The underdamped Langevin equation is a standard model in statistical mechanics and a versatile tool to sample the Gibbs-Boltzmann distribution, or its path integral generalization by adding appropriate friction and random forcing terms to a Hamiltonian system of equations of motion. In most implementations of the model, the friction coefficient is position-independent and Markovian, and the random force is a Gaussian white noise.

More recently, different groups have suggested to adopt generalized Langevin equations (GLE) for a variety of purposes, such as atomistic and multiscale simulations of biomolecules,1 materials modelling,2 turbulence modelling,3 or the simulation of open quantum systems.4 In these approaches, two major variations on the Langevin theme occur: the friction becomes non-local or non-Markovian, and the noise process is not a white noise process, but typically coloured or fractional noise.5 In spite of their agreed-upon relevance in various applications, GLE models are much less understood with regard to their theoretical properties or their algorithmic implementation, as compared to the standard Langevin model.6

In this discussion meeting, we shall gather experts from applied mathematics, physics and chemistry to address these aspects in different contexts. We plan in-depth discussion of the state-of-the-art and the open issues, in particular

1. Markovian approximations of the GLE (e.g. low-rank approximations of memory kernels, GLE driven by fractional Brownian motion, etc.)
2. asymptotic analysis of the GLE (e.g. slow-fast systems, high-friction limits, realisation of holonomic and non-holonomic constraints, etc.)
3. memory effects in complex physical systems (kernel estimation, coarse-graining far from equilibrium, non-linear effects, etc.)