Split-operator technique for propagating phase space functions: Exploring chaotic, dissipative and relativistic dynamics

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Abstract

We conduct a comprehensive analysis of the split-operator method for propagating phase space distribution functions in different scenarios of classical mechanics. A numerical method based on Fast Fourier Transform allows to propagate almost any sampled or exact localized initial state, as well as the direct calculation of current densities in phase space. In order to demonstrate the potential of the proposed numerical scheme some simulations involving chaotic, dissipative and relativistic dynamics are performed. In the conducted simulations, dynamical functions like autocorrelations as well as the detailed structures in phase space are discussed. We find that the split-operator technique demonstrates the effectiveness for studying time evolution of interacting one-dimensional classical systems.

Introduction

The split-operator technique (SOT) has been considered one of the most plausible alternatives to perform the time-evolution of localized initial states in quantum systems. It has been successfully applied for the solution of the Schrödinger equation in coordinates as well as phase space representations  [1], [2], [3], [4]. It has become an ideal for propagating initial states in many branches of physics, for instance, applications involving wave-packet dynamics in molecular systems  [5], [6], [7] and frequent use of quantum molecular dynamics  [8], [9]. In the recent past, Finite difference has adopted the SOT scheme to simulate propagation phenomena in quantum mechanics and quantum field theory  [10], [11]. Moreover, the versatility of this hybrid model becomes an advantage of using both modeling schemes; Noticeably, to simulate the scattering of sound in the time-dependent domain  [12], and the propagation of waves through optical fibers  [13]. Recently, the SOT has exploited the power of high performance computing and it has been implemented in modern hardware architectures for solving time-dependent Schrödinger equations and time-dependent Dirac equations  [14] as well as the Klein–Gordon equation  [15]. In contrast to quantum mechanics, little is known about the benefits of this approach in the context of classical mechanics, in fact, there is only one work in the literature where this issue has been addressed in detail. More precisely, the pioneering work of Dattoli et al.  [16] has developed a numerical method based on SOT for propagating the Liouville equation, and it was successfully applied for the case of a one-dimensional Hamiltonian system. Nevertheless, the numerical approach has not been widely noticed in many significant situations of classical dynamics. Possibly, it is due to their work having a limitation of implementation, the initial state should be known in an exact form as an inevitable step to perform the propagation in phase space. Furthermore, the traditional propagation methods in classical mechanics continue being the favorites to propagate initial states, since they involve mainly three stages, namely: (i) the initial state is frequently sampled on phase space accordingly with a particular distribution function, (ii) the corresponding classical trajectories are propagated at final time using a numerical integrator for solving the classical equations of motion; thus, those propagated trajectories contribute to the final state.(iii) In general, the propagated state should be smoothed in a grid, since all classical trajectories contribute in a different way to each phase space point. This methodology is commonly used in classical molecular dynamics  [17], [18], semiclassical propagation of wave-packets and Wigner functions  [19], [20], [21], as well as studies about quantum–classical correspondence where a comparison with classical calculations is necessary  [22], [23]. However, it is worth mentioning that some complications can arise in the traditional propagation methods, for instance, the sampling method cannot always cover completely the initial state, and the convergence of methods as well as the propagated state will be a compromise. The numerical integration generally is time consuming when non-linearities appear in the classical equations of motion, and the propagated state requires a lot of classical trajectories in order to obtain a good resolution in phase space.

The present paper is concerned with classical phase space propagation based on the split-operator technique, and one of the motivations of this work is to fill the knowledge gap on this isolated topic. In order to achieve this, we will adopt the SOT to different scenarios of classical (relativistic) mechanics involving; regular, chaotic motion, and dissipative dynamics, and subsequently, propagation in high-dimensional phase spaces. We propose a numerical method for propagating phase space distribution functions with several advantages in contrast to the traditional methods. More precisely, (i) our approach is directly based on computations on a grid of size N×N representing the whole phase space of classical systems, thus the resolution in phase space is defined at the beginning of the calculation, and it can be managed with the actual computational resources. (ii) The initial state should be storage on grid, but it can be computed in exact form or sampled as is done by the traditional methods, and finally (iii) the numerical propagation is accomplished by the fast Fourier transform where the computation method takes the order of O(NlogN) operations that will reduce significantly the computational cost of the numerical simulations.

This paper is organized as follows: Section  2 explains the theoretical background of propagation phase space distribution functions according to the Liouville equation, and the numerical solution based on fast Fourier transform. In Section  3 we describe the method for propagating Hamiltonian systems with δ-kick interaction. In Sections  4 Propagation of non-Hermitian quantum dynamics in the classical limit, 5 Propagation of a relativistic particle is considered for the time evolution of phase space distribution functions for dissipative dynamics associated with the classical limit of non-Hermitian quantum systems, and relativistic systems, respectively. The Section  6 deal with the phase space propagation for an interacting one-dimensional classical systems. Finally, we conclude in Section  7.

