The properties of a statistical complexity measure that are characterized by nonextensivity in entropy have been investigated, which is of so-called disequilibrium type. Considering the composition law for two systems with different nonextensivities (quasi-multiplicativity), a nontrivial relation between the nonextensive parameters and the fluctuating bit number in information theory has been mentioned. To see the time evolution of the nonextensive complexity measure, we examine systems having a lognormal distribution, the underlying dynamics for which is known to obey a random multiplicative process in the presence of a boundary constraint.

We generalize Shannon’s information theory in a nonadditive way by focusing on the source coding theorem. The nonadditive information content we adopted is consistent with the concept of the form invariance structure of the nonextensive entropy. Some general properties of the nonadditive information entropy are studied, in addition, the relation between the nonadditivity q and the codeword length is pointed out.

We discuss how the thermodynamical Legendre transform structure can be retained not only for the arbitrary entropic form but also for the arbitrary form of the energy constraints by following the discussion of Plastino and Plastino. The thermodynamic relation between the expectation values and the conjugate Lagrange multipliers are seen to be universal. Furthermore, Gibbs' fundamental equation is shown to be unaffected by the choice of the entropy and the definition of the mean values due to the robustness of the Legendre transform structure.