Initial or final values for semiclassical evolutions in the Weyl–Wigner representation

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

Trajectory design of UAV-aided energy-harvesting relay networks in the terahertz band

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

The on-line modelling system BOLCHEM consists of a meteorological limited area hydrostatic model (BOLAM) coupled with different gas chemistry models (SAPRC90 and CB-IV chemical mechanisms), an aerosol model (M7, work in progress) and a Lagrangian dispersion model for computation of forward and backward trajectories. The model is under development, but recent applications have shown its reliability for air quality studies. Future developments will focus on data assimilation of atmospheric composition and on feedback effects of composition on radiation and dynamics.KeywordsAerosol Optical DepthSaharan DustFriction Velocity ThresholdOzone SensitivityLagrangian TransportThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Observables in nonlinear and multidimensional infrared spectroscopy may be calculated from nonlinear response functions. Numerical challenges associated with the fully quantum-mechanical calculation of these dynamical response functions motivate the development of semiclassical methods based on the numerical propagation of classical trajectories. The Herman-Kluk frozen Gaussian approximation to the quantum propagator has been demonstrated to produce accurate linear and third-order spectroscopic response functions for thermal ensembles of anharmonic oscillators. However, the direct application of this propagator to spectroscopic response functions is numerically impractical. We analyze here the third-order response function with Herman-Kluk dynamics with the two related goals of understanding the origins of the success of the approximation and developing a simplified representation that is more readily implemented numerically. The result is a semiclassical approximation to the nth-order spectroscopic response function in which an integration over n pairs of classical trajectories connected by distributions of discontinuous transitions is collapsed to a single phase-space integration, in which n continuous trajectories are linked by deterministic transitions. This significant simplification is shown to retain a full description of quantum effects.

Air trajectory model applied to an in-depth diagnosis of potential diamondback moth infestations on the Canadian Prairies

Air backward trajectory calculations are commonly used in a variety of atmospheric analyses, in particular for source attribution evaluation. The accuracy of backward trajectory analysis is mainly determined by the quality and the spatial and temporal resolution of the underlying meteorological data set, especially in the cases of complex terrain. This work describes a new tool for the calculation and the statistical elaboration of backward trajectories. To take advantage of the high-resolution meteorological database of the Italian national air quality model MINNI, a dedicated set of procedures was implemented under the name of M-TraCE (MINNI module for Trajectories Calculation and statistical Elaboration) to calculate and process the backward trajectories of air masses reaching a site of interest. Some outcomes from the application of the developed methodology to the Italian Network of Special Purpose Monitoring Stations are shown to assess its strengths for the meteorological characterization of air quality monitoring stations. M-TraCE has demonstrated its capabilities to provide a detailed statistical assessment of transport patterns and region of influence of the site under investigation, which is fundamental for correctly interpreting pollutants measurements and ascertaining the official classification of the monitoring site based on meta-data information. Moreover, M-TraCE has shown its usefulness in supporting other assessments, i.e., spatial representativeness of a monitoring site, focussing specifically on the analysis of the effects due to meteorological variables.

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple dynamics in which a set of N coherent state coordinates evolves in forward and backward trajectories, while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward trajectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed.

A method for computing single trajectories representing boundary layer transport

In the context of high-energy particle physics, a reliable theory-experiment confrontation requires precise theoretical predictions. This translates into accessing higher-perturbative orders, and when we pursue this objective, we inevitably face the presence of complicated multi-loop Feynman integrals. There are serious bottlenecks to compute them with classical tools: the time to explore novel technologies has arrived. In this work, we study the implementation of quantum algorithms to optimize the integrands of scattering amplitudes. Our approach relies on the manifestly causal Loop-Tree Duality (LTD), which re-casts the loop integrand into phase-space integrals and avoids spurious non-physical singularities. Then, we codify this information in such a way that a quantum computer can understand the problem, and build Hamiltonians whose ground state are directly related to the causal representation. Promising results for generic families of multi-loop topologies are presented.

Though privacy-preserving consensus of multi-agent systems has been extensively studied, one issue remains unsolved. That is, the final consensus value may not be exactly the average of all the agents' initial values. In this paper, we investigate the privacy-preserving consensus problem of multiagent systems, where the final consensus value is the weighted average of all the agents' initial values with predefined weights. A noise generation mechanism is designed, and a novel privacy-preserving consensus algorithm is proposed. Theoretical analysis shows that the proposed algorithm can realize appointed-weight consensus while achieving privacy preserving.

The contact process is a particular case of birth-and-death processes on infinite particle configurations. We consider the contact processes on locally compact separable metric spaces. We prove the existence of a one-parameter set of invariant measures in the critical regime under the condition imposed on the associated Markov jump process. This condition means that any pair of independent trajectories of this jump process run away from each other. The general scheme can be applied to the contact process on the lattice in a heterogeneous and random environments as well as to the contact process on graphs and on manifolds.