4,567 research outputs found
Minimum energy states of the plasma pinch in standard and Hall magnetohydrodynamics
Axisymmetric relaxed states of a cylindrical plasma column are found
analytically in both standard and Hall magnetohydrodynamics (MHD) by complete
minimization of energy with constraints imposed by invariants inherent in
corresponding models. It is shown that the relaxed state in Hall MHD is the
force-free magnetic field with uniform axial flow and/or rigid azimuthal
rotation. The relaxed states in standard MHD are more complex due to the
coupling between velocity and magnetic field. Application of these states for
reversed-field pinches (RFP) is discussed
Janssen effect and the stability of quasi 2-D sandpiles
We present the results of three dimensional molecular dynamics study of
global normal stresses in quasi two dimensional sandpiles formed by pouring
mono dispersed cohesionless spherical grains into a vertical granular Hele-Shaw
cell. We observe Janssen effect which is the phenomenon of pressure saturation
at the bottom of the container. Simulation of cells with different thicknesses
shows that the Janssen coefficient is a function of the cell
thickness. Dependence of global normal stresses as well as on the
friction coefficients between the grains () and with walls () are
also studied. The results show that in the range of our simulations
usually increases with wall-grain friction coefficient. Meanwhile by increasing
while the other system parameters are fixed, we witness a gradual
increase in to a parameter dependent maximal value
Exponential renormalization
Moving beyond the classical additive and multiplicative approaches, we
present an "exponential" method for perturbative renormalization. Using Dyson's
identity for Green's functions as well as the link between the Faa di Bruno
Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the
composition of formal power series is analyzed. Eventually, we argue that the
new method has several attractive features and encompasses the BPHZ method. The
latter can be seen as a special case of the new procedure for renormalization
scheme maps with the Rota-Baxter property. To our best knowledge, although very
natural from group-theoretical and physical points of view, several ideas
introduced in the present paper seem to be new (besides the exponential method,
let us mention the notions of counterfactors and of order n bare coupling
constants).Comment: revised version; accepted for publication in Annales Henri Poincar
Renormalization: a quasi-shuffle approach
In recent years, the usual BPHZ algorithm for renormalization in perturbative
quantum field theory has been interpreted, after dimensional regularization, as
a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs,
with values in a Rota-Baxter algebra of amplitudes. We associate in this paper
to any such algebra a universal semi-group (different in nature from the
Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes
associated to Feynman graphs produces the expected operations: Bogoliubov's
preparation map, extraction of divergences, renormalization. In this process a
key role is played by commutative and noncommutative quasi-shuffle bialgebras
whose universal properties are instrumental in encoding the renormalization
process
Mixable Shuffles, Quasi-shuffles and Hopf Algebras
The quasi-shuffle product and mixable shuffle product are both
generalizations of the shuffle product and have both been studied quite
extensively recently. We relate these two generalizations and realize
quasi-shuffle product algebras as subalgebras of mixable shuffle product
algebras. As an application, we obtain Hopf algebra structures in free
Rota-Baxter algebras.Comment: 14 pages, no figure, references update
Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
In this article we continue to explore the notion of Rota-Baxter algebras in
the context of the Hopf algebraic approach to renormalization theory in
perturbative quantum field theory. We show in very simple algebraic terms that
the solutions of the recursively defined formulae for the Birkhoff
factorization of regularized Hopf algebra characters, i.e. Feynman rules,
naturally give a non-commutative generalization of the well-known Spitzer's
identity. The underlying abstract algebraic structure is analyzed in terms of
complete filtered Rota-Baxter algebras.Comment: 19 pages, 2 figure
What is the trouble with Dyson--Schwinger equations?
We discuss similarities and differences between Green Functions in Quantum
Field Theory and polylogarithms. Both can be obtained as solutions of fixpoint
equations which originate from an underlying Hopf algebra structure. Typically,
the equation is linear for the polylog, and non-linear for Green Functions. We
argue though that the crucial difference lies not in the non-linearity of the
latter, but in the appearance of non-trivial representation theory related to
transcendental extensions of the number field which governs the linear
solution. An example is studied to illuminate this point.Comment: 5 pages contributed to the proceedings "Loops and Legs 2004", April
2004, Zinnowitz, German
Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
We describe a unification of several apparently unrelated factorizations
arisen from quantum field theory, vertex operator algebras, combinatorics and
numerical methods in differential equations. The unification is given by a
Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff
formula in our study of the Hopf algebra approach of Connes and Kreimer to
renormalization in perturbative quantum field theory. There we showed that the
Birkhoff decomposition of Connes and Kreimer can be obtained from a certain
Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter
operator. We will explain how the same decomposition generalizes the
factorization of formal exponentials and uniformization for Lie algebras that
arose in vertex operator algebra and conformal field theory, and the even-odd
decomposition of combinatorial Hopf algebra characters as well as to the Lie
algebra polar decomposition as used in the context of the approximation of
matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy
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