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Journal of Mathematical Physics

On the binary collision expansion and resummations of it

Antoine Royer

Published Oct 1, 1983• 0 citations
Publisher

Abstract

It is pointed out that when there are only two interaction terms in the Hamiltonian, the binary collision expansion (BCE) for the scattering operator can be resummed into a closed form. When there are more than two interaction terms, one can resum the BCE into a continued-fraction-like form by using repeatedly the two interactions resummation. Concise derivations of the BCE are first given, the time-dependent BCE being obtained in a form applicable to both quantum evolution operators and classical frequency modulated oscillators.

DOI
10.1063/1.525610

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It is pointed out that when there are only two interaction terms in the Hamiltonian, the binary collision expansion (BCE) for the scattering operator can be resummed into a closed form. When there are more than two interaction terms, one can resum the BCE into a continued-fraction-like form by using repeatedly the two interactions resummation. Concise derivations of the BCE are first given, the time-dependent BCE being obtained in a form applicable to both quantum evolution operators and classical frequency modulated oscillators.
The binary collision expansion for the density matrix of a system of NNN particles with pair interaction was derived by Huang, Lee, and Yang by expansion in the interaction and subsequent summation over terms represented by certain classes of diagrams. A simpler derivation has been obtained by the use of the $\frac{N(N\ensuremath{-}1)}{2}$ integral equations which relate the density matrix of the system to the density matrices of systems in which only one pair of particles interacts. Successive substitution of the integral equations into each other yields an expansion which by a trivial additional step becomes the binary collision expansion. Our derivation shows also that the binary collision expansion is the Laplace inverse of the coordinate representation of the expansion in terms of scattering operators.
摘要 应用李政道和杨振宁提出的二元碰撞展开法,对于有强短程排斥互作用的多粒子系发展了一种按图形计算双时间-温度格临函数的规则,其各级近似由二体散射相移的各次幂所表征。文中得出了玻色和费米统计情形的计算规则,并且研究了有凝聚相存在的玻色系统。本文主要讨论了单粒子格临函数的图形技术,但容易将它们推广应用于计算各种时间-温度格临函数。 Abstract The method of binary collision expansion has been employed to obtain series developments in powers of the scattering length between two particles for double-time-temperature-dependent Green's functions of statistical mechanics. We have studied the cases of Bose and Fermi statistics as well as the case of a Bose system in the presence of a condensed phase, and have established rules for computation of the successive terms of expansion with the aid of diagrams. The present method is applicable to many particle systems with strong and short-ranged mainly repulsive two-body interactions. 作者及机构信息 蔡建华1, 徐宏华2 (1)南京大学物理系; (2)南京大学物理系,1964年应届毕业生 Authors and contacts C. H. TSAI1, H. H. HSU2 (1)南京大学物理系; (2)南京大学物理系,1964年应届毕业生 参考文献 [1] 施引文献
In the spin-boson model, a continued fraction form is proposed to systematically resum high-order quantum kinetic expansion (QKE) rate kernels, accounting for the bath relaxation effect beyond the second-order perturbation. In particular, the analytical expression of the sixth-order QKE rate kernel is derived for resummation. With higher-order correction terms systematically extracted from higher-order rate kernels, the resummed quantum kinetic expansion approach in the continued fraction form extends the Pade approximation and can fully recover the exact quantum dynamics as the expansion order increases.
In the spin-boson model, a continued fraction form is proposed to systematically resum high-order quantum kinetic expansion (QKE) rate kernels, accounting for the bath relaxation effect beyond the second-order perturbation. In particular, the analytical expression of the sixth-order QKE rate kernel is derived for resummation. With higher-order correction terms systematically extracted from higher-order rate kernels, the resummed quantum kinetic expansion approach in the continued fraction form extends the Pade approximation and can fully recover the exact quantum dynamics as the expansion order increases.
In this paper we report a new method for resummation of operator expansions of the evolution operator. These resummation techniques are applied to the Feynman propagator and then to the derivation of an effective Schrödinger equation for general system–bath problems. This paper presents a significant advance over work previously reported by us [J. Chem. Phys. 96, 5952 (1992)], and it provides a highly accurate way to calculate quantum mechanical data for many dimensional systems. We then apply this new analytic resummation technique to the calculation of flux–flux correlation functions for two, three, and four dimensional problems in order to study the range of applicability of the approach. For moderate frequencies and coupling strengths, not only is the resummed operator calculational method essentially exact, it also requires a fraction of the computer time that the exact calculation consumes. For truly many dimensional problems this approach should provide the first accurate quantum mechanics of rate processes.