  The relation between noise and Fick's diffusion coefficient in barrier
limited transport associated with hopping or tunneling mechanisms of particles
obeying the asymmetric simple exclusion processes (ASEP) is physically assessed
by Monte Carlo simulations. For a closed ring consisting of a large number of
barriers the diffusion coefficient is related explicitly to the current noise
thus revealing the existence of a generalized Nyquist-Einstein relation. Both
diffusion and noise are confirmed to decrease as the square root of the number
of barriers as a consequence of the correlation induced by ASEP. By contrast,
for an open linear chain of barriers the diffusion coefficient is found to be
no longer related to current noise. Here diffusion depends on particle
concentration but is independent of the number of barriers.
