNutt, and A. Coley

J. Math. Phys. 54, 022502 (2013); http://dx.doi.org/10.1063/1.4791691 (29 pages)

Online Publication Date: 20 February 2013

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We solve the equivalence problem for vacuum pp-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G2 and G3 sub-classes in terms of these invariants. It is known [J. M. Collins, “The Karlhede classification of type N vacuum spacetimes,” Class. Quantum Grav. 8, 1859–1869 (1991)10.1088/0264-9381/8/10/011] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q ⩽ 4 bound is sharp and explicitly describe all such maximal order solutions.
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04.20.Jb Exact solutions
02.10.Ud Linear algebra
04.20.Gz Spacetime topology, causal structure, spinor structure
back to top Dynamical Systems

Dynamics of the Heisenberg model and a theorem on stability

Leonidas Pantelidis

J. Math. Phys. 54, 022701 (2013); http://dx.doi.org/10.1063/1.4791703 (18 pages)

Online Publication Date: 20 February 2013

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We consider the general discrete classical Heisenberg model (HM) with z axis anisotropy and external magnetic field and show that its phase space is foliated into a family of invariant manifolds (the leaves) diffeomorphic to (S2)Λ, where Λ is the number of spins. We also show that the flow on each leaf S is Hamiltonian. Subsequently, we focus on the isotropic HM in the absence of external field. We discuss the rotational symmetry of the model and further analyze its phase space structure. We prove that the manifold F of longitudinal fixed points intersects each leaf S orthogonally. For a real local flow with a continuous symmetry, we show that the Lyapunov stability of invariant sets on an invariant subspace can be extended to the whole phase space. This general theorem is the main result of the article. We use it to show that, in the case of the isotropic HM, the ferromagnetic state and the antiferromagnetic state with non-zero total spin are both stable fixed points. The theorem does not apply at total spin zero, and indeed the AF state on an equal-spins leaf is found to be unstable.
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75.10.Jm Quantized spin models, including quantum spin frustration
back to top Classical Mechanics and Classical Fields

Two charges on a plane in a magnetic field: Special trajectories

M. A. Escobar-Ruiz and A. V. Turbiner

J. Math. Phys. 54, 022901 (2013); http://dx.doi.org/10.1063/1.4792478 (15 pages)

Online Publication Date: 26 February 2013

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The classical mechanics of two Coulomb charges on a plane (e1, m1) and (e2, m2) subject to a constant magnetic field perpendicular to the plane is considered. Special “superintegrable” trajectories (circular and linear) for which the distance between charges remains unchanged are indicated and their constants of motion are presented. The number of the independent constants of motion for the special trajectory is larger than for generic ones and hence they can be called “superintegrable.” A classification of pairs of charges for which special trajectories occur is given. The special trajectories are analyzed for three particular cases, namely that of two electrons, an electron-positron pair, and an electron-α-particle pair.
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45.05.+x General theory of classical mechanics of discrete systems
back to top Fluids

Extended thermodynamics of charged gases with many moments

M. C. Carrisi and S. Pennisi

J. Math. Phys. 54, 023101 (2013); http://dx.doi.org/10.1063/1.4789544 (18 pages)

Online Publication Date: 1 February 2013

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Recently a model with many moments for the description of relativistic gases has been studied and an exact closure has been found, depending on an arbitrary set of single variable functions. In the case of a charged gas and when the electromagnetic field acts as an external force, the exploitation of the entropy principle produces an additional condition. A closure compatible with this further condition has been found, when the highest order moment has an even number of free indexes. It amounts in restrictions on the arbitrary single variable functions appearing in the general case. They are polynomials of increasing degree with respect to equilibrium, which coefficients are arbitrary constants. When the highest order moment has an odd number M of free indexes the further condition is different from that appearing in the case M even and alternative techniques must be used to find a closure compatible with it. In this paper we take into account this last model and we find a closure compatible with the further condition. As well as in the case M even, also in the case M odd we find that the arbitrary single variable functions of the general theory are polynomials of increasing degree with respect to equilibrium, which coefficients are arbitrary constants.
Show PACS
05.70.Ce Thermodynamic functions and equations of state
02.10.Ab Logic and set theory
02.10.De Algebraic structures and number theory
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