A law of order estimation and leading-order terms for a family of averaged quantities on a multibaker chain system

In this study a family of local quantities defined on each partition and its averaging on a macroscopic small region, site, are defined on a multibaker chain system. On its averaged quantities, a law of order estimation (LOE) in the bulk system is proved, making it possible to estimate the order of the quantities with respect to the representative partition scale parameter $\mathrm{\Delta }$. Moreover, the form of the leading-order terms of the averaged quantities is obtained, and the form enables us to have the macroscopic quantity in the continuum limit, as $\mathrm{\Delta }\to 0$, and to confirm its partitioning independency. These deliverables fully explain the numerical results obtained by Ishida, consistent with the irreversible thermodynamics.