8.3.1 Hoover-Cupido

[Phys.Rev.A 22(1980)1690]

Reverting to the $q,p$ notation we introduce an appropriate perturbative term in the Hamiltonian:

$$\Delta H = \sum_{i} \left( \vec{q}_{i} \vec{p}_{i}\right):\left( \vec{\nabla}\vec{u}\right)^{T}$$ (8.23)

where $\vec{u}(\vec{q})$ is the hydrodynamic velocity field, and

$$\mbox{${\bf A}$}:\mbox{${\bf B}$}^{T} \equiv A_{\alpha \beta} \left( B^{T}\right)_{\beta \alpha}$$ (8.24)

(inner product.)

For reasons of his own, W. Hoover dubbed the construct $(qp)$ Doll's tensor.

Written in dummy index notation we have (omitting the particle index $i$)

$$\Delta H = q_{\alpha} p_{\beta} \partial_{\alpha} u_{\beta}$$ (8.25)

and the resulting equations of motion are

$$\dot{q}_{\gamma} = \frac{1}{m}p_{\gamma} +q_{\alpha}\partial_{\alpha} u_{\gamma}$$ (8.26)

$$\dot{p}_{\gamma} = K_{\gamma} -p_{\beta}\partial_{\gamma} u_{\beta}$$ (8.27)

To avoid gradual heating of the sample, a Gaussian (or other) thermostat may be added.

For the simple homogeneous shear with $\gamma \equiv \partial_{z} u_{x}$ as the only non-vanishing element of $\nabla \vec{u}$, the above equations read

$$\dot{x} = \frac{1}{m} p_{x}+\gamma z$$ (8.28)

$$\dot{y} = \frac{1}{m} p_{y}$$ (8.29)

$$\dot{z} = \frac{1}{m} p_{z}$$ (8.30)

$$\dot{p}_{x} = K_{x}$$ (8.31)

$$\dot{p}_{y} = K_{y}$$ (8.32)

$$\dot{p}_{z} = K_{z}- \gamma p_{x}$$ (8.33)

Shearing boundary conditions as introduced by Lees and Edwards may be applied. But now the flow is driven by the additional terms in the equations of motion.

Again, the stress tensor element $\sigma_{xz}$ is evaluated, and from its average the viscosity is determined as $\eta=-<\sigma_{xz}>/\gamma$.

The method may used with an oscillatory applied shear, yielding the frequency-dependent viscosity. Also, a volume dilation may be applied to determine the bulk visosity.