On the approximation of Feynman-Kac path integrals for quantum statistical mechanics

Stephen D. Bond; Brian B. Laird; Benedict J. Leimkuhler

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On the approximation of Feynman-Kac path integrals for quantum statistical mechanics

Stephen D. Bond·Brian B. Laird·Benedict J. Leimkuhler

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Abstract

Discretizations of the Feynman-Kac path integral representation of the quantum mechanical density matrix are investigated. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional function space, by restricting the integration to a subspace of all admissible paths. Using this process, a wide class of methods can be derived, with each method corresponding to a different choice for the approximating subspace. The traditional ``short-time'' approximation and ``Fourier discretization'' can be recovered from this approach, using linear and spectral basis functions respectively. As an illustration, a novel method is formulated using cubic elements and is shown to have improved convergence properties when applied to a simple model problem.

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Stephen D. Bond, Brian B. Laird, Benedict J. Leimkuhler. 2000-07-06. On the approximation of Feynman-Kac path integrals for quantum statistical mechanics. https://arxiv.org/abs/cond-mat/0007112

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