Vortex dynamics in R⁴

Banavara N. Shashikanth

The author studies the vortex dynamics of Euler's equations for a constant density fluid flow in R⁴ with special focus on singular Dirac delta distributions of the vorticity supported on two-dimensional surfaces (membranes). The self-induced velocity field of a membrane has a logarithmic divergence. A regularization done via the LIA then shows that the regularized velocity field is proportional to the mean curvature vector field of the membrane rotated by 90° in the plane of normals. The author also presents a Hamiltonian structure for the regularized self-induced motion of the membrane. The dynamics of the four-form ω ∧ ω is examined and it is shown that Ertel's vorticity theorem in R³ , for the constant density case, is a special case of this dynamics.

J. Math. Phys. 53, 013103 (2012)

Composite parameterization and Haar measure for all unitary and special unitary groups

Christoph Spengler, Marcus Huber, and Beatrix C. Hiesmayr

The authors adopt the concept of the composite parameterization of the unitary group U(d) to obtain a novel parameterization of the special unitary group SU(d), and furthermore, consider the Haar measure in terms of the introduced parameters. The authors show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d).

J. Math. Phys. 53, 013501 (2012)

The stochastisation hypothesis and the spacing of planetary systems

Jacky Cresson

The stochastisation hypothesis aims to provide a framework to deal with physical systems in random environment.It is applied here in two different cases: in the study of the dynamics of a protoplanetary nebula and in the chaotic long-term behaviour of a generic planetary system using previous works of Albeverio et al.

J. Math. Phys. 52, 113502 (2011)

On a generalization of Jacobi's elliptic functions and the double sine-Gordon kink chain

Michael Pawellek

The special properties of these functions are discussed, addition theorems are presented, and a list of indefinite integrals are given. As a physical application, it is shown that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in terms of generalized Jacobi functions.

J. Math. Phys. 52, 113701 (2011)

Fractal structure of ferromagnets: The singularity structure analysis

Victor K. Kuetche, Thomas B. Bouetou, and Timoleon C. Kofane

Following the Weiss-Tabor-Carnevale approach designed for studying the integrability properties of nonlinear partial differential equations, the singularity structure of a (2+1)-dimensional wave-equation describing the propagation of polariton solitary waves in a ferromagnetic slab were investigated.

J. Math. Phys. 52, 092903 (2011)

On polygonal relative equilibria in the N-vortex problem

M. Celli, E. A. Lacomba, and E. Pérez-Chavela

It is first shown that a relative equilibrium formed of a regular polygon and a possible vortex at the center, with more than three vertices on the polygon (two if there is a vortex at the center), requires equal vorticities on the polygon. We also provide an 8-vortex configuration, formed of two concentric squares making an angle of 45°, with uniform vorticity on each square, which is in relative equilibrium for any value of the vorticities.

J. Math. Phys. 52, 103101 (2011)

Internal heating driven convection at infinite Prandtl number

Jared P. Whitehead and Charles R. Doering

Derive here is an improved rigorous lower bound on the space and time averaged temperature of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries driven by uniform internal heating. A singular stable stratification is introduced as a perturbation to a non-singular background profile yielding ≥ 0.419 [R log R]^−1/4.

J. Math. Phys. 52, 093101 (2011)

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap

V. Achilleos, G. Theocharis, P. G. Kevrekidis, N. I. Karachalios, F. K. Diakonos, and D. J. Frantzeskakis

The model studied here, describing, an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, global bifurcation theory is employed to rigorously prove that—in the discrete regime—all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases).

J. Math. Phys. 52, 092701 (2011)

Twisted hierarchies associated with the generalized sine-Gordon equation

Hui Ma and Derchyi Wu

Derived in this paper are explicit interesting first and higher flows of twisted -hierarchies, justify that the one-dimensional systems of twisted -hierarchies for J = I_{q, n − q}(1 ≤ q ≤ n − 1), called the generalized sinh-Gordon equations, are the Gauss-Codazzi equations for n-dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo-Euclidean (2n − 1)-dimensional space 2q−12n−1 with index 2q − 1.

J. Math. Phys. 52, 093704 (2011)

Asymptotic evolution of quantum walks with random coin

A. Ahlbrecht, H. Vogts, A. H. Werner, and R. F. Werner

The authors study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain.

J. Math. Phys. 52, 042201 (2011)

When do measures on the space of connections support the triad operators of loop quantum gravity?

Hanno Sahlmann

Under what conditions do Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux? A precise mathematical formulation investigation is given. The flux-observables we consider play an important role in loop quantum gravity since they can be defined without recurse to a background geometry and they might also be of interest in the general context of quantization of non-Abelian gauge theories.

J. Math. Phys. 52, 012503 (2011)

The Χ²-divergence and mixing times of quantum Markov processes

K. Temme, M. J. Kastoryano, M. B. Ruskai4, M. M. Wolf, and F. Verstraete

The mixing time of a classical Markov chain is the time it takes for the chain to be close to its steady state distribution, starting from an arbitrary initial state. The ability to bound the mixing time is important, for example in the field of computer science, where the bound can be used to give an estimate for the running time of some probabilistic algorithm such as the Monte Carlo algorithm. A detailed analysis of quantum versions of the Χ²-divergence is introduced, which is applied to the investigation of mixing times of quantum Markov processes. In this context we analyze different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.

