# Nose-Hoover thermostat: real-time average problem?

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?

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.