Transport of a quantum particle in a time-dependent white-noise potential

Peter D. Hislop; Kay Kirkpatrick; Stefano Olla; Jeffrey Schenker

doi:10.1063/1.5054017

Transport of a quantum particle in a time-dependent white-noise potential

Peter D. Hislop, Kay Kirkpatrick, Stefano Olla, Jeffrey Schenker

Mathematics

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4 Scopus citations

Abstract

We show that a quantum particle in Rd, for d ≥ 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement ²¢(x, x, t) dx grows like t³ in any dimension. The white-noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. Similar results were established in one dimension by Jayannavar and Kumar [Phys. Rev. Lett. 48(8), 553-556 (1982)], and for any dimension using different methods by Fischer et al. [Phys. Rev. Lett. 73(12), 1578-1581 (1994)]. We also prove that for the same white-noise potential model on the lattice Zd, for d ≥ 1, the mean square displacement is diffusive growing like t¹. This behavior on the lattice is consistent with the diffusive behavior observed for similar models on the lattice Zd with a time-dependent Markovian potential by Kang and Schenker [J. Stat. Phys. 134, 1005-1022 (2009)].

Original language	English
Article number	083303
Journal	Journal of Mathematical Physics
Volume	60
Issue number	8
DOIs	https://doi.org/10.1063/1.5054017
State	Published - Aug 1 2019

Bibliographical note

Funding Information:
P.D.H. is thankful to P. Müller for several discussions on Ref. 17 and on Refs. 7 and 8. P.D.H. is also thankful to S. De Bièvre for discussions on classical systems and J. Marzuola for discussions on stochastic PDEs. P.D.H. was partially supported by Grant No. NSF DMS 11-03104; K.K. was partially supported by Grant Nos. NSF DMS-1106770 and CAREER DMS-1254791 and a Simons Sabbatical Fellowship; S.O. was partially supported by the Grant No. ANR-15-CE40-0020-01 grant LSD; and J.S. was partially supported by Grant No. NSF DMS-1500386, while some of this work was done.

Publisher Copyright:
© 2019 Author(s).

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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title = "Transport of a quantum particle in a time-dependent white-noise potential",

abstract = "We show that a quantum particle in Rd, for d ≥ 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement 2¢(x, x, t) dx grows like t3 in any dimension. The white-noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. Similar results were established in one dimension by Jayannavar and Kumar [Phys. Rev. Lett. 48(8), 553-556 (1982)], and for any dimension using different methods by Fischer et al. [Phys. Rev. Lett. 73(12), 1578-1581 (1994)]. We also prove that for the same white-noise potential model on the lattice Zd, for d ≥ 1, the mean square displacement is diffusive growing like t1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models on the lattice Zd with a time-dependent Markovian potential by Kang and Schenker [J. Stat. Phys. 134, 1005-1022 (2009)].",

author = "Hislop, {Peter D.} and Kay Kirkpatrick and Stefano Olla and Jeffrey Schenker",

note = "Funding Information: P.D.H. is thankful to P. M{\"u}ller for several discussions on Ref. 17 and on Refs. 7 and 8. P.D.H. is also thankful to S. De Bi{\`e}vre for discussions on classical systems and J. Marzuola for discussions on stochastic PDEs. P.D.H. was partially supported by Grant No. NSF DMS 11-03104; K.K. was partially supported by Grant Nos. NSF DMS-1106770 and CAREER DMS-1254791 and a Simons Sabbatical Fellowship; S.O. was partially supported by the Grant No. ANR-15-CE40-0020-01 grant LSD; and J.S. was partially supported by Grant No. NSF DMS-1500386, while some of this work was done. Publisher Copyright: {\textcopyright} 2019 Author(s).",

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N2 - We show that a quantum particle in Rd, for d ≥ 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement 2¢(x, x, t) dx grows like t3 in any dimension. The white-noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. Similar results were established in one dimension by Jayannavar and Kumar [Phys. Rev. Lett. 48(8), 553-556 (1982)], and for any dimension using different methods by Fischer et al. [Phys. Rev. Lett. 73(12), 1578-1581 (1994)]. We also prove that for the same white-noise potential model on the lattice Zd, for d ≥ 1, the mean square displacement is diffusive growing like t1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models on the lattice Zd with a time-dependent Markovian potential by Kang and Schenker [J. Stat. Phys. 134, 1005-1022 (2009)].

AB - We show that a quantum particle in Rd, for d ≥ 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement 2¢(x, x, t) dx grows like t3 in any dimension. The white-noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. Similar results were established in one dimension by Jayannavar and Kumar [Phys. Rev. Lett. 48(8), 553-556 (1982)], and for any dimension using different methods by Fischer et al. [Phys. Rev. Lett. 73(12), 1578-1581 (1994)]. We also prove that for the same white-noise potential model on the lattice Zd, for d ≥ 1, the mean square displacement is diffusive growing like t1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models on the lattice Zd with a time-dependent Markovian potential by Kang and Schenker [J. Stat. Phys. 134, 1005-1022 (2009)].

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Transport of a quantum particle in a time-dependent white-noise potential

Abstract

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