J. Math. Phys. 51, 122201 (2010)

Localization of multidimensional Wigner distributions

Elliott H. Lieb and Yaron Ostrover

A well known result of P. Flandrin states that a Gaussian uniquely maximizes the integral of the Wigner distribution over every centered disk in the phase plane. While there is no difficulty in generalizing this result to higher-dimensional polydisks, the generalization to balls is less obvious. In this note we provide such a generalization.

J. Math. Phys. 51, 102101 (2010)

The principle of locality: Effectiveness, fate, and challenges

Sergio Doplicher

The special theory of relativity and quantum mechanics merge in the key principle of quantum field theory, the principle of locality. We review some examples of its “unreasonable effectiveness” in giving rise to most of the conceptual and structural frame of quantum field theory, especially in the absence of massless particles.

J. Math. Phys. 51, 015218 (2010)

Statics and dynamics of magnetic vortices and of Nielsen–Olesen (Nambu) strings

S. Gustafson, I. M. Sigal, and T. Tzaneteas

We review recent works on statics and dynamics of magnetic vortices in the Ginzburg–Landau model of superconductivity and of Nielsen–Olesen (Nambu) strings in the Abelian–Higgs model of particle physics.

J. Math. Phys. 51, 015217 (2010)

Maximal violation of Bell inequalities by position measurements

J. Kiukas and R. F. Werner

We show that it is possible to find maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using only position measurements on a pair of entangled nonrelativistic free particles. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the two-time correlations between distant particle trajectories have a joint distribution, and hence cannot violate any Bell inequality.

J. Math. Phys. 51, 072105 (2010)

Spectral gap, and split property in quantum spin chains

Taku Matsui

We consider a class of ground states with spectral gap for quantum spin chains on an integer lattice and we prove that the factorization lemma of Hastings [“Topology and phases in fermionic systems,” J. Stat. Mech.: Theory Exp. 2008, L01001] implies split property (weak statistical independence) of left and right semi-infinite subsystems.

J. Math. Phys. 51, 015216 (2010)

Algebraic complementarity in quantum theory

Dénes Petz

This paper is an overview of the concept of complementarity, the relation to state estimation, to Connes–Størmer conditional (or relative) entropy, and to uncertainty relation. Complementary Abelian and noncommutative subalgebras are analyzed.

J. Math. Phys. 51, 015215 (2010)

Formulas for joint probabilities for the asymmetric simple exclusion process

Craig A. Tracy and Harold Widom

In earlier work, the authors obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process. Here, formulas are obtained for joint probabilities for several particles.

J. Math. Phys. 51, 063302 (2010)

Almost commuting matrices, localized Wannier functions, and the quantum Hall effect

Matthew B. Hastings and Terry A. Loring

For models of noninteracting fermions moving within sites arranged on a surface in three-dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are K-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry.

J. Math. Phys. 51, 015214 (2010)

Batalin–Vilkovisky integrals in finite dimensions

C. Albert, B. Bleile, and J. Fröhlich

Homological perturbation theory is introduced and used to develop the integration theory behind BV and to describe the BV quantization of a Lagrangian system with symmetries. Localization (illustrated in terms of Duistermaat–Heckman localization) as well as anomalous symmetries are discussed in the framework of BV.

J. Math. Phys. 51, 015213 (2010)

Thermodynamic limit for isokinetic thermostats

G. Gallavotti and E. Presutti

Thermostat models in space dimension d = 1,2,3 for nonequilibrium statistical mechanics are considered and it is shown that, in the thermodynamic limit, the motions of frictionless thermostats and isokinetic thermostats coincide.

J. Math. Phys. 51, 053303 (2010)

Developments in the theory of universality

Vieri Mastropietro

Recently, a rigorous foundation of several aspects of the theory of universality for statistical mechanics models with continuously varying has been reached; it has its root in the mapping of such systems into fermionic interacting theories and uses the modern renormalization group methods developed in the context of constructive quantum field theory.

J. Math. Phys. 51, 015212 (2010)

Exact results for ionization of model atomic systems

O. Costin, J. L. Lebowitz, C. Stucchio, and S. Tanveer

We review recent rigorous results concerning the ionization of model quantum systems by time-periodic external fields. The systems we consider consist of a single particle (electron) with a reference Hamiltonian H₀ = −Δ+V₀(x) (x ∊ ^d)  having both bound and continuum states.

J. Math. Phys. 51, 015211 (2010)

Twenty five years of two-dimensional rational conformal field theory

Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert

We give a condensed panoramic view of the development of two-dimensional rational conformal field theory in the last 25 years.

J. Math. Phys. 51, 015210 (2010)

From operator algebras to superconformal field theory

Yasuyuki Kawahigashi

We survey the operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.

J. Math. Phys. 51, 015209 (2010)

A quantum central limit theorem for sums of independent identically distributed random variables

V. Jakšić, Y. Pautrat, and C.-A. Pillet

We formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables.

