Long time coarse grain leads to large deviation statistics and statistical mechanics

The long time coarse grain of the observables often shows the large deviation statistics. A reason for this is that the long time coarse grained observables lose their correlations or memories. The establishment of the large deviation statistics leads to the statistical mechanics. In this paper, we show examples to prove that the probability distribution functions are well-defined for the nonlinear dynamical systems as a result of the coarse grained time being longer than the correlation time of the systems