The effects of introducing a small amount of non-thermal distribution (NTD) of elements in big bang nucleosynthesis (BBN) are studied by allowing a fraction of the NTD to be time-dependent so that it contributes only during a certain period of the BBN evolution. The fraction is modeled as a Gaussian-shaped function of $log(T)$, where $T$ is the temperature of the cosmos, and thus the function is specified by three parameters; the central temporal position, the width and the magnitude. The change in the average nuclear reaction rates due to the presence of the NTD is assumed to be proportional to the Maxwellian reaction rates but with temperature $T_{rm NTD} equiv zeta T$, $zeta$ being another parameter of our model. By scanning a wide four-dimensional parametric space at about half a million points, we have found about 130 points with $chi^2< 1$, at which the predicted primordial abundances of light elements are consistent with the observations. The magnitude parameter $varepsilon_0$ of these points turns out to be scattered over a very wide range from $varepsilon_0 sim 10^{-19}$ to $sim 10^{-1}$, and the $zeta$-parameter is found to be strongly correlated with the magnitude parameter $varepsilon_0$. The temperature region with $0.3times 10^9 mbox{K} lesssim T lesssim 0.4times 10^9 mbox{K}$ or the temporal region $tsimeq 10^3$ s seems to play a central role in lowering $chi^2$.
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