A RENORMALIZATION APPROACH TO KINETIC TRANSPORT IN CRYSTALS Jens Marklof School of Mathematics EPSRC Leadership Interview, 10 June 2008 1
Key aspects • Derivation of novel kinetic equations for transport in crystals • Development of new powerful tools in measure rigidity and renormal- ization flow dynamics with wide-ranging applications • Training of researchers on all levels through advanced lecture courses, supervision, research workshops and a distinguished visitors programme 2
Kinetic theory Since the pioneering work of Boltzmann in the 1870s, a central objective in mathe- matical physics is the derivation of macro- scopic evolution equations from the underly- ing fundamental microscopic laws , classical or quantum. In his 1872 paper, Boltzmann derived the famous Boltzmann equation , assuming that the dynamics of the colliding gas molecules Ludwig Boltzmann (1844-1906) is chaotic. 3
The Lorentz gas In an attempt to describe the evolution of a dilute electron gas in a metal, Lorentz pro- posed in 1905 a model, where the heavier scatterers are assumed to be fixed, whereas the electrons are interacting with the scatter- ers but not with each other. Lorentz assumed that the macroscopic dy- namics is governed by the linear Boltzmann equation . . . Hendrik Lorentz (1853-1928) 4
. . . which was rigorously proved for a ran- dom scatterer configuration in the 1970s and 1980s by Gallavotti, Spohn and Boldrighini- Bunimovich-Sinai. These results have more recently been extended to the full quantum problem. My recent work ∗ with A. Str¨ ombergsson (Uppsala) shows that, perhaps surprisingly, the linear Boltzmann equation does not hold in a periodic scatterer configuration . The re- sulting kinetic transport equation is substan- tially more complicated. *Annals of Mathematics, at press 5
Renormalization dynamics and measure rigidity 2 r 2 r ℓr 1 − d ℓr 1 − d ℓ The theory of measure rigidity implies that averages over the initial velocity can be replaced by averages over the space of lattices. We are the first to use measure rigidity techniques in kinetic theory. They have proved superior over all previous approaches and will form the key technical tools in the proposed analysis. 6
Expected research output For more than a century the kinetic laws for transport in crystals have been assumed to be governed by variants of the linear Boltzmann equation, one of the basic evolution equations in statistical physics. Using fundamentally new techniques, the proposed research will show that this is generally not the case, and establish rigorously the correct macro- scopic transport equations. Key outcomes will be. . . 7
• Fundamental kinetic laws for transport in crystals , including – long-range potentials with Coulomb-type singularities – crystal defects, vibrational modes, electromagnetic fields – quasicrystals – quantum effects • Technical advances in measure rigidity and theory of renormalization flows • Exploration of further applications , e.g., dynamics in billiards and wave- guides with small dents, small-denominator problems in the theory of PDE 8
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