Coarse-graining effects on time-dependent information on invariant sets

In this paper a family of information I(β) with a parameter β is introduced which is generalized from the entropy functional proposed by Goldstein et al. Using this information, the effects of coarse graining on the time-dependent characteristics of probability density on invariant sets of dissipative system were examined by comparing theoretical time-dependent characteristics of the information with coarse-grained ones. As a result, it was found that Ic(β), the coarse-grained information of I(β), collapses into a single line of Ic(2) and becomes to be independent of β near t = 0. This identification indicates that the probability density is uniformized on the invariant set, and the phenomena was found to be caused by the information production originated from the coarse graining. Thus the initial value of Ic(2) determines the macroscopic initial state which is far from steady state in the coarse-grained system. The uniformization is expected to explain the mechanism that makes the mesoscopic level difference of initial distribution lose and realizes the same macroscopic unsteady state if its macroscopic initial state is identical as irreversible processes go.