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:
Let

$$H = \sum_{i}\frac{p_{i}^{2}}{2m}+\frac{1}{2}\sum_{i}\sum_{j}u_{ij} -\sum_{i}c_{i}x_{i}F$$ (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

$$A(t)= \sum_{i}c_{i}x_{i}(t); \;\;\;\;\;\;\; \dot{A}(t)=V j_{c}(t) = \sum_{i}c_{i} \dot{x}_{i}$$ (8.44)

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

Again, the external field does work on the system, heating it:

$$\frac{dH}{dt}=-F\, V\, j_{c}$$ (8.45)

The heating may once more be avoided by Gaussian terms in the eqs. of motion which now read, in 2 dimensions,

$$\dot{x}_{i} = \frac{1}{m} p_{ix}$$ (8.46)

$$\dot{y}_{i} = \frac{1}{m} p_{iy}$$ (8.47)

$$\dot{p}_{ix} = K_{ix}+c_{i} F$$ (8.48)

$$\dot{p}_{iy} = K_{iy} \;\;\;\;\;\;\;\;\;\;[- \frac{\lambda}{m}p_{iy}]$$ (8.49)

In this way a ``color conductivity''

$$\sigma_{c}\equiv \lim_{F \rightarrow 0}\frac{<j_{c}>}{F}$$ (8.50)

may be evaluated. Applying Linear Response Theory one may show that $\sigma_{c}$ is closely related to the diffusion constant:

$$D=\frac{(N-1)VkT}{N^{2}} \sigma_{c}$$ (8.51)