Thermostats for
flexible control of a statistical ensemble

Benedict Leimkuhler,
University of Edinburgh (UK)

Abstract:

Many complex systems are subject to uncertainty in the initial data, chaotic internal mixing, and unresolved interactions with an environment. For these reasons a statistical perspective is often taken: trajectories are treated as tools for computing averages with respect to some statistical ensemble (defined by a suitable phase space density). I will discuss models for computing statistics in a generalized canonical ensemble, where the density is a smooth function of the energy of the restricted system (an equilibrium state). A stochastic-dynamic "thermostat" (actually a wide family of methods) can be used to define a dynamics that characterizes the system's embedding within the larger energetic bath, which leaves the desired target distribution invariant. The advantage of these techniques is that they provide an elegant control of the desired distribution: a small perturbation is often all that is needed to achieve correct sampling, and the perturbations can be introduced in restricted phase space directions, limiting the impact on dynamical observables. Although the thermostat method does not provide a proper dynamical closure, it is very straightforward to implement in a wide range of situations. Under certain assumptions, these methods can be shown to be ergodic, meaning that almost every extended dynamics trajectory samples the equilibrium measure. I will discuss applications of thermostats in molecular dynamics and to the classical model for vortex dynamics on the disk.

Many complex systems are subject to uncertainty in the initial data, chaotic internal mixing, and unresolved interactions with an environment. For these reasons a statistical perspective is often taken: trajectories are treated as tools for computing averages with respect to some statistical ensemble (defined by a suitable phase space density). I will discuss models for computing statistics in a generalized canonical ensemble, where the density is a smooth function of the energy of the restricted system (an equilibrium state). A stochastic-dynamic "thermostat" (actually a wide family of methods) can be used to define a dynamics that characterizes the system's embedding within the larger energetic bath, which leaves the desired target distribution invariant. The advantage of these techniques is that they provide an elegant control of the desired distribution: a small perturbation is often all that is needed to achieve correct sampling, and the perturbations can be introduced in restricted phase space directions, limiting the impact on dynamical observables. Although the thermostat method does not provide a proper dynamical closure, it is very straightforward to implement in a wide range of situations. Under certain assumptions, these methods can be shown to be ergodic, meaning that almost every extended dynamics trajectory samples the equilibrium measure. I will discuss applications of thermostats in molecular dynamics and to the classical model for vortex dynamics on the disk.