Generalized Langevin Equations in classical and quantum simulations

Posted Pascal Frey

Extended Discussion Meeting

Date: 4-6 October 2021

Location: ISCD – Sorbonne Université – 4 Place Jussieu, Paris

Organizers: S. Bonella (EPF-Lausanne), C. Hartmann (BTU Cottbus-Senftenberg), S. Huppert (Sorbonne Université), F. Pietrucci (Sorbonne Université)

Underdamped Langevin dynamics is a well-established tool to sample the classical Boltzmann distribution, or its path integral generalization by adding appropriate friction and random forcing terms to the underlying Hamiltonian equations of motion. In most implementations, the friction coefficient is position- independent and Markovian and the random force is a Gaussian white noise. Well-established theoretical results on the properties of the dynamics exist, together with effective algorithms to implement it in computer simulations of general systems.

More recently, different groups have suggested adopting Generalized Langevin equations (GLE) for a variety of purposes in atomistic and multiscale simulations. Of particular interest for this meeting are:

  • Projection of high-dimensional dynamics on collective variables
  • The quantum thermal bath/quantum thermostat methods
  • Generalised thermostats in classical or path integral calculations.
GLEs represent an effective mathematical approach to model dynamical trajectories of high-dimensional complex atomic systems projected on a low- dimensional space of collective variables. Due to the projection, Hamilton’s equations of motion can be approximated by a stochastic differential equation that in general is non-Markovian. Compared to alternative approaches to model kinetics, a clear physical picture of the process is preserved. An other category of methods, like the quantum thermal bath, make use of a non-Markovian GLEs to approximate zero-point energy effects in complex systems. In spite of several applications of GLEs, considerably less formal and algorithmic works exists compared to the standard case. Suggestions for additional topics of discussion from participants are also welcome.

In this extended discussion meeting, following on a previous meeting in June 2019 in Lausanne with a similar format, we shall gather experts from applied mathematics, physics, and chemistry to address the formal properties of GLEs in different contexts. We plan in-depth discussion of the state-of- the-art and the open issues. In addition to its theoretical interest, this discussion would pave the way for improved numerical implementations, fully establishing GLEs as a simulation tool and enabling to further explore their potential in new challenging applications. For example, specific items of interest will be the definition of numerical methods to reconstruct optimal GLE models from simulation (or experimental) data, and the control of the accuracy of quantum thermostat methods.

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