Generalized Phase Space Research Articles

Reaction dynamics through kinetic transition states.

The transformation of a system from one state to another is often mediated by a bottleneck in the system's phase space. In chemistry, these bottlenecks are known as transition states through which the system has to pass in order to evolve from reactants to products. The chemical reactions are usually associated with configurational changes where the reactants and products states correspond, e.g., to two different isomers or the undissociated and dissociated state of a molecule or cluster. In this Letter, we report on a new type of bottleneck which mediates kinetic rather than configurational changes. The phase space structures associated with such kinetic transition states and their dynamical implications are discussed for the rotational vibrational motion of a triatomic molecule. An outline of more general related phase space structures with important dynamical implications is given.

SU‐E‐T‐505: Monte Carlo Simulations for TrueBeam Using 4D IAEA Phase Spaces

Purpose: To provide a tool for Monte Carlo simulations of dynamic beam delivery in the highly relevant situations where the full linac model is unavailable (such as is currently the case of the TrueBeam linac), which also includes the capability of reading and scoring 4D phase spaces. Methods: We have redesigned DOSXYZnrc source 20 to enable it to dynamically load a BEAMnrc shared library at run time and preserve synchronization between dynamic linac components and the beam motion in the patient geometry, even when starting from a phase space. We have enabled BEAMnrc and DOSXYZnrc to use the actual z coordinate of each particle, when reading a 3D IAEA phase space. Moreover, these codes can now read and output 4D phase spaces by reading/scoring the MU index of each particle. These 4D phase spaces can be used for further synchronization. Results: The updated source 20 has been validated, using film and ion chamber measurements (typically, differences around 1%), and comparisons with a commercial treatment planning system (3D gamma pass rates over 97% for a 3%‐3 mm criterion), for a variety of interesting clinical applications, including jaw‐tracking, non‐coplanar, and multiple‐arc VMAT, for both standard and flattening filter free (FFF) photon beams, starting from curved or general 4D IAEA phase space files. Conclusion: It is now possible to use source 20 for Monte Carlo simulations of TrueBeam and other modern radiotherapy technologies, starting from phase spaces (including curved or, more generally, 4D IAEA phase spaces). The simulations are performed in a single run, continuously, regardless of the complexity of the beam delivery. The new features have been extensively tested in clinical applications, with excellent results. The new 4D phase space capability could prove invaluable in other applications, such as in vivo portal dosimetry or adaptive planning. This work was supported, in part, by research grants provided by British Columbia Cancer Agency and Varian Medical Systems.

Kinetic modeling for dye removal using polyelectrolyte enhanced ultrafiltration

A generalized kinetic model is proposed for the first time for dye removal using polyelectrolytes in application of polyelectrolyte enhanced ultrafiltration. Three polyelectrolyte-dye systems, reported in the literature have been taken up for case studies. Different cases, namely, nature of dye and polyelectrolyte system and their concentration, effect of solution pH and electrolyte concentration have been included in the general framework of the modeling. The equilibrium constants are evaluated by minimizing the errors involved in the measured and experimental values of dye retention data. The matching between the calculated and the experimental data is found to be adequate. A general phase space analysis involving the equilibrium constants has also been carried out to determine the region of feasible solution, in order to facilitate dye removal using engineered polyelectrolyte.

Rapid Short-Time Path Planning for Phase Space

This paper addresses optimal motion for general machines. Approximation for optimal motion requires a global path planning algorithm that precisely calculates the whole dynamics of a machine in a brief calculation. We propose a path planning algorithm that consists of path searching and pruning algorithms. The pruning algorithmis based on our analysis of state resemblance in general phase space. To confirm precision, calculation cost, optimality and applicability of the proposed algorithm, we conducted several shortest time path planning experiments for the dynamic models of double inverted pendulums. Precision to reach the goal states of the pendulums was better than other algorithms. Calculation cost was 58 times faster at least. We could tune optimality of proposed algorithm via resolution parameters. A positive correlation between optimality and resolutions was confirmed. Applicability was confirmed in a torque based position and velocity feedback control simulation. As a result of this simulation, the double inverted pendulums tracked planned motion under noise while keeping within torque limitations.

