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8.4 Diffusion by NEMD

To determine the diffusion constant $D$ Ciccotti et al. [J. Stat. Phys. 21(1979)1] suggest the following Hamiltonian non-equilibrium method:
H = \sum_{i}\frac{p_{i}^{2}}{2m}+\frac{1}{2}\sum_{i}\sum_{j}u_{ij}
\end{displaymath} (8.43)

where $c_{i}=\pm 1$ for even/odd numbered particles is a ``color charge'' coupling to the applied ``color field'' $F$.

In other words, particles are drawn in opposite directions depending on their number; otherwise, they are completely identical.

Keeping $F$ constant we have
A(t)= \sum_{i}c_{i}x_{i}(t); \;\;\;\;\;\;\;
\dot{A}(t)=V  j_{c}(t) = \sum_{i}c_{i}  \dot{x}_{i}
\end{displaymath} (8.44)

where $j_{c}$ is the ``color current''.

Again, the external field does work on the system, heating it:
\frac{dH}{dt}=-F\, V\, j_{c}
\end{displaymath} (8.45)

The heating may once more be avoided by Gaussian terms in the eqs. of motion which now read, in 2 dimensions,
$\displaystyle \dot{x}_{i}$ $\textstyle =$ $\displaystyle \frac{1}{m}  p_{ix}$ (8.46)
$\displaystyle \dot{y}_{i}$ $\textstyle =$ $\displaystyle \frac{1}{m}  p_{iy}$ (8.47)
$\displaystyle \dot{p}_{ix}$ $\textstyle =$ $\displaystyle K_{ix}+c_{i}  F$ (8.48)
$\displaystyle \dot{p}_{iy}$ $\textstyle =$ $\displaystyle K_{iy} \;\;\;\;\;\;\;\;\;\;[- \frac{\lambda}{m}p_{iy}]$ (8.49)

In this way a ``color conductivity''
\sigma_{c}\equiv \lim_{F \rightarrow 0}\frac{<j_{c}>}{F}
\end{displaymath} (8.50)

may be evaluated. Applying Linear Response Theory one may show that $\sigma_{c}$ is closely related to the diffusion constant:
D=\frac{(N-1)VkT}{N^{2}}  \sigma_{c}
\end{displaymath} (8.51)

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F. J. Vesely / University of Vienna