Jens Marklof's publications
Macroscopic bodies-a chair, glass of water, a cloud of gas-are made up of microscopic particles: from molecules and atoms to electrons and a whole zoo of other elementary particles. Although the idea of an atomic structure of matter goes back to the ancient Greeks, its general acceptance is remarkably recent. The great pioneer of statistical physics, Ludwig Boltzmann (1844-1906), was still ridiculed by many of his peers for using the atomic hypothesis in his historic work-and this at the end of the 19th century, only a few years before Max Planck (1858-1947) set the foundations for the discovery of quantum mechanics!
Given that the number of microscopic particles is enormous (a glass of water contains around 1025=10,000,000,000,000,000,000,000,000 molecules!), it is fair to ask whether it is at all possible to derive the evolution of a macroscopic system (such as a drop of milk diffusing in a cup of tea) from the dynamics of atoms and molecules governed by quantum theory. The only hope one can have is that the microscopic world is sufficiently random, or chaotic, so that details of the microscopic dynamics average out to produce a coherent large-scale evolution.
To illustrate this mechanism with an analogue example, imagine we have a group of 60 students, and their task is to produce a sequence of 60 numbers between 1 and 6. It is of course impossible to predict how many 1's, 2's, etc. will be produced, especially if the students are allowed to communicate and construct sequences with, say, artificially many 1's. If on the other hand, each student throws a dice and the sequence is determined in this way, then most likely there will be approx. ten 1's, ten 2's, etc., i.e., the same fraction of each number. If the number of students would be larger, 100, 1000, 10000, and so on, the ratio of 1's will converge to 1:6 precisely (this is called the law of large numbers). Thus the behaviour of a "macroscopic" group of students is completely predictable, although (or rather, because) each individual acts randomly. In the same way, the random microscopic dynamics of gas molecules results in a well-predicable macroscopic evolution of the gas cloud.
The characterization of chaos and randomness in microscopic dynamics is one of the central objectives in my research. I am particularly excited about our recent progress in the investigation of an electron gas travelling through a crystal, where the atoms of the crystal are arranged at the vertices of a cubic lattice. In a joint paper with Andreas Strombergsson at Uppsala University, we explain that the randomness created by the collisions at the crystal lattice determines the evolution of a macroscopic electron cloud. This solves a puzzle posed by Dutch physicist Hendrik Lorentz (1853-1928) more than a century ago.
The breakthrough in this long-standing problem was made possible by "measure rigidity", a relatively new branch of pure mathematics that has led to a number of important applications in several areas of pure and applied mathematics. In simplistic terms, measure rigidity may be viewed as a technical tool that can detect and characterize random phenomena without the need of heavy analysis. It is an extremely effective technique, and I am convinced we will see more spectacular applications in the future. The current focus of my research is to remove some of the simplifying assumptions in Lorentz' model of the electron gas. In particular the incorporation of quantum theory is a crucial challenge, and it will be interesting to see whether our measure rigidity techniques can be adapted to this setting.
My research programme contributes to the understanding of the laws of nature, the emergence of chaos and randomness and their effect on large-scale particle dynamics. It uses tools from many diverse areas in mathematics and physics, and fosters the interaction between the subjects. As fundamental research, it provides the theoretical foundation for technological applications.