Szilard's Perpetuum Mobile

In a previous article, we have demonstrated by a general phase space argument that a Maxwellian Demon is compatible with statistical mechanics. In this article, we show how this idea can be put to work in the prevalent model of the Demon, namely, a particle-in-a-box, used, for example, by Szilard and Bennett. In the literature, this model is used in order to show that a Demon is incompatible with statistical mechanics, either classical or quantum. However, we show that a detailed phase space analysis of this model illustrates that a Maxwellian Demon is compatible with statistical mechanics.

Occupation measure functionals in merging phase space

This paper deals with the averaging and diffusion approximation for occupation time of general phase space semi-Markov processes under split with simple and double merging in a series scheme. These results seem to be particularly interesting since the variance coefficient is obtained in a simple form. The method of proof of weak convergence results here is based on the construction of the compensating operator of the extended Markov renewal processes and on the solution of the singular perturbation problem of this operator acting on test functions.

TOPOLOGICAL ASPECTS OF THE SPECIFIC HEAT

We describe how the specific heat of a quantum system is related to a positive definite metric defined on the generalized phase space in which the dynamics and thermodynamics of the system take place. This relationship is given through the components of a second-rank covariant metric tensor, enhancing a topological nature of the specific heat. We also present two examples where it can be seen how the uncertainty principle imposes strong constraints on the values achieved by the specific heat showing its inherent quantum nature.

Coherent state approach to the cross-collisional effects in the population dynamics of a two-mode Bose–Einstein condensate

We reanalyze the non-linear population dynamics of a Bose–Einstein condensate (BEC) in a double well trap considering a semiclassical approach based on a time dependent variational principle applied to coherent states associated to SU(2) group. Employing a two-mode local approximation and hard sphere type interaction, we show in the Schwinger’s pseudo-spin language the occurrence of a fixed point bifurcation that originates a separatrix of motion on a sphere. This separatrix corresponds to the borderline between two dynamical regimes of Josephson oscillations and mesoscopic self-trapping. We also consider the effects of interaction between particles in different wells, known as cross-collisions. Such terms are usually neglected for traps sufficiently far apart, but recently it has been shown that they contribute to the effective tunneling constant with a factor growing linearly with the particle number. This effect changes considerably the effective tunneling of the system for sufficiently large number of trapped atoms, in perfect accord with experimental data. Finally, we identify analytically the transition parameter associated to the bifurcation in the generalized phase space of the model with cross-collision terms, and show how the dynamical regime depends on the initial conditions of the system and the collisional parameters values.

Spin-like current from phase space distributions

The spin 0 generalized phase space approach provides a general expression for local current which depends on the choice of the distribution function and generally deviates from the Schrödinger current. It is shown that the continuity equation restricts the admissible bilinear distributions such that the current has a unique dependence on the wavefunction and coincides with the non-relativistic limit of the relativistic spin current for a spin eigenstate, up to a constant vector. Examples of non-bilinear distributions that have the latter property are given.

PHASE-SPACE CONSTRAINTS ON NEUTRINO LUMINOSITIES

While the importance of phase space constraints for gravitational clustering of neutrinos (which are fermions) is well recognized, the explicit use of such constraints to limit neutrino emission from ultra energetic sources has not been stressed. Special and general relativistic phase space constraints are shown to limit neutrino luminosities in compact sources in various situations.

The disorder problem for stochastic jump processes

A Bayes formulation for the problem of rapid detecting appeared disorders in the probability characteristics of the observed stochastic jump process with values in a sufficiently general phase space is presented. It is supposed that the time of disorder appearance does not depend on the evolution of the observed process and coincides with the moment of one of the process jumps. Recurrent equations for a posteriori disorder probability are obtained. The lower bound for the optimal moment of alarming the happened disorder for minimizing the sum of false alarm probability (without disorder) and the mean time of alarm delay is found.

DEFORMATION QUANTIZATION FOR COUPLED HARMONIC OSCILLATORS ON A GENERAL NONCOMMUTATIVE SPACE

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two-coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.