Section snippets

Liouville equation and the numerical schema

The time-evolution of phase space distribution function for a one-dimensional system is governed by the Liouville equation ∂ρ(q,p;t)∂t=Lˆρ(q,p;t), where ρ(q,p;t) is the phase space distribution function and Lˆ is the Liouville operator. The latter is given by Lˆ=[∂H∂q∂∂p−∂H∂p∂∂q] where H is the classical Hamiltonian function. The solution to Eq. (1) can formally be written as ρ(q,p;t)=Uˆ(t)ρ(q0,p0;0), where ρ(q0,p0;0) is the phase space distribution function at initial time, and the evolution

Propagation of δ-kicked systems

In the literature, impulsively driven systems are often referred to δ-kicked systems, due to the fact that the duration of each impulse is infinitely small compared to the period of the unperturbed system. These systems have become an essential tool for studying many phenomena from both theoretical and experimental perspectives, for instance, one of the primary approaches is quantum chaos in cold atoms  [24], [25], which ensures that it can provide valuable insights into non-linear-Hamiltonian

Propagation of non-Hermitian quantum dynamics in the classical limit

For decades considerable attention has been paid to describe the non-Hermitian Hamiltonian systems in the context of quantum mechanics. In particular, existing theoretical approaches provide an instructive description of open quantum systems, specifically these are subjected to systems involving decay, scattering and transport phenomena  [30], [31], [32]. However, little is known about the quantum–classical correspondence of non-Hermitian Hamiltonian systems, and recent theoretical studies have

Propagation of a relativistic particle

Recently, the problem of phase space evolution for a relativistic system has been addressed in the framework of Hamiltonian mechanics  [40], more precisely, for the relativistic harmonic oscillator. In this case, the propagation is achieved by considering the method proposed in Ref.  [16] and it has successfully applied for an initially Gaussian distribution. Nonetheless, it is in general not amenable for non analytical initial states. In order to illustrate our numerical method, we propagate

Propagation of an interacting Hamiltonian system

In the recent past, many theoretical efforts have been carried out to understand one of the most intriguing quantum phenomena, entanglement, and many authors have explored different measures in order to characterize it. The most common measures for interacting (biparticle) systems are the entropies, which gives the measure of non-separability of individual subsystems. In spirit, to understand the classical counterpart of this phenomenon very few studies have been carried out in the past  [41],

Conclusions

We have adapted the SOT for propagating phase space distribution functions in scenarios previously not considered in the literature. The numerical approach is rather general and allows to propagate almost any localized initial state, and it is not necessary to know the state in exact form as is the case of Ref.  [16]. We have considered the time evolution of initial states in the framework of classical and relativistic mechanics, in particular we have shown the performance of the method in

Acknowledgments

The first author thanks the Grupo de Materiales Magnéticos & Nanoestructuras of Universidad del Quindío for their generous provision of computer facilities, and he acknowledges the financial support from Universidad del Quindío.

References (44)

  • M.D. Feit et al.

    J. Comput. Phys.

    (1982)
  • G. Dattoli et al.

    Physica D

    (1998)
  • H. Bauke et al.

    Comput. Phys. Commun.

    (2011)
  • M. Ruf et al.

    J. Comput. Phys.

    (2009)
  • A. García-Vela

    Chem. Phys.

    (2002)
  • P. Szriftgiser et al.

    Commun. Nonlinear Sci. Numer. Simul.

    (2003)
  • D. Binosi et al.

    Comput. Phys. Commun.

    (2009)
  • R.M. Angelo et al.

    Physica A

    (2004)
  • G. Torres-Vega et al.

    Phys. Rev. Lett.

    (1991)
  • M.D. Feit et al.

    J. Chem. Phys.

    (1982)
  • M.D. Feit et al.

    J. Chem. Phys.

    (1984)
  • M. Brauna et al.

    Comput. Phys. Commun.

    (1996)
  • Y. Wu et al.

    J. Chem. Phys.

    (2006)
  • X. Chen et al.

    J. Chem. Phys.

    (2006)
  • R. Kosloff

    Annu. Rev. Phys. Chem.

    (1994)
  • G. Dattoli et al.

    La Rivista del Nuovo Cimento

    (1997)
  • D. Bosquetti et al.

    Z. Kristallogr.

    (2005)
  • O.V. Sinkin et al.

    J. Lightwave Technol.

    (2003)
  • G. Dattoli et al.

    Phys. Rev. E

    (1995)
  • P. Manikandan et al.

    J. Chem. Phys.

    (2012)
  • W.H. Miller

    J. Phys. Chem. A

    (2001)
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