Centroid Dynamics Research Articles

Combined centroid-envelope dynamics of intense, magnetically focused charged beams surrounded by conducting walls

In this paper we analyze the combined envelope-centroid dynamics of magnetically focused high-intensity charged beams surrounded by conducting walls. Similar to the case where conducting walls are absent, it is shown that the envelope and centroid dynamics decouple from each other. Mismatched envelopes still decay into equilibrium with simultaneous emittance growth, but the centroid keeps oscillating with no appreciable energy loss. Some estimates are performed to analytically obtain characteristics of halo formation seen in the full simulations.

Centroid motion in periodically focused beams

The role of the centroid dynamics in the transport of periodically focused particle beams is investigated. A Kapchinskij–Vladimirskij equilibrium distribution for an off-axis beam is derived. It is shown that centroid and envelope dynamics are uncoupled and that unstable regions for the centroid dynamics overlap with previously stable regions for the envelope dynamics alone. Multiparticle simulations validate the findings. The effects of a conducting pipe encapsulating the beam are also investigated. It is shown that the charge induced at the pipe may generate chaotic orbits which can be detrimental to the adequate functioning of the transport mechanism.

Combining the Semiclassical Initial Value Representation with Centroid Dynamics

A semiclassical initial value representation (SC-IVR) for centroid dynamics (CD) is derived to provide a more accurate description of quantum time correlation functions (TCFs). The time-dependent quasi-density operator (QDO) in SC-IVR-CD is shown to exhibit more exact behavior than that of the CMD approximation, thereby improving the calculated quantum TCFs. Three typical one-dimensional model potentials (weak anharmonic, double well, and quartic) were studied as examples in calculating TCFs with the SC-IVR-CD method. Importantly, for intermediate temperatures, SC-IVR-CD gives a more exact description of the coherent behavior of the quantum TCFs than does centroid molecular dynamics (CMD).

On the calculation of single-particle time correlation functions from Bose-Einstein centroid dynamics.

The calculation of single-particle time correlation functions using the Bose-Einstein centroid dynamics formalism is discussed. A new definition of the quasidensity operator is used to calculate the centroid force on a given particle for an anharmonic system. The force includes correlation effects due to quantum statistics and is used for the calculation of the classical-like dynamics of phase-space centroid variables within the centroid molecular dynamics approximation. Time correlation functions are then obtained for single-particle quantities. These correspond to the double-Kubo transform of exact quantum-mechanical correlation functions. The centroid dynamics results are compared to those of exact basis-set calculations and a good agreement is found. The level of accuracy is in fact the same as what was observed earlier for the calculation of center-of-mass correlation functions for Fermi-Dirac and Bose-Einstein statistics, and for any correlation function for Boltzmann statistics. These results show that it is now possible to use Bose-Einstein centroid molecular dynamics to calculate single-particle correlation functions for systems where quantum exchange effects are present.

Basic sound generation mechanisms in inviscid vortex interactions at low Mach number

The sound generation mechanisms during finite core vortex interactions at low Mach number are investigated in the present study. The theoretical deductions show clearly that the basic sound generation mechanisms are associated with the vortex core deformation and the vorticity centroid dynamics, independent of the vortex system. Such deductions are substantiated by numerical experiments with the interactions of two-dimensional vortices, vortex pairs and vortex rings. Detailed discussions on the similarities and differences between the sound generation processes of the two-dimensional and axisymmetric vortex systems are given. The relative importance of the two sound generation mechanisms in these vortex systems, their characteristics and interactions, which are hardly found in existing literature, are also examined. The present findings have also generalized and substantiated the previous results of the authors on the topic.

Centroid dynamics with quantum statistics

AbstractThis paper discusses the recent developments of a new molecular dynamics approach for the study of quantum dynamics for systems obeying Bose—Einstein statistics. The formalism is based on the mapping of a quantum mechanical system onto a set of phase space variables. These phase space variables are associated with the centroid (center‐of‐mass) of a Feynman path. We present the essential features of the formalism for the representation of operators and correlation functions and discuss some aspects of its practical implementation. We also present a recently developed simplified model for superfluid environments, based on an effective centroid potential approximation to the partition function. In the context of this model, we introduce a new expression for the calculation of the condensate fraction based on the off‐diagonal one‐particle density matrix. We perform calculations for a many‐particle system where the masses and pair interactions correspond to those of helium. The results show that this simplified model exhibits Bose—Einstein condensation below a certain characteristic temperature.

Direct experimental access to microscopic dynamics in liquid hydrogen.

