Pure and Applied Analysis

Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows

Jory Griffin and Jens Marklof

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We show that the time evolution of a quantum wavepacket in a periodic potential converges in a combined high-frequency/Boltzmann–Grad limit, up to second order in the coupling constant, to terms that are compatible with the linear Boltzmann equation. This complements results of Eng and Erdős for low-density random potentials, where convergence to the linear Boltzmann equation is proved in all orders. We conjecture, however, that the linear Boltzmann equation fails in the periodic setting for terms of order 4 and higher. Our proof uses Floquet–Bloch theory, multivariable theta series and equidistribution theorems for homogeneous flows. Compared with other scaling limits traditionally considered in homogenisation theory, the Boltzmann–Grad limit requires control of the quantum dynamics for longer times, which are inversely proportional to the total scattering cross-section of the single-site potential.

Article information

Pure Appl. Anal., Volume 1, Number 4 (2019), 571-614.

Received: 2 November 2018
Revised: 20 March 2019
Accepted: 6 June 2019
First available in Project Euclid: 29 October 2019

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Zentralblatt MATH identifier

Primary: 37A17: Homogeneous flows [See also 22Fxx] 82C10: Quantum dynamics and nonequilibrium statistical mechanics (general)

theta functions homogeneous dynamics quantum transport


Griffin, Jory; Marklof, Jens. Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows. Pure Appl. Anal. 1 (2019), no. 4, 571--614. doi:10.2140/paa.2019.1.571. https://projecteuclid.org/euclid.paa/1572314426

