The computation of transport coefficients (e.g., mobility, diffusion coefficient, thermal conductivity, and shear viscosity), which measure how rapidly a perturbed system returns to equilibrium, is vitally important in molecular dynamics and mesoscale modelling. For instance, mobility (or the diffusion coefficient) has applications in ionic batteries, while the thermal conductivity of atom chains is mostly used for theoretical physics applications.

One standard approach is based on fluctuations of equilibrium dynamics, through the celebrated Green-Kubo formula based on the time integral of a suitable time correlation function. However, it is well documented that the equilibrium approach is subject to large statistical fluctuations in measuring the transport coefficient, due to the difficulty of accurately evaluating long time correlations; therefore, the signal-to-noise ratio is particularly unfavorable at long times, where there may be a significant (but unrelated) contribution to the integral defining the transport coefficient.

Alternative popular approaches include the nonequilibrium molecular dynamics (NEMD) method, where the system is subjected to external perturbations and relaxed to its steady state, in order to artificially induce larger fluctuations, thereby dramatically improving the signal-to-noise ratio of the measured response. One should however limit the magnitude of the forcing in order to avoid nonlinear effects in the response. There are various extensions/modifications of NEMD, including dual approaches where the value of the flux is fixed and the average magnitude of the forcing needed to induce it is measured [1]. There are also other nonequilibrium methods based on transient relaxations to the equilibrium state from perturbed initial conditions.

The Summer School is intended primarily for PhD students and postdocs working in the following fields:

• applied mathematics, in particular numerical methods for molecular dynamics;
• statistics and machine learning using sampling methods;
• numerical methods for physics, chemistry, materials science and computational biology.

The program is divided in two parts. The first part is devoted to general lectures (including error estimates for the computation of transport coefficients [2]) and hands-on sessions for both statisticians and applied mathematicians, and researchers from physics, chemistry, materials science and computational biology. The second part is dedicated to a training on the molecular dynamics package Molly [3], written in the Julia language, which allows an easy implementation of new numerical methods, while being computationally efficient. It is possible to attend only the first or second part of the Summer School.

IMPORTANT NOTE: In order to participate in the Summer School, please also ask your advisor to send a letter of recommendation to the organisers via email: Xiaocheng Shang (x.shang.1@bham.ac.uk) and Gabriel Stoltz (gabriel.stoltz@enpc.fr).