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

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

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 languageEnglish
Article number083303
JournalJournal of Mathematical Physics
Issue number8
StatePublished - 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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