Many articles as well as the following presentation

MD Ensembles and Thermostats, page 23,

claim that the principle of the nose-hoover thermostat is "removing the real-time average"-problem.

I can see that for a canonical ensemble we need to extend the classical equations of motion according to Newton's 2nd law with some kind of a heat-bath. Further, the extended system must ensure ergodicity to be useful for MD simulations.

But what is meant by the "problem of real-time average"? How does the Sundman time-transformation help here?

Any ideas?


1 Answer 1


This point arises because of the history associated with the derivation of this method.

The original paper of Nosé involved a transformation of variables, including a scaling of the time, according to $\mathrm{d}t'=s \mathrm{d}t$ where $s$ is a dynamical variable. Therefore, solving the equations (in $t'$) and sampling the trajectories at regular intervals of $t'$ does not correspond to sampling at regular intervals of "real" time $t$: the interval depends on the value of $s$. To sample from a canonical ensemble, you need regular intervals of $t$, independent of the value of $s$, and therefore should sample the scaled-variable trajectories at varying intervals of $t'$, which is somewhat inconvenient. This is the problem being referred to.

The subsequent reformulation by Hoover eliminated this problem: there is no longer a scaled time variable. Solution of the resulting equations as a function of time has been shown to generate states sampled from the correct ensemble (assuming no ergodicity problems, which may be tackled in many cases by Nosé-Hoover chains, but this is peripheral to the question). There is a slight drawback, in that the resultant dynamics is non-Hamiltonian, but our understanding of non-Hamiltonian statistical mechanics has progressed a lot since these algorithms were formulated, and it is not a major problem.

As I understand it, the Sundman time transformation is the one I mentioned above, so I guess it is part of the problem, not part of the solution. However, an alternative method of solution to the problem, the Nosé-Poincaré algorithm, was proposed: you can read up on it here, and the original paper is by SD Bond, BJ Leimkuhler and BB Laird, J Comput Phys, 151, 114 (1999).

  • $\begingroup$ Confusion resolved. Excellent answer. Thanks a lot. $\endgroup$
    – OD IUM
    Commented Apr 8, 2019 at 16:23
  • $\begingroup$ As a general statement, nowadays I do not see good reasons to prefer Nosé-Hoover to Nosé-Poincaré thermostat. But for historical reasons. $\endgroup$ Commented Apr 8, 2019 at 16:29

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