Stochastic methods for non-equilibrium dynamical systems

June 1 to June 5, 2015

at the

American Institute of Mathematics, San Jose, California

organized by

Mark Demers, Renato Feres, Matthew Nicol, and Hongkun Zhang

Original Announcement

This workshop will be devoted to the study of the statistical properties of dynamical systems of physical interest. Such systems include mathematical billiards and their perturbations, more general Hamiltonian mechanical systems on manifolds with boundary, the Lorentz flow, intermittent maps which model systems with slowly mixing or ''sticky'' regions, and random dynamical systems from non-equilibrium statistical physics. In particular, this workshop will seek to exploit recent advances in techniques to study non-uniformly hyperbolic systems in order to expand our understanding of statistical properties such as decay of correlations, large deviations, return time statistics and related limit laws. The workshop will bring together experts in spectral techniques, coupling, statistical mechanics and probabilistic dynamics.

The main topics for the workshop are:

  1. Investigate properties of dynamical systems of physical interest, including systems out of equilibrium. This focus includes billiards and their perturbations, including systems under external forces and time dependent systems that lack a stationary distribution. A main thrust for this class of systems is to develop a rigorous theory for such issues as entropy production, transport and diffusion coefficients.
  2. Develop a broader understanding of advanced statistical properties, such as extreme value theory, return time statistics, and large deviations theory for a broad class of dynamical systems of physical interest. In particular, some recently constructed slowly mixing billiards systems are expected to obey stable limit laws.
  3. Explore new model systems of interest to applied scientists and engineers, such as Knudsen diffusion in nano-structured channels and thermo-mechanical behavior of nano devices. The interaction of theoretical and applied mathematicians will be enhanced by the physical relevance of the systems under consideration and is a crucial aspect of the workshop.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

A spectral approach for quenched limit theorems for random expanding dynamical systems
by  Davor Dragicevic, Gary Froyland, Cecilia Gonzalez-Tokman, and Sandro Vaienti,  Comm. Math. Phys. 360 (2018), no. 3, 1121–1187  MR3803820
Stochastic dynamics: Markov chains and random transformations
by  Felix X.-F. Ye, Yue Wang and Hong Qian,  Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 7, 2337–2361  MR3543636
Almost sure invariance principle for random piecewise expanding maps
by  D. Dragicevic, G. Froyland, C. González-Tokman and S. Vaienti