Thermostats for
flexible control of a statistical ensemble
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.