Correlation factor, velocity autocorrelation function and frequency-dependent tracer diffusion coefficient

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Abstract
The correlation factor, defined as the ratio between the tracer diffusion coefficient in lattice gases and the diffusion coefficient for a corresponding uncorrelated random walk, is known to assume a very simple form under certain conditions. A simple derivation of this is given with the aid of the Green-Kubo formula for the tracer diffusion coefficient and it is generalised to a frequency-dependent correlation factor. The application to lattice gases with nearly vanishing vacancy concentration is critically discussed. The derivation is also extended to more complex hopping processes where the correlation factor is a tensor. The velocity autocorrelation function of the tracer particle is demonstrated to exhibit a t-(d2+1)/ long-time tail, in agreement with mode-coupling predictions.