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Elliptic functions are of prime importance for physics and mathematics students. We study the Weierstrass elliptic family of functions in the real domain as the solution to an initial value problem. Basic features of the Weierstrass elliptic family and the relation with the Jacobi elliptic functions are derived from the fundamental theory and methods of differential equations. Our approach allows introducing these functions in the undergraduate curriculum very early. reducing time and effort. [ABSTRACT FROM AUTHOR]

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds. [ABSTRACT FROM AUTHOR]

For discrete classical Molecular Dynamics (MD) obtained by the 'Verlet' algorithm (VA) with the time increment h there exists (for sufficiently small h) a shadow Hamiltonian H with energy Ẽ(h), for which the discrete particle positions lie on the analytic trajectories for H. The first order estimate of Ẽ(h) is employed to determine the relation with the corresponding energy, E, for the analytic dynamics with h = 0 and the zero-order estimate E0(h) of the energy for discrete dynamics, appearing in the literature for MD with VA. We derive a corresponding time reversible VA algorithm for canonical dynamics for the (NVT(h)) ensemble and determine the relations between the energies and temperatures for the different ensembles, including the (NVE0(h)) and (NVT0(h)) ensembles. The differences in the energies and temperatures are proportional with h2 and they are of the order of a few tenths of a percent for a traditional value of h. The relations between (NVẼ(h)) and (NVE), and (NVT(h)) and (NVT) are easily determined for a given density and temperature, and allow for using larger time increments in MD. The accurate determinations of the energies are used to determine the kinetic degrees of freedom in a system of N particles. It is 3N - 3 for a three dimensional system. The knowledge of the degrees of freedom is necessary when simulating small system, e.g., at nucleation. [ABSTRACT FROM AUTHOR]

Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie–Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the spherical midpoint method on s o (3) . In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie–Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature. [ABSTRACT FROM AUTHOR]

We investigate the current noise autocorrelations and cross correlations through two interacting quantum dots coupled to four external leads. The results indicate sign reversal of the cross correlation in different tunneling regions, respect to competing or supporting processes. By adjusting the dot-lead coupling in one dot, shot noise in another system can be modulated from sub- to super-Poissonian in the Coulomb blockade regime. Furthermore, spin injection can induce super-Poissonian shot noises in both dots due to the corporation of spin blockade and Coulomb blockade. [ABSTRACT FROM AUTHOR]

We investigate the electron spin relaxation of n-type InAs quantum wires by numerically solving the fully microscopic kinetic spin Bloch equations with the relevant scattering explicitly included. We find that the quantum-wire size and the growth direction influence the spin relaxation time by modulating the spin-orbit coupling. Due to intersubband scattering in connection with the spin-orbit interaction, spin-relaxation in quantum wires can show different characteristics from those in bulk or quantum wells and can be effectively manipulated by various means. [ABSTRACT FROM AUTHOR]

We calculate the shear piezocoefficient π44 in p-type Si with a 6×6 k·p Hamiltonian model using the Boltzmann transport equation in the relaxation-time approximation. Furthermore, we fabricate and characterize p-type silicon piezoresistors embedded in a (001) silicon substrate. We find that the relaxation-time model needs to include all scattering mechanisms in order to obtain correct temperature and acceptor density dependencies. The k·p results are compared to results obtained using a recent tight-binding (TB) model. The magnitude of the π44 piezocoefficient obtained from the TB model is a factor of 4 lower than experimental values; however, the temperature and acceptor density dependencies of the normalized values agree with experiments. The 6×6 Hamiltonian model shows good agreement between the absolute value of π44 and the temperature and acceptor density dependencies when compared to experiments. Finally, we present a fitting function of temperature and acceptor density to the 6×6 model that can be used to predict the piezoresistance effect in p-type silicon. [ABSTRACT FROM AUTHOR]

Incoherence in the controlled Hamiltonian is an important limitation on the precision of coherent control in quantum information processing. Incoherence can typically be modeled as a distribution of unitary processes arising from slowly varying experimental parameters. We show how it introduces artifacts in quantum process tomography and we explain how the resulting estimate of the superoperator may not be completely positive. We then go on to attack the inverse problem of extracting an effective distribution of unitaries that characterizes the incoherence via a perturbation theory analysis of the superoperator eigenvalue spectra. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross–Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross–Pitaevskii equation, but not for any other cases. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with 2p-order Hamiltonian systems with impulsive effects. We investigate the variational structure associated to this system. In addition, we obtain some results of multiple solutions for asymptotically linear 2p-order Hamiltonian systems via variational methods and critical point theorems. Meanwhile, some examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

Examines the classical, semiclassical and quantum mechanics of a chaotic Hamiltonian system. Adiabatic separation of the x and y motion; Connection between integrable approximation and the exact dynamics; Agreement between energy levels and energy spectrum.