We have obtained the double-differential incoherent neutron scattering cross section of liquid and solid parahydrogen in various thermodynamic conditions using TOSCA, a time-of-flight, inverse geometry, crystal analyzer spectrometer, operating at the pulsed neutron source ISIS. The measured cross section provides direct experimental access to the self part of the center-of-mass inelastic structure factor of the parahydrogen molecules in the system. Data have been corrected for the experimental effects and then analyzed in the framework of the Young-Koppel model and the Gaussian approximation. The velocity autocorrelation functions and their energy spectra have been obtained from a fitting procedure, making use of the quantum generalized Langevin equation and of model memory functions, and finally compared to the most recent results of both molecular centroid dynamics and self-consistent quantum mode-coupling theory. Some dynamic quantities were also related to simple equilibrium properties and simulated through a standard path integral Monte Carlo code. Results are very interesting but still urge for further developments of theoretical and dynamic simulation approaches, as well as for more extensive experimental efforts.

An effective centroid Hamiltonian and its associated centroid dynamics for indistinguishable particles in a harmonic trap

We show that incorporating the effects of Bose–Einstein or Fermi–Dirac quantum statistics within the centroid molecular dynamics formalism leads to additional correlations in the system due to exchange effects. In the case of Bose–Einstein statistics they appear as an additional attraction between physical particles while an additional repulsion is observed for Fermi–Dirac statistics. We show that we can account for these correlations through the effective centroid Hamiltonian. Within the approach based on the phase space centroid density, this Hamiltonian depends on centroid momenta in a nonclassical way. We illustrate the above findings using a simple model of two bosons and fermions in a harmonic potential. The average of a centroid variable along centroid trajectories based on such an effective Hamiltonian can be used to study the equilibrium properties of quantum systems. Is is also shown that the dynamics of the centroid variables derived from the quantum mechanical dynamics of the corresponding physical observables does not depend on exchange effects for a harmonic system.

Centroid-based methods for calculating quantum reaction rate constants: Centroid sampling versus centroid dynamics

A new method was recently introduced for calculating quantum mechanical rate constants from centroid molecular dynamics (CMD) simulations [E. Geva, Q. Shi, and G. A. Voth, J. Chem. Phys. 115, 9209 (2001)]. This new method is based on a formulation of the reaction rate constant in terms of the position-flux correlation function, which can be approximated in a well defined way via CMD. In the present paper, we consider two different approximated versions of this new method, which enhance its computational feasibility. The first approximation is based on propagating initial states which are sampled from the initial centroid distribution, on the classical potential surface. The second approximation is equivalent to a classical-like calculation of the reaction rate constant on the centroid potential, and has two distinct advantages: (1) it bypasses the problem of inefficient sampling which limits the applicability of the full CMD method at very low temperatures; (2) it has a well defined TST limit which is directly related to path-integral quantum transition state theory (PI-QTST). The approximations are tested on a model consisting of a symmetric double-well bilinearly coupled to a harmonic bath. Both approximations are quite successful in reproducing the results obtained via full CMD, and the second approximation is shown to provide a good estimate to the exact high-friction rate constants at very low temperatures.

Operator formulation of centroid dynamics for Bose–Einstein and Fermi–Dirac statistics

This paper is devoted to the development of an operator formulation of the recent extension of the centroid molecular dynamics method [J. Chem. Phys. 110, 3647 (1999); 111, 5303 (1999)] to boson and fermion systems. An operator calculus is used to rederive the basic equations of centroid dynamics. The following generalization to the case of systems of many indistinguishable particles is based on the use of a projection operator. Two different definitions of the quasi-density operator for bosonic and fermionic systems are suggested. The first definition allows an exact evaluation of equilibrium properties for systems with exchange effects using classical-like molecular dynamics calculations. The second one provides a formal justification of Bose–Einstein/Fermi–Dirac centroid dynamics with the same set of approximations as for Boltzmann statistics, and can be used to extract quantum dynamical information. In this case, the corresponding centroid correlation function can be related to a double Kubo transformed quantum mechanical one.

Path integral formulation of centroid dynamics for systems obeying Bose–Einstein statistics

This paper presents a formal foundation for the recent extension [J. Chem. Phys. 110, 3647 (1999)] of the centroid molecular dynamics (CMD) method to systems obeying Bose–Einstein statistics. It is shown that the introduction of centroid phase space coordinates corresponding to individual physical particles allows one to obtain (exact) canonical averages within the framework of the bosonic CMD method. It is also shown that formally exact expressions for quantum mechanical Kubo transformed correlation functions can be written in terms of individual particle centroids and that a CMD approximation can be introduced. Calculations for a bosonic trimer are used as an illustration of the new concepts introduced in this work.

A path integral centroid molecular dynamics method for Bose and Fermi statistics

We propose a path integral centroid molecular dynamics (CMD) method extended to Bose and Fermi statistics. An extended method of path integral molecular dynamics (PIMD) for such statistics is also developed as a technique of calculations of static properties. Bose PIMD and CMD simulations have been performed for bulk liquid 4 He and ideal Bose gas, respectively. The remnant of λ transition is observed for bulk liquid 4 He, while the effect of Bose statistics on the centroid dynamics spanning several nanoseconds is observed for ideal Bose gas.

