Diffusion in a quenched heterogeneous environment in the presence of bias is considered analytically. The first-passage-time statistics can be applied to obtain the drift and the diffusion coefficient in periodic quenched environments. We show several transition points at which sample-to-sample fluctuations of the drifts or the diffusion coefficients remain large even when the system size becomes large, i.e., non-self-averaging. Moreover, we find that the disorder average of the diffusion coefficient diverges or becomes 0 when the corresponding annealed model generates superdiffusion or subdiffusion, respectively. This result implies that anomalous diffusion in an annealed model is traced by anomaly of the diffusion coefficients in the corresponding quenched model.
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