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The Journal of Chemical Physics

Stochastic theory for molecular collisions in the perturbation limit

Stuart D. Augustin, HERSCHEL A. RABITZ

Published Sep 1, 1977• 17 citations
Publisher

Abstract

Previous stochastic theory research has described individual molecular collisions by a master equation or a Fokker–Planck equation. The perturbation (weak coupling) limit of this theory is examined more carefully in this paper. It is shown how the formulation reduces to exact quantum mechanics in the perturbation regime. The methods are applied to the collinear forced quantum oscillator model for purposes of comparison with other theories. The results of this work and previous research show that stochastic collision theory firmly connects up to both the weak and strong coupling limits. It is suggestive that computationally attractive schemes which afford reasonable accuracy can now be developed for the intermediate coupling regime as well.

DOI
10.1063/1.435094

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Previous stochastic theory research has described individual molecular collisions by a master equation or a Fokker–Planck equation. The perturbation (weak coupling) limit of this theory is examined more carefully in this paper. It is shown how the formulation reduces to exact quantum mechanics in the perturbation regime. The methods are applied to the collinear forced quantum oscillator model for purposes of comparison with other theories. The results of this work and previous research show that stochastic collision theory firmly connects up to both the weak and strong coupling limits. It is suggestive that computationally attractive schemes which afford reasonable accuracy can now be developed for the intermediate coupling regime as well.
Taking the Wigner function as the basic N-body distribution function, quantum-mechanical modifications to the Fokker—Planck equation are studied using the two different approaches introduced, respectively, by Kirkwood and Prigogine. In Sec. II the weak coupling limit is studied using the time-smoothing technique first introduced by Kirkwood; the results of this section can perhaps form the basis of an approximate treatment of the transport phenomena of pure liquids. In Sec. III we treat the problem of the Brownian motion of a heavy particle employing the perturbation techniques developed by Prigogine and co-workers; in this section it is found that the form of the quantum-mechanical Fokker—Planck equation is identical to the classical equation, but that the friction coefficient contains quantum-mechanical corrections. Finally, it is noted that the weak coupling result obtained by the time-smoothing technique is identical to the approximation of small momentum transfer (Sec. III), if in the latter case the scattering cross section is represented in the Born approximation.
A maximum entropy approach to time-dependent phenomena is employed to construct a quantum stochastic theory for adiabatic processes in condensed phases. This theory is illustrated by considering the relative motion of a two-particle system interacting with a heat bath. We present a fully quantum mechanical phase space master equation, which is new. In the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion, our master equation reduces to a simpler form, and the principle of detailed balance emerges. This limit is shown to be equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2. The simpler master equation is utilized in the development of a quantum mechanical nonlinear Fokker–Planck equation. For cases in which correlations involving spatial and momentum fluxes are short ranged, the nonlinear Fokker–Planck equation reduces to the usual linear form, with ’’collision’’ and ’’streaming’’ terms that contain quantum corrections to all orders in ℏ. The relationship between the classical limit of the present theory and Brownian motion theory based on the Chapman–Kolmogorov equation is established.
Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker–Planck equation and a quantum master equation from a system–bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.
A quantum stochastic theory for nonadiabatic processes in condensed phases and on surfaces is given. The present theory is primarily concerned with processes involving the fragmentation of two-particle systems and/or the ’’collision’’ of two particles with internal structure, i.e., electronic, spin, and nuclear degrees of freedom. A fully quantum mechanical joint energy space–quantum phase space master equation for two-particle systems is presented. A simpler form of this equation emerges in the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion. (This limit is equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2.) The simpler master equation is converted into a nonlinear quantum Fokker–Planck equation for nonadiabatic systems. Introducing simplifying assumptions, we obtain a quantun analog of the stochastic Liouville equation augmented with irreversible kinetic terms, which describes nonadiabatic transitions and the interruption of coherence in energy space at different points in quantum phase space. The final results are appropriately modified to incorporate the influence of applied time-dependent electric and magnetic fields. The present work provides a formal theoretical framework for examining the role of quantum effects that are neglected in ’’semiclassical’’ kinetic and ’’diffusional’’ theories for nonadiabatic processes.
Abstract. We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics. In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling between the system and the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain emission and absorption as well as scattering interactions. 1.
We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer's equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.