Dynamical Systems Seminar

The classical planar periodic Lorentz gas is a deterministic model for gas particle motion in statistical mechanics. It consists of noninteracting molecules scattered by a doubly periodic array of circular obstacles. The trajectories of molecules form a 3-dimensional volume preserving flow. It is well-known (Sinai et al, 1980s) that the flow is mixing, but the rate of mixing (decay of correlations) was unknown, though conjectured by Mathematical Physicists to be of order 1/t.

In this talk, I will describe the first results on decay of correlations for such slowly mixing flows, including the verification of the conjectured 1/t decay rate for the Lorentz gas.

In the second part of the talk, I will indicate certain aspects of the proof. A key role is played by operator renewal sequences. The renewal equation is familiar from undergraduate courses treating recurrence/transience of simple random walks. In our case however the the coefficients of the renewal equations will be linear operators instead of probabilities.

The talk will include an introduction to the ideas needed from analysis and probability theory.