TY - JOUR
T1 - Transport of a quantum particle in a time-dependent white-noise potential
AU - Hislop, Peter D.
AU - Kirkpatrick, Kay
AU - Olla, Stefano
AU - Schenker, Jeffrey
N1 - 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).
PY - 2019/8/1
Y1 - 2019/8/1
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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U2 - 10.1063/1.5054017
DO - 10.1063/1.5054017
M3 - Article
AN - SCOPUS:85070415550
SN - 0022-2488
VL - 60
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 8
M1 - 083303
ER -