Profile-Unbiased Thermostat and String Phases

As we have already demonstrated, the SLLOD equations have been highly successful for studying moderate shear rate systems. To review, the equations of motion for planar Couette flow, with Nose-Hoover thermostats, - are  

Evans and Morriss wrote a new set of equations of motion for NEMD that allow for secondary flows  

The original scheme utilized Gaussian isokinetic thermostats, whereas in Eqs. [217] we have replaced it with a Nose-Hoover thermostat. In this equation, the true local streaming velocity is given by iyy, + Usi(q,). In principle, there are no restrictions on u i, so the steady state velocity can be of any form hence, Evans and Morriss refer to Eqs. [217] as profile-unbiased thermostats (PUT). The PUT scheme requires only a reasonable prescription for determining the true local streaming velocity. 

In most of the early PUT attempts, the streaming velocities did not evolve continuously over time instead, they were instantaneous quantities that were freshly reassigned at each time step of the simulation. Such PUT procedures, however, might introduce other problems. At one extreme, PBTs interpret all secondary flows as thermal fluctuations and therefore weaken them. In the other extreme, if the u j s in a PUT scheme are heavily influenced by the 

It is interesting to observe that the original Evans-Morriss PUT equations of motion [Eqs. (217)] place the momenta in an unusual frame the p s are neither in the laboratory frame, nor are they truly thermal. The quantity being thermostated, hence the true thermal momentum, is p - mUgi. Since it is at least aesthetically desirable to have direct access to the peculiar momenta, a simple change of variables can be applied. In the process, the phase space is extended to include a dynamical variable for the local streaming velocity. The new equations of motion become  

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