Dynamics-Based Thermostat Techniques

Deterministic trajectories can sometimes be used to sample the canonical measure. A very simple technique of this type is Hamiltonian dynamics itself. 

Recall from Chap. 5 that, as the dynamics automatically preserves functions of H, the Liouvillian satisfies Ccp(H) = 0. We also know that for Hamiltonian dynamics is skew-adjoint, i.e., = — which implies that = 0. Hence any 

If we select a set of points at random from the canonical distribution, then initiate trajectories of Hamiltonian dynamics from each of these points, the points will remain Gibbs-distributed over time. If the paths themselves are ergodic on the surfaces of constant energy H = E then the collection of paths may provide a usable sampling of the canonical distribution. Such a sampling technique relies on having the means of choosing initial points from the canonical distribution as this is the 

Alternative families of methods exist which are based on extending the phase space of the system in such a way as to (a) disturb the conserved quantity H = E, while (b) nonetheless preserving the phase space distribution. In general these 

Assuming ergodicity of the coupled system it is then possible to calculate averages with respect to the Gibbs-Boltzmann measure by direct averaging along trajectories of the extended system. 

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