Gauge-invariant poisson brackets for chromohydrodynamics

Noncanonical hamiltonian structures are presented both for Yang-Mills Vlasov plasmas, and for ideal fluids interacting with Yang-Mills fields. These hamiltonian structures are given Lie-algebraic interpretations.     

Hydrodynamical poisson brackets and local lie algebras

Zakharov's and Gibbons' observations on the relation between the Benney equation in hydrodynamics and the Vlasov equation in kinetic theory is put into general framework of infinite-dimensional Lie algebras associated to Lie algebra structures on rings of functions on finite-dimensional manifolds.     

On Hamilton systems with even and odd poisson brackets

We present a new bound for the decay rate of a metastable state (in the context of the Fokker-Planck equation) and indicate circumstances under which this bound provides a good estimate. The estimate is used to study the conjectured relation =k Im F between decay rate and free energy analytic continuation. A counterexample as well as an explanation for the relation's usual validity are provided.     

An extension of the fourier convolution theorem in terms of poisson brackets

The Fourier convolution theorem is extended to cover nonstationary and inhomogeneous phenomena. The Fourier transforms of input and transfer functions, F and K, are assumed to be slowly varying functions of x and t. The first-order corrections to the usual convolution theorem are given by Poisson brackets of F and K. These are calculated over k, ω, x and t. The method is applied to study induced currents in a plasma.     

Dirac structures and poisson reductions on poisson groupoids

The notion of characteristic pairs of Dirac structures was introduced by Liu in 2000. In this paper, the invariant Dirac structures on Poisson actions and pullback Dirac structures are characterized in terms of their characteristic pairs. Poisson homogeneous spaces and Poisson reduction are discussed.     

On poisson structure and curvature

We consider a curved space-time whose algebra of functions is the commutative limit of a noncommutative algebra and which has therefore an induced Poisson structure. In a simple example we determine a relation between this structure and the Riemann tensor.     

Parastatistics and Dirac brackets

Parastatistics (parafields) has been used in relation to several models of physical systems like the quark and the nuclear shell models. However, the physics of parafields is not completely clear. If classical para-Bose or para-Fermi variables could be constructed, then because of the correspondence principle some traces of the corresponding quantum properties could be found at the classical limit. In this way, by studying the simplestc-number systems some hints for the quantum of parafields could be expected.We introduce and discuss classical paravariables. We constructc-number para-Fermi variables in terms of coupled classical oscillators. Several similarities to the corresponding quantum case are observed. The results support Cusson's remark that systems described in terms of parastatistics may really be composite systems.     

A solvable problem in poisson statistics

Among the wide field of interest of Pierre Résibois, the exact solution of various problems in nonequilibrium statistical mechanics took a large place. Recently he used¹ the model of hard rods moving on a line to study some properties in kinetic theory. As a tribute to his memory, I present in this paper the derivation of the exact solution of a problem of Poisson noise.     

On the variational noncommutative poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.     

Dual Peierls brackets for strings

A new type of Peierls bracket which provides an additional representation of the Virasoro algebra, and which may be of relevance for a manifestly dual string field theory is constructed for the closed and open bosonic strings.     

Identity for harmonic oscillator brackets

An identity satisfied by the harmonic oscillator (Talmi-Moshinsky) brackets is derived from two equivalent methods for evaluating an integral often encountered in cluster model studies.     

On a poisson reduction for Gel'fand-Zakharevich manifolds

We formulate and discuss a reduction theorem for Poisson pencils associated with a class of integrable systems, defined on bi-Hamiltonian manifolds, recently studied by Gel'fand and Zakharevich. The reduction procedure is suggested by the bi-Hamiltonian approach to the separation of variables problem.     

Phase transitions driven by state-dependent poisson noise

Nonlinear systems driven by state-dependent Poisson noise are introduced to model the persistence of climatic anomalies in land-atmosphere interaction caused by the soil-moisture dependence of the frequency of rainfall events. It is found that these systems may give rise to bimodal probability distributions, while the state variable randomly persists around the preferential states because of transient dynamics that are opposite to the long-term behavior. Mean-field analysis and numerical simulations of the spatially distributed systems reveal a symmetry-breaking bifurcation for sufficiently strong spatial diffusive couplings and intermediate noise intensities. In such conditions, the initial development of spatial patterns is followed by a stable configuration, selected on the bases of the initial conditions in correspondence of the remnants of the modes of the uncoupled system.     

Degenerate odd poisson bracket on grassmann variables

A linear degenerate odd Poisson bracket (antibracket) realized solely on Grassmann variables is proposed. It is revealed that this bracket has at once three Grassmann-odd nilpotent Δ-like differential operators of the first, second and third orders with respect to the Grassmann derivatives. It is shown that these Δ-like operators, together with the Grassmann-odd nilpotent Casimir function of this bracket, form a finite-dimensional Lie superalgebra.     

On braided poisson and quantum inhomogeneous groups

The well known incompatibility between inhomogeneous quantum groups and the standardq-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantumISO(p, N - p) containingSO _q (p, N - p) with |q|=1 are constructed forN=2p, 2p + 1, 2p + 2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Poisson brackets in condensed matter physics

A general method of deriving nonlinear equations of hydrodynamics for both normal liquid and superfluid ⁴He and ³He, equations of the elasticity theory, equations for spin waves in magnets and spin glasses, liquid crystals, and so on is described. The method is based on the use of the Poisson “hydrodynamic” brackets. Hydrodynamic brackets are on the one hand, a classical limit of quantum commutators, on the other hand, Poisson brackets of certain symmetry groups inherent in the given problem: groups of general coordinate transformations for hydrodynamics and elasticity theory, groups of local spin rotations for spin waves, etc. Along with well-known examples nonlinear equations of the elasticity theory for bodies with impurities, dislocations and disclinations, and equations of motion for spin glasses and multisublattice magnets are studied.     

An operator approach to Poisson brackets

The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments.     

Poisson brackets,strings and membranes

We construct Poisson brackets at the boundaries of open strings and membranes with constant background fields which are compatible with their boundary conditions. The boundary conditions are treated as primary constraints which give infinitely many secondary constraints. We show explicitly that we need only two (the primary one and one of the secondary ones) constraints to determine the Poisson brackets of strings. We apply this to membranes by using canonical transformations. Received: 2 May 2002 / Revised version: 29 May 2002 / Published online: 16 August 2002 RID="a" ID="a" e-mail: tezuka@physics.s.chiba-u.ac.jp