6. Effects of the thermostat

In the above simulations of shear flow and elongational flow, the velocity gradient does work on the system that is converted to heat, which must be removed in order to keep the temperature constant and to maintain a steady state. In a real macroscopic system, this takes place by heat conduction to the system boundaries and this could in principle be arranged in a microscopic simulation cell as well. Unfortunately, this is inconvenient because a temperature gradient of molecular dimensions would make the system inhomogeneous, and thus make it difficult to define the thermodynamic state. Therefore, the temperature is kept constant by forcing the kinetic energy to be a constant of motion by applying a Gaussian thermostat, see Eq. (5). This thermostat was originally devised independently by Hoover et al. [32–34] and by Evans [22]. The equilibrium ensemble averages of the phase functions and time correlation functions generated when this thermostat is applied are essentially canonical [35]. Away from equilibrium, it can be shown that the effect of the Gaussian thermostat on the ensemble averages is proportional to the square of the external field, whereas the thermodynamic fluxes driven by the field are directly proportional to the field in a linear transport process. Thus, the corresponding linear transport coefficients that are equal to the ratio of the flux and the field in the limit of zero field are independent of the thermostat. Therefore, transport coefficients obtained from the simulations of shear flow and elongational flow are independent of the thermostat since there is a linear relation between the velocity gradient and the shear stress in the Newtonian regime and since we are interested in the limit of zero velocity gradient. Neither is the correctness of the variational principle affected by the thermostat since it is valid in the linear regime.

The situation is different in the heat flow simulations because here we actually want a temperature gradient. This gradient is obtained by applying two bar thermostats at different temperatures acting over a limited range and separated by a distance that is long compared to this range, see Figure 7 and Eq. (14). Therefore, the movements of only a small fraction of the molecules are affected by the thermostats, whereas the movements of the majority of the molecules away from the bar thermostats are governed by the ordinary Newtonian equations of motion. Thus, it is reasonable to assume that the influence of the details of the thermostat on the ensemble averages of the phase functions is limited in this case too.
