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Title: Kinetic theory for classical and quantum many-body chaos

Abstract

For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase-space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth is a front propagating into an unstable state in phase space. As in conventional Boltzmann transport, which follows from the dynamics of the net particle number density exchange, the kernel of this kinetic integral equation for chaos is also set by the 2-to-2 scattering rate. This provides a mathematically precise statement of the known fact that in dilute weakly coupled gases, transport and scrambling (or ergodicity) are controlled by the same physics.

Authors:
 [1];  [2];  [2]
+ Show Author Affiliations
  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
  2. Leiden Univ. (Netherlands)
Publication Date:
Research Org.:
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Nuclear Physics (NP)
OSTI Identifier:
1637335
Alternate Identifier(s):
OSTI ID: 1489861; OSTI ID: 1611575
Grant/Contract Number:  
SC0011090
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review. E
Additional Journal Information:
Journal Volume: 99; Journal Issue: 1; Journal ID: ISSN 2470-0045
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; finite temperature field theory; kinetic theory; quantum chaos; quantum field theory; quantum transport; Physics

Citation Formats

Grozdanov, Sašo, Schalm, Koenraad, and Scopelliti, Vincenzo. Kinetic theory for classical and quantum many-body chaos. United States: N. p., 2019. Web. doi:10.1103/PhysRevE.99.012206.
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Grozdanov, Sašo, Schalm, Koenraad, & Scopelliti, Vincenzo. Kinetic theory for classical and quantum many-body chaos. United States. https://doi.org/10.1103/PhysRevE.99.012206
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Grozdanov, Sašo, Schalm, Koenraad, and Scopelliti, Vincenzo. Tue . "Kinetic theory for classical and quantum many-body chaos". United States. https://doi.org/10.1103/PhysRevE.99.012206. https://www.osti.gov/servlets/purl/1637335.
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@article{osti_1637335,
title = {Kinetic theory for classical and quantum many-body chaos},
author = {Grozdanov, Sašo and Schalm, Koenraad and Scopelliti, Vincenzo},
abstractNote = {For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase-space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth is a front propagating into an unstable state in phase space. As in conventional Boltzmann transport, which follows from the dynamics of the net particle number density exchange, the kernel of this kinetic integral equation for chaos is also set by the 2-to-2 scattering rate. This provides a mathematically precise statement of the known fact that in dilute weakly coupled gases, transport and scrambling (or ergodicity) are controlled by the same physics.},
doi = {10.1103/PhysRevE.99.012206},
journal = {Physical Review. E},
number = 1,
volume = 99,
place = {United States},
year = {Tue Jan 08 00:00:00 EST 2019},
month = {Tue Jan 08 00:00:00 EST 2019}
}
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Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1103/PhysRevE.99.012206
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Citation Metrics:
Cited by: 24 works
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Figures / Tables:

FIG. 1 FIG. 1: The spectra of the kernel $\mathfrak{L}$(p, l) for the linearized Boltzmann equation (and also of $\langle$T xy(kz), T xy(−kz)$rangle$R, cf. Eq. (34)) (top left) and of the kernel $\mathfrak{L}$EX(p, l) for the kinetic equation for the OTOC (top right) are plotted over the complex ω plane and inmore » the limit of βm → 0. In the lower half of the complex ω plane, there is a dense sequence of numerically obtained poles. In both spectra, these poles are believed to be the signature of a branch cut. See [42] and also [34, 44–46]. In the upper half of the complex ω plane, only the kernel $\mathfrak{L}$EX(p, l) has distinct poles which are identified with the Lyapunov exponents, as explained below equation (11). The dependence of these two Lyapunov exponents and the branch cuts on βm is depicted in the inlay (bottom). For large values of βm, the Lyapunov exponents decay exponentially. The plots are obtained by diagonalizing the kernels of the integral equations (20) and (32) after a discretization with N = 1000 grid points on the domain p ∈ [m/N,N ×m]. The discretization is not uniform. This is done in order for the diagonalization to appropriately account for the contributions of both the soft momenta and collinear momenta p ≈ l, which are not negligible even when both p and l are large. The finite size of the branch cuts, i.e. its end point for large Im(ω), is related to finite domain of the discretization procedure.« less
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Works referenced in this record:

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journal, August 2017

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On thermostats and entropy production
text, January 2000

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    Works referencing / citing this record:

    Reentrant superconductivity in a quantum dot coupled to a Sachdev-Ye-Kitaev metal
    journal, December 2019

    Recent Developments in the Holographic Description of Quantum Chaos
    journal, March 2019

    Recent Developments in the Holographic Description of Quantum Chaos
    journal, March 2019

    Scrambling and Lyapunov Exponent in Unitary Networks with Tunable Interactions
    text, January 2020

    Scrambling with conservation law
    text, January 2021

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      Figures / Tables found in this record:

      FIG. 1 (p. 6)figure
      Diagram 1 (p. 8)figure
      Diagram 10 (p. 9)figure
      Diagram 11 (p. 9)figure
      Diagram 12 (p. 9)figure
      Diagram 2 (p. 9)figure
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      Diagram 4 (p. 9)figure
      Diagram 5 (p. 9)figure
      Diagram 6 (p. 9)figure
      Diagram 7 (p. 9)figure
      Diagram 8 (p. 9)figure
      Diagram 9 (p. 9)figure
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