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  • G. Allaire and A. Piatnitski, “Homogenization of the Schrödinger equation and effective mass theorems”, Comm. Math. Phys. 258:1 (2005), 1–22.
  • G. Bal, A. Fannjiang, G. Papanicolaou, and L. Ryzhik, “Radiative transport in a periodic structure”, J. Stat. Phys. 95:1-2 (1999), 479–494.
  • G. Bal, G. Papanicolaou, and L. Ryzhik, “Radiative transport limit for the random Schrödinger equation”, Nonlinearity 15:2 (2002), 513–529.
  • G. Bal, T. Komorowski, and L. Ryzhik, “Asymptotics of the solutions of the random Schrödinger equation”, Arch. Ration. Mech. Anal. 200:2 (2011), 613–664.
  • A. Benoit and A. Gloria, “Long-time homogenization and asymptotic ballistic transport of classical waves”, preprint, 2017. To appear in Ann. Sci. École Norm. Sup.
  • A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
  • M. V. Berry, “Quantizing a classically ergodic system: Sinai's billiard and the KKR method”, Ann. Physics 131:1 (1981), 163–216.
  • M. S. Birman and T. A. Suslina, “Periodic second-order differential operators: threshold properties and averaging”, Algebra i Analiz 15:5 (2003), 1–108. In Russian; translated in St. Petersburg Math. J. 15:5 (2004), 639–714.
  • C. Boldrighini, L. A. Bunimovich, and Y. G. Sinai, “On the Boltzmann equation for the Lorentz gas”, J. Stat. Phys. 32:3 (1983), 477–501.
  • E. Caglioti and F. Golse, “On the Boltzmann–Grad limit for the two dimensional periodic Lorentz gas”, J. Stat. Phys. 141:2 (2010), 264–317.
  • F. Castella, “On the derivation of a quantum Boltzmann equation from the periodic von Neumann equation”, M2AN Math. Model. Numer. Anal. 33:2 (1999), 329–349.
  • F. Castella, “From the von Neumann equation to the quantum Boltzmann equation in a deterministic framework”, J. Stat. Phys. 104:1-2 (2001), 387–447.
  • F. Castella, “From the von Neumann equation to the quantum Boltzmann equation, II: Identifying the Born series”, J. Stat. Phys. 106:5-6 (2002), 1197–1220.
  • F. Castella and A. Plagne, “Non derivation of the quantum Boltzmann equation from the periodic von Neumann equation”, Indiana Univ. Math. J. 51:4 (2002), 963–1016.
  • R. V. Craster, J. Kaplunov, and A. V. Pichugin, “High-frequency homogenization for periodic media”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466:2120 (2010), 2341–2362.
  • P. Dahlqvist and G. Vattay, “Periodic orbit quantization of the Sinai billiard in the small scatterer limit”, J. Phys. A 31:30 (1998), 6333–6345.
  • D. Eng and L. Erdős, “The linear Boltzmann equation as the low density limit of a random Schrödinger equation”, Rev. Math. Phys. 17:6 (2005), 669–743.
  • L. Erdős and H.-T. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation”, Comm. Pure Appl. Math. 53:6 (2000), 667–735.
  • A. Eskin, G. Margulis, and S. Mozes, “Quadratic forms of signature $(2,2)$ and eigenvalue spacings on rectangular 2-tori”, Ann. of Math. $(2)$ 161:2 (2005), 679–725.
  • G. B. Folland, Harmonic analysis in phase space, Ann. of Math. Studies 122, Princeton Univ. Press, 1989.
  • G. Gallavotti, “Divergences and the approach to equilibrium in the Lorentz and the wind-tree models”, Phys. Rev. 185:1 (1969), 308–322.
  • P. Gérard, “Mesures semi-classiques et ondes de Bloch”, pp. [exposé] 16 in Séminaire sur les Équations aux Dérivées Partielles: 1990–1991, École Polytech., Paris, 1991.
  • P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, “Homogenization limits and Wigner transforms”, Comm. Pure Appl. Math. 50:4 (1997), 323–379.
  • F. Golse, “On the periodic Lorentz gas and the Lorentz kinetic equation”, Ann. Fac. Sci. Toulouse Math. $(6)$ 17:4 (2008), 735–749.
  • J. Griffin, Quantum dynamics in highly localised periodic potentials, Ph.D. thesis, University of Bristol, 2017.
  • J. Griffin and J. Marklof, “Probabilistic model for quantum transport in a low-density periodic potential”, in preparation.
  • D. Harutyunyan, G. W. Milton, and R. V. Craster, “High-frequency homogenization for travelling waves in periodic media”, Proc. A. 472:2191 (2016), art. id. 20160066.
  • H. A. Lorentz, “Le mouvement des électrons dans les métaux”, Arch. Néerl. 10 (1905), 337–371.
  • G. Margulis and A. Mohammadi, “Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms”, Duke Math. J. 158:1 (2011), 121–160.
  • J. Marklof, “Pair correlation densities of inhomogeneous quadratic forms, II”, Duke Math. J. 115:3 (2002), 409–434. Correction in 120:1 (2003), 227–228.
  • J. Marklof, “Pair correlation densities of inhomogeneous quadratic forms”, Ann. of Math. $(2)$ 158:2 (2003), 419–471.
  • J. Marklof and A. Strömbergsson, “Kinetic transport in the two-dimensional periodic Lorentz gas”, Nonlinearity 21:7 (2008), 1413–1422.
  • J. Marklof and A. Strömbergsson, “The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems”, Ann. of Math. $(2)$ 172:3 (2010), 1949–2033.
  • J. Marklof and A. Strömbergsson, “The Boltzmann–Grad limit of the periodic Lorentz gas”, Ann. of Math. $(2)$ 174:1 (2011), 225–298.
  • J. Marklof and A. Strömbergsson, “The periodic Lorentz gas in the Boltzmann–Grad limit: asymptotic estimates”, Geom. Funct. Anal. 21:3 (2011), 560–647.
  • P. A. Markowich, N. J. Mauser, and F. Poupaud, “A Wigner-function approach to (semi)classical limits: electrons in a periodic potential”, J. Math. Phys. 35:3 (1994), 1066–1094.
  • F. Nier, “Asymptotic analysis of a scaled Wigner equation and quantum scattering”, Transport Theory Statist. Phys. 24:4-5 (1995), 591–628.
  • F. Nier, “A semi-classical picture of quantum scattering”, Ann. Sci. École Norm. Sup. $(4)$ 29:2 (1996), 149–183.
  • G. Panati, H. Spohn, and S. Teufel, “Effective dynamics for Bloch electrons: Peierls substitution and beyond”, Comm. Math. Phys. 242:3 (2003), 547–578.
  • H. Spohn, “Derivation of the transport equation for electrons moving through random impurities”, J. Stat. Phys. 17:6 (1977), 385–412.
  • H. Spohn, “The Lorentz process converges to a random flight process”, Comm. Math. Phys. 60:3 (1978), 277–290.
  • A. Strömbergsson and P. Vishe, “An effective equidistribution result for ${\rm SL}(2,\mathbb{R})\ltimes(\smash{\mathbb{R}^2)^{\oplus k}}$ and application to inhomogeneous quadratic forms”, preprint, 2018.