8.3.2 Evans-Morriss

[Computer Phys. Rep. 1/6(1984) 297]

These authors remarked that the in the stationary laminar shear case the Hoover-(qp) algorithm does not produce the correct flow pattern. As evident from equ. 8.33 the shear force acts on the ``wrong'' component of the momentum - $p_{z}$ instead of $p_{x}$. This does not destroy the relation between $\sigma_{xz}$ and $\eta$, but it results in an unphysical flow geometry.

Evans and Morris suggested a simple solution. Instead of 8.26 -8.27 they wrote

$$\dot{q}_{\gamma} = \frac{1}{m}p_{\gamma} +q_{\alpha}\partial_{\alpha} u_{\gamma}$$ (8.34)

$$\dot{p}_{\gamma} = K_{\gamma} -p_{\alpha}\partial_{\alpha} u_{\gamma}$$ (8.35)

For simple Couette flow we now have

$$\dot{x} = \frac{1}{m} p_{x}+z \gamma$$ (8.36)

$$\dot{y} = \frac{1}{m} p_{y}$$ (8.37)

$$\dot{z} = \frac{1}{m} p_{z}$$ (8.38)

$$\dot{p}_{x} = K_{x}- p_{z} \gamma \;\;\left[ -\frac{\lambda}{m} p_{x} \right]$$ (8.39)

$$\dot{p}_{y} = K_{y}\;\;\;\;\; \left[-\frac{\lambda}{m} p_{y} \right]$$ (8.40)

$$\dot{p}_{z} = K_{z}\;\;\;\;\;\left[ -\frac{\lambda}{m} p_{z}\right]$$ (8.41)

where in $\left[ \dots \right]$ we have added optional Gaussian constraint forces to keep $T=const$. It is clear that equ. 8.39 has now the right form to produce Couette flow. In addition, these authors demonstrated that their method produces the correct viscosity up to much higher (non-linear regime) shear values than the Doll's tensor method.

Evans' and Morriss' equations of motion cannot be derived from a perturbed Hamiltonian. However, they fulfill the necessary linear response condition 8.12.

The above equations of motion are known as S'LLOD equations, as they make use of the tensor $(\vec{p}\vec{q})$, the transpose of the DOLL'S tensor $(\vec{q}\vec{p})$.