Erratum: “A relationship between centroid dynamics and path integral quantum transition state theory” [J. Chem. Phys. 112, 8747 (2000)

Erratum: “A relationship between centroid dynamics and path integral quantum transition state theory” [J. Chem. Phys. <b>112</b>, 8747 (2000)

A Feynman path centroid dynamics approach for the computation of time correlation functions involving nonlinear operators

The centroid dynamics formalism is extended to the calculation of time correlation functions of nonlinear operators. It is shown that centroid correlation functions can be related to quantum mechanical ones via higher order Kubo-type transforms. A key step is the construction of the correlation functions from a mixed classical/semiclassical centroid representation of the operators. A general methodology is developed to relate these Kubo-type transforms to the desired quantum correlation functions. The approach is tested using a one-dimensional anharmonic potential for which the 〈x2x2(t)〉 and the 〈x3x3(t)〉 correlation functions are computed. Applications of this new approach are also outlined.

A relationship between centroid dynamics and path integral quantum transition state theory

The theory of Feynman path centroid dynamics is applied to the calculation of quantum barrier crossing rates. The formulation starts from the exact definition of the quantum survival probability of the reactant state, and the reaction rate is then defined as the steady-state limit of the decay rate of the survival probability. A formulation is given in terms of exact centroid dynamics. Then, based on an approximation for the initial reactant state and the centroid molecular dynamics (CMD) approximation for the dynamics, a new approximate rate expression is obtained which is equal to the path integral quantum transition state theory (PI-QTST) expression multiplied by a transmission factor of order unity. This factor varies with the choice of the dividing surface in the low temperature limit, but it is invariant to that choice at higher temperatures. It is then shown that the PI-QTST rate expression results from the quadratic barrier approximation for the calculation of the transmission factor only. The potential to use the new rate expression as an improved version of the PI-QTST is also tested for model systems. For certain choices of the dividing surface, it is shown that the new reaction rate expression results in improvement over the PI-QTST results. The overall formulation also yields a better understanding of the barrier crossing dynamics viewed from the centroid perspective and the rigorous origin of the PI-QTST formula.

Molecular dynamics algorithms for path integrals at constant pressure

Extended system path integral molecular dynamics algorithms have been developed that can generate efficiently averages in the quantum mechanical canonical ensemble [M. E. Tuckerman, B. J. Berne, G. J. Martyna, and M. L. Klein, J. Chem. Phys. 99, 2796 (1993)]. Here, the corresponding extended system path integral molecular dynamics algorithms appropriate to the quantum mechanical isothermal–isobaric ensembles with isotropic-only and full system cell fluctuations are presented. The former ensemble is employed to study fluid systems which do not support shear modes while the latter is employed to study solid systems. The algorithms are constructed by deriving appropriate dynamical equations of motions and developing reversible multiple time step algorithms to integrate the equations numerically. Effective parallelization schemes for distributed memory computers are presented. The new numerical methods are tested on model (a particle in a periodic potential) and realistic (liquid and solid para-hydrogen and liquid butane) systems. In addition, the methodology is extended to treat the path integral centroid dynamics scheme, [J. Cao and G. A. Voth, J. Chem. Phys. 99, 10070 (1993)], a novel method which is capable of generating semiclassical approximations to quantum mechanical time correlation functions.

Effect of fast velocity on the beam dynamics in a modified betatron accelerator

In this paper we have derived the equations that describe the slow motion (bounce) of the reference electron that is located at the beam centroid, taking into account its fast motion, for an electron beam that is confined in the magnetic field of a Modified Betatron Accelerator. The two equations for the guiding center (g.c.) of the centroid have been derived by averaging the centroid orbit over two time periods that are distinctly different. These two periods are associated with the normal modes of the system. Although our approach is not exact, it has the advantage that the fast velocity appears explicitly in the various expressions and thus its significance on the centroid dynamics can be rather easily assessed. A general conclusion of our results is that the fast velocity does not have a profound impact on the dynamics of the centroid, provided that the ratio ${\mathrm{u}}_{\mathrm{\ensuremath{\perp}}}$/${\mathrm{u}}_{0}$\ensuremath{\lesssim}0.5, where ${\mathrm{u}}_{\mathrm{\ensuremath{\perp}}}$ is the transverse and ${\mathrm{u}}_{0}$ is the total velocity. A very interesting conclusion of our work is that the centroid equilibrium position is insensitive to ${\mathrm{u}}_{\mathrm{\ensuremath{\perp}}}$, provided that ${\mathrm{u}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\lesssim}${\mathrm{u}}_{0}$/2. We explain this counter-intuitive result.