Canonical sampling

Zuckerman, D.M., Lyman, E. A second look at canonical sampling of biomolecules using replica exchange simulation. J. Chem. Theory Comput. 2006, 2, 1200-2. 

In summary, a wide variety of tools are available to solve canonical sampling problems by using information from different generalized ensembles. Such... 

Example C Liquid and fee Phases of Hard Spheres. The ESPS strategy can also be applied when one of the phases is a liquid [58]. A configuration selected at random from those explored in canonical sampling of the liquid phase will serve as a reference state. Since the liquid and solid phases generally have... 

In contrast to canonical sampling distributions whose form can be written down [Eq. (1)] the Extended Sampling (ES) distributions discussed in Section III.C... 

It would be more circumspect to say that the partition function cannot in general be naturally or usefully expressed as a canonical average since that average inevitably falls into the category for which canonical sampling is inadequate [13]. 

We do not have to make the choice Ps = Pq- In many cases the obvious choice Ps = Po, though not strictly optimal, is adequate. But for some observables the choice of a canonical sampling distribution is so sub-optimal as to be useless [6]. The problem of determining phase coexistence is a case in point. 

In contrast to canonical sampling distributions whose form can be written down (1) the Extended Sampling (ES) distributions discussed in Sect. 4.3 have to be built. There is a large literature devoted to the building techniques, extending back at least as far as reference [16]. We restrict our attention to relatively recent developments (those that seem to be reflected in current practices) and we shall focus on those aspects which are most relevant to ES distributions facilitating two-phase sampling. 

The procedure just described may be referred to as a crude micro-canonical sampling procedure and is likely to be praaical only for low-dimensional systems. Much more sophisticated and efficient procedures have been developed and are in use. Rather than attempting to completely survey these methods, we refer the interested reader to the literature for more information on this topic. > i > ... 

Bussi, G., Donadio, D., Pairinello, M. Canonical sampling through velocity rescaling. J. Chem. Phys. 126, 014101(1-7) (2007)... 

Recall that for canonical sampling, the (stationary or invariant) average of an observable (j> q,p) is given as... 

When choosing the two noises to be independent, the BBK scheme will produce an error independent of stepsize h. Modifying the given noise processes in (c) such that ER j/2 = a A > 0 that recovers canonical sampling... 

Computer Investigation] Consider canonically sampling the planar system with Hamiltonian... 

We compare the trajectories and invariant distributions of the non-Newtonian line sampling thermostat (8.33)-(8.35) with three Newtonian-based thermostats Langevin dynamics (7.4), Nos6-Hoover (8.2)-(8.4) and Nos6-Hoover-Langevin (8.27)-(8.29) dynamics. We seek to canonically sample a one-dimensional system with double-well potential energy function given as... 

If the total momentum is chosen to be zero, then it will formally remain zero along trajectories (see [78, 254] for a discussion on the affect this has on canonical sampling). However, in practice, a small numerical artifact may be introduced. Moreover, under numerical discretization, bias due to finite stepsize may be present allowing a build up of total momentum in long simulations. 

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