Fractal diffusion processes in particle physics

A series of consistent measurements of kinematic variables for pion diffraction production processes by pions with an initial momentum of about 4 GeV/c were analyzed: π+ + p → p + 2π+ + π− and π− + p → p + 2π− + π+. The Hurst method analysis discovered the presence of the memory effect for both data arrays. The distributions of the transition probability density appeared to seek some equilibrium shape, characteristic of the fractal Brownian motion (FBM). The process can be defined by the special diffusion Fokker-Planck equation (FPE). The obtained values of Hurst coefficient 0.5 < H < 1, which is a parameter of FPE, mean that the processes explored are realized in fractal generalized phase space with fractional dimension.

A FURTHER STUDY ON NON-ABELIAN PHASE SPACES: LEFT-SYMMETRIC ALGEBRAIC APPROACH AND RELATED GEOMETRY

The notion of non-abelian phase space of a Lie algebra was first formulated and then discussed by Kuperschmidt. In this paper, we further study the non-abelian phase spaces in terms of left-symmetric algebras. We interpret the natural appearance of left-symmetric algebras from the intrinsic algebraic properties and the close relations with the classical Yang–Baxter equation. Furthermore, using the theory of left-symmetric algebras, we study some interesting geometric structures related to phase spaces. Moreover, we also discuss the generalized phase spaces with certain non-trivial algebraic structures on the dual spaces.

Generalized phase space projection for nonlinear noise reduction

Improved phase space projection methods, adapted from related work in the linear signal processing field based on subspace decomposition, are presented for application to the problem of additive noise reduction in the context of phase space analysis. These methods improve upon existing methods such as Broomhead–King singular spectrum analysis projection by minimizing overall signal distortion subject to constraints on the residual error, rather than using a direct least-squares fit. This results in a range of weighted projections which estimate and compensate for the portion of the principal component's singular values corresponding to noise rather than signal energy, and which include least-squares (LS) and least minimum mean square error (LMMSE) as subcases. The nature of phase space covariance, the key element in construction of the projection matrix, is examined across global phase spaces as well as within local neighborhood regions. The resulting algorithm, illustrated on a noisy Henon map as well as on the task of speech enhancement, is applicable to a wide variety of nonlinear noise reduction tasks.

Kac–Moody theories for colored phase space (quantum Hall) droplets

We derive the canonical structure and Hamiltonian for arbitrary deformations of a higher-dimensional (quantum Hall) droplet of fermions with spin or color on a general phase space manifold. Gauge fields are introduced via a Kaluza–Klein construction on the phase space. The emerging theory is a nonlinear higher-dimensional generalization of the gauged Kac–Moody algebra. To leading order in ℏ this reproduces the edge state chiral Wess–Zumino–Witten action of the droplets.

Adiabatic charge transport and the Kubo formula for Landau‐type Hamiltonians

AbstractThe adiabatic charge transport is investigated in a two‐dimensional Landau model perturbed by a bounded potential at zero temperature. We show that if the Fermi level lies in a spectral gap, then in the adiabatic limit the accumulated excess Hall charge is given by the linear response Kubo‐Štreda formula. The proof relies on the expansion of Nenciu, some generalized phase space estimates, and a bound on the speed of propagation. © 2004 Wiley Periodicals, Inc.

Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H , as canonically determined subsets of the space T s of symmetric trace-class operators, is investigated. The space T s is identified with the B -space predual of the Lie-algebra L ( H ) s of the Lie group U . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P( H ) . Closeness of all the orbits lying in T s is also proved.

Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H , as canonically determined subsets of the space T s of symmetric trace-class operators, is investigated. The space T s is identified with the B-space predual of the Lie-algebra L( H) s of the Lie group U . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “one-dimensional” orbit, i.e. for the projective Hilbert space P( H) . Closeness of all the orbits lying in T s is also proved.

On Markov-Additive Jump Processes

In 1995, Pacheco and Prabhu introduced the class of so-called Markov-additive processes of arrivals in order to provide a general class of arrival processes for queueing theory. In this paper, the above class is generalized considerably, including time-inhomogeneous arrival rates, general phase spaces and the arrival space being a general vector space (instead of the finite-dimensional Euclidean space). Furthermore, the class of Markov-additive jump processes introduced in the present paper is embedded into the existing theory of jump processes. The best known special case is the class of BMAP arrival processes.