PMID: 15244971Jul 13, 2004Paper

Existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Cristian Predescu

Abstract

In this paper I provide significant mathematical evidence in support of the existence of direct short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is "experimental," I only provide a "physicist's" proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in Appendix B, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present paper is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.

References

Dec 1, 1986·Physical Review A: General Physics·R P Feynman, H Kleinert
Jun 1, 1986·Physical Review. B, Condensed Matter·R Giachetti, V Tognetti
Jul 1, 1985·Physical Review Letters·J D DollD L Freeman
Mar 15, 2003·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cristian Predescu, J D Doll

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Citations

Jul 30, 2005·The Journal of Chemical Physics·Charusita Chakravarty
Feb 2, 2010·The Journal of Chemical Physics·Robert E ZillichSiu A Chin
Apr 21, 2007·The Journal of Chemical Physics·Thomas F Miller, Cristian Predescu
Jul 11, 2006·The Journal of Chemical Physics·Shaji ChempathAlexis T Bell
Jun 11, 2005·The Journal of Chemical Physics·Cristian PredescuVladimir A Mandelshtam
Feb 9, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cristian Predescu
May 21, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cristian Predescu
Dec 31, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cristian Predescu
Apr 28, 2009·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Antun BalazAxel Pelster
Mar 10, 2012·Physical Review Letters·R Rota, J Boronat
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·M A VoznesenskiyA P Lyubartsev
May 21, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Cristian Predescu
Feb 14, 2006·The Journal of Physical Chemistry. B·Cristian Predescu

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