J. Math. Phys. 51, 015208 (2010)

Correlation inequalities for quantum spin systems with quenched centered disorder

Pierluigi Contucci and Joel L. Lebowitz

It is shown that random quantum spin systems with centered disorder satisfy correlation inequalities previously proved in the classical case.

J. Math. Phys. 51, 023302 (2010)

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

Guido Gentile

Applications to both quasi-integrable Hamiltonian systems [Kolmogorov-Arnold-Moser theory] and non-Hamiltonian dissipative systems are discussed. The method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged.

J. Math. Phys. 51, 015207 (2010)

Hard-sphere fluids with chemical self-potentials

M. K.-H. Kiessling and J. K. Percus

The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container and in contact with either a heat and a matter reservoir, or just a heat reservoir.

J. Math. Phys. 51, 015206 (2010)

Nonadditivity of Rényi entropy and Dvoretzky's theorem

Guillaume Aubrun, Stanisław Szarek, and Elisabeth Werner

The goal of this note is to show that the analysis of the minimum output p-Rényi entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoretzky's theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden–Winter, disproving the additivity conjecture for the minimal output p-Rényi entropy (for p>1).

J. Math. Phys. 51, 022102 (2010)

Singularities of complex-valued solutions of the two-dimensional Burgers system

Dong Li and Yakov G. Sinai

We consider the two-dimensional viscous Burgers system without external forcing. For complex-valued solutions, due to the loss of maximum principle and energy estimates, smooth solutions can develop finite-time singularities. We construct an open set of six-parameter families of initial conditions such that the corresponding solutions exhibit blowups in finite time.

J. Math. Phys. 51, 015205 (2010)

Landau damping

C. Mouhot and C. Villani

In this note we present the main results from the recent work of Mouhot and Villani (“On the Landau damping,” arXiv:0904.2760), which for the first time establish Landau damping in a nonlinear context.

J. Math. Phys. 51, 015204 (2010)

Time asymptotics and entanglement generation of Clifford quantum cellular automata

Johannes Gütschow, Sonja Uphoff, Reinhard F. Werner, and Zoltán Zimborás

We study the time evolution of different classes of Clifford quantum cellular automata (CQCAs). We then identify invariant states and study convergence properties of classes of states, such as quasifree and stabilizer states. Finally, we consider the generation of entanglement analytically and numerically for stabilizer and quasifree states.

J. Math. Phys. 51, 015203 (2010)

Nonequilibrium, thermostats, and thermodynamic limit

G. Gallavotti and E. Presutti

The relation between thermostats of “isoenergetic” and “frictionless” kind is studied and their equivalence in the thermodynamic limit is proven in space dimension d=1,2 and for special geometries, d=3.

J. Math. Phys. 51, 015202 (2010)

Space and time from translation symmetry

A. Schwarz

We show that the notions of space and time in algebraic quantum field theory arise from translation symmetry if we assume asymptotic commutativity. We argue that this construction can be applied to string theory.

J. Math. Phys. 51, 015201 (2010)

Deligne–Beilinson cohomology and Abelian link invariants: Torsion case

F. Thuillier

For the Abelian Chern–Simons field theory, we consider the quantum functional integration over the Deligne–Beilinson cohomology classes and present an explicit path-integral nonperturbative computation of the Chern–Simons link invariants in SO(3) P3, a toy example of a 3-manifold with torsion.

J. Math. Phys. 50, 122301 (2009)

Total current fluctuations in the asymmetric simple exclusion process

Craig A. Tracy and Harold Widom

A limit theorem for the total current in the asymmetric simple exclusion process (ASEP) with step initial condition is proven. This extends the result of Johansson on TASEP to ASEP.

J. Math. Phys. 50, 095204 (2009)

A universal magnification theorem. II. Generic caustics up to codimension five

A. B. Aazami and A. O. Petters

We prove that for a generic family of general mappings between planes exhibiting singularities up to codimension five, and for a point in the target lying anywhere in the region giving rise to the maximum number of real preimages (lensed images), the total signed magnification of the preimages will always sum to zero. The wide field imaging surveys slated to be conducted by the Large Synoptic Survey Telescope are expected to find observational evidence for many of these higher-order caustic singularities.

J. Math. Phys. 50, 082501 (2009)

A mathematical theory of stochastic microlensing. I. Random time delay functions and lensing maps

A. O. Petters, B. Rider, and A. M. Teguia

From first principles, we initiate the development of a mathematical theory of stochastic microlensing—a central tool in probing dark matter on galactic scales.  The results of this paper are relevant to the theory of random fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing.

J. Math. Phys. 50, 072503 (2009)

Variational principle for the Wheeler–Feynman electrodynamics

Jayme De Luca

We adapt the formally defined Fokker action into a variational principle for the electromagnetic-two-body problem. We introduce properly defined boundary conditions to construct a functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an end point along each trajectory plus the respective segment of trajectory for the other particle inside the light cone of each end point.

J. Math. Phys. 50, 062701 (2009)

Abstract cluster expansion with applications to statistical mechanical systems

Suren Poghosyan and Daniel Ueltschi

We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions.

J. Math. Phys. 50, 053509 (2009)