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arXiv:1111.0751

Continuous Limits of Classical Repeated Interaction Systems

Julien Deschamps

Published Nov 3, 2011• 0 citations
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Abstract

We consider the physical model of a classical mechanical system (called "small system") undergoing repeated interactions with a chain of identical small pieces (called "environment"). This physical setup constitutes an advantageous way of implementing dissipation for classical systems, it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions ([2]), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations in the limit. In particular we recover the usual Langevin equations associated to the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton's equations and the repeated interactions. We embed it into a continuous-time dynamical system and we compute the limit when the time step goes to 0. This way we obtain a discrete-time approximation of stochastic differential equation, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the Lp and almost sure convergence of this scheme. We end up with applications to concrete physical exemples such as a charged particule in an uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.

DOI
10.48550/arxiv.1111.0751

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All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. 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Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Reservoir Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Harmonic Oscillator Analytic Expectation Values . . . . . . . . . . . . . . . 42 4.2.1 Ohmic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Drude Regularized Bath . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Sampling Strategies and Analytic Comparison . . . . . . . . . . . . . . . . . 44 4.4 Limits of the Markovian Approximation . . . . . . . . . . . . . . . . . . . . 45 5 Dissipative Vibrational Dynamics of Diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Molecular Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Anharmonic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Transient Non-Markovian Effects . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 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. . . . . . . . . . . . . . . . . 92 D.4 Computation of Matrix Elements and Expectation Values . . . . . . . . . . 92 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

•Sep 9, 2010•1 citation

Stochastic Master Equations in Thermal Environment

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Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1.

•Sep 1, 2008•0 citations
We consider the physical model of a classical mechanical system (called "small system") undergoing repeated interactions with a chain of identical small pieces (called "environment"). This physical setup constitutes an advantageous way of implementing dissipation for classical systems, it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions ([2]), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations in the limit. In particular we recover the usual Langevin equations associated to the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton's equations and the repeated interactions. We embed it into a continuous-time dynamical system and we compute the limit when the time step goes to 0. This way we obtain a discrete-time approximation of stochastic differential equation, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the Lp and almost sure convergence of this scheme. We end up with applications to concrete physical exemples such as a charged particule in an uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.
We consider a repeated quantum interaction model describing a small system $\Hh_S$ in interaction with each one of the identical copies of the chain $\bigotimes_{\N^*}\C^{n+1}$, modeling a heat bath, one after another during the same short time intervals [0,h][0,h][0,h]. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order hhh. After giving the GNS representation, we establish the relation between the time scale hhh and the classical low density limit. We introduce a chemical potential μμμ related to the time hhh as follows: h2=eβμh^2=e^{βμ}h2=eβμ. We further prove that the solution of the associated discrete evolution equation converges strongly, when hhh tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.
We consider a repeated quantum interaction model describing a small system [Formula: see text] in interaction with each of the identical copies of the chain ⊗ℕ* ℂn+1, modeling a heat bath, one after another during the same short time intervals [0, h]. We suppose that the repeated quantum interaction Hamiltonian is split into two parts: a free part and an interaction part with time scale of order h. After giving the GNS representation, we establish the connection between the time scale h and the classical low density limit. We introduce a chemical potential µ related to the time h as follows: h2 = eβµ. We further prove that the solution of the associated discrete evolution equation converges strongly, when h tends to 0, to the unitary solution of a quantum Langevin equation directed by the Poisson processes.
In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. The resulting stochastic Liouville-von Neumann equation describes the full non-Markovian dynamics without explicit memory but instead accounts for it implicitly through the correlations of the complex-valued noise forces. The present thesis provides a first application of the Stockburger-Grabert stochastic Liouville-von Neumann equation to the computation of the dynamics of anharmonic, continuous open systems. In particular, it is demonstrated that trajectory based propagators allow for the construction of a numerically stable propagation scheme. With this approach it becomes possible to achieve the tremendous increase of the noise sample count necessary to stochastically converge the results when investigating such systems with continuous variables. After a test against available analytic results for the dissipative harmonic oscillator, the approach is subsequently applied to the analysis of two different realistic, physical systems. As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.:1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theory of Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Influence Functional Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Quantum Brownian Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Stochastic Unraveling of the Influence Functional . . . . . . . . . . . . . . . 20 2.4 Improved Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Modified Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Guide Trajectory Transformation . . . . . . . . . . . . . . . . . . . . 24 2.5 Obtaining Properly Correlated Stochastic Samples from Filtered White Noise 24 3 Unified Stochastic Trajectory Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Semiclassical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Guide Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Real Coherent State Center Coordinates . . . . . . . . . . . . . . . . 31 3.1.3 Propagation Scheme Including Stochastic Forces . . . . . . . . . . . 32 3.2 Stochastic Bohmian Mechanics with Complex Action . . . . . . . . . . . . . 33 3.2.1 Hydrodynamic Formulation of Bohmian Mechanics . . . . . . . . . . 33 3.2.2 Bohmian Mechanics with Complex Action . . . . . . . . . . . . . . . 34 3.2.3 Stochastic BOMCA Trajectories . . . . . . . . . . . . . . . . . . . . 38 3.3 Noise Distribution Preserving Removal of Adverse Samples . . . . . . . . . . 39 4 Dissipative Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Reservoir Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Harmonic Oscillator Analytic Expectation Values . . . . . . . . . . . . . . . 42 4.2.1 Ohmic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Drude Regularized Bath . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Sampling Strategies and Analytic Comparison . . . . . . . . . . . . . . . . . 44 4.4 Limits of the Markovian Approximation . . . . . . . . . . . . . . . . . . . . 45 5 Dissipative Vibrational Dynamics of Diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Molecular Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Anharmonic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Transient Non-Markovian Effects . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Trapping by Dissipation and Thermalization . . . . . . . . . . . . . . . . . . 53 6 Tunneling with Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1 Eckart Barrier Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 Dissipative Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3 Investigation of Markovianity . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Appendix A Conventions for Constants, Reservoir Kernels, and Influence Phases 69 Appendix B Stochastic Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 B.1 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 Position Verlet Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 B.3 Runge-Kutta Fourth Order Scheme . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix CMorse Oscillator Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix DPrerequisites of a Successful Stochastic Propagation . . . . . . . . . . . . . . 79 D.1 Hubbard-Stratonovich Transformation . . . . . . . . . . . . . . . . . . . . . 79 D.2 Kernels of the Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 D.2.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 89 D.3 Guide Trajectory Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 D.3.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 92 D.4 Computation of Matrix Elements and Expectation Values . . . . . . . . . . 92 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1.