Generally, the adiabatic approximation is defined via a projection onto an adiabatically moving electronic base. The Hamiltonian then is separated into three constituents (‘‘adiabatic,’’ ‘‘diagonal-nonadiabatic,’’ ‘‘nondiagonal-nonadiabatic’’). A model system is considered for which the correct zero-order eigenfunctions are known. It is shown that with these any single of the three Hamiltonian constituents yields diverging expectation values; only their sum yields convergence. From this the conclusion seems stringent that any result pertaining to electronic transitions, which is based on the adiabatic approximation, is an artifact. [ABSTRACT FROM AUTHOR]

The order conditions for modified Runge-Kutta methods are derived via the rooted trees. Symmetry and symplecticity conditions and exponential fitting conditions for modified diagonally implicit Runge-Kutta (DIRK) are considered. Three new exponentially fitted symmetric and symplectic diagonally implicit Runge-Kutta (EFSSDIRK) methods of respective second order and fourth order are constructed. Phase properties of the new methods are analyzed. The new EFSSDIRK methods are applied to several Hamiltonian problems and compared to the results obtained by the existing symplectic DIRK methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present a basic theory of fractional dynamics, i.e., the fractional conformal invariance of Mei symmetry, and find a new kind of conserved quantity led by fractional conformal invariance. For a dynamical system that can be transformed into fractional generalized Hamiltonian representation, we introduce a more general kind of single-parameter fractional infinitesimal transformation of Lie group, the definition and determining equation of fractional conformal invariance are given. And then, we reveal the fractional conformal invariance of Mei symmetry, and the necessary and sufficient condition whether the fractional conformal invariance would be the fractional Mei symmetry is found. In particular, we present the basic theory of fractional conformal invariance of Mei symmetry and it is found that, using the new approach, we can find a new kind of conserved quantity; as a special case, we find that an autonomous fractional generalized Hamiltonian system possesses more conserved quantities. Also, as the new method’s applications, we, respectively, find the conserved quantities of a fractional general relativistic Buchduhl model and a fractional Duffing oscillator led by fractional conformal invariance of Mei symmetry. [ABSTRACT FROM AUTHOR]

We derive a maximum principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables. [ABSTRACT FROM AUTHOR]

Transition probabilities for a class of two level systems described by explicitly time dependent Hamiltonians are considered. Provided only that the approach to the infinite time limit is non-trivial falling at least as fast as 1/t for large t, the transition probability takes a particularly simple form depending only on the value of Hamiltonian parameters in this limit [ABSTRACT FROM AUTHOR]

We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present an overview of the active area via the spectral linearization method for solving integrable systems. New examples of integrable systems, which have been discovered, are based on the so called Lax representation of the equations of motion. Through the Adler-Kostant-Symes construction, however, we can produce Hamiltonian systems on coadjoint orbits in the dual space to a Lie algebra whose equations of motion take the Lax form. We outline an algebraic-geometric interpretation of the ows of these systems, which are shown to describe linear motion on a complex torus. These methods are exemplified by several problems of integrable systems of relevance in mathematical physics. [ABSTRACT FROM AUTHOR]

The basic idea of a microscopic understanding of thermodynamics is to derive its main features from a microscopic probability distribution. In such a vein, we investigate the thermal statistics of quasi-probabilities's semiclassical analogs in phase space for the important case of quadratic Hamiltonians, focusing attention in the three more important instances, i.e. those of Wigner, - and Husimi distributions. Introduction of an effective temperature permits one to obtain a unified thermodynamic description that encompasses and unifies the three different quasi-probability distributions. This unified description turns out to be classical. [ABSTRACT FROM AUTHOR]

We study the existence of analytic first integrals of the complex Hamiltonian systems of the form H = 1 2 ∑ i = 1 2 p i 2 + V l ( q 1 , q 2 ) with the homogeneous polynomial potential V l ( q 1 , q 2 ) = α ( q 2 − i q 1 ) l ( q 2 + i q 1 ) k − l , l = 0 , … , k , α ∈ C ∖ { 0 } of degree k called exceptional potentials . In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials V 0 , V 1 , V k − 1 , V k and V k / 2 when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials . Here we prove that the exceptional potentials with k even different from V 0 , V 1 , V k − 1 , V k and V k / 2 , have no independent analytic first integral different from the Hamiltonian one. [ABSTRACT FROM AUTHOR]