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

Revaluation of the first-order upwind difference scheme to solve coarse-grained master equations

“…It is clear that such a relation cannot be found when the continuity equation (1) is discretized by higher-order upwind difference schemes. Hence it should be emphasized that this coincidence is only numerical and has been verified by authors [21] (see also Appendix A).…”

“…Moreover it provides the time-dependent measure on the most probable region that affects the behavior of macroscopic observables [21]. In the (non-equilibrium) steady state the entropy agrees with the conventional coarse-grained entropy times À1 [7,12].…”

“…The entropy differs from the conventional one in that P j is partially replaced with P Ã j and is not multiplied by À1. The entropy corresponds to a member of a family of information for the coarse-grained system proposed by authors [21], generalized from an entropy functional on the invariant set proposed by Goldstein et al [1]. The entropy does not involve the identification of the repelling area in the computational domain and is easy to compute its time variations.…”

“…The scheme is very superior to the Monte-Carlo method in computer time and storage for solving the CGMEs and is effective even in multi-dimensional cases provided the partition size D is sufficiently small. Therefore, we authors has been numerically verified the agreement and used the UDS to evaluate the above-mentioned time-dependent information on invariant sets [21]. However the explanation for the agreement was entirely insufficient from the theoretical and computational viewpoints.…”

“…Therefore, the fluctuation of a real state point (''realization'' [15]) that corresponds to the change of the unsteady probability measure, or that of the probability density (measure) due to the uncertainty of the state stray from the main topic of this study; the fluctuation or noise raised by the Monte-Carlo simulations [16][17][18][19][20] are not involved in this study. In addition, if we apply the Monte-Carlo integration to evaluate the transition probability every time when needed for saving storage, the fluctuation of the computed probability is inevitable, and the ''noise'' essentially reduces the accuracy of lower probability measure and, therefore, they are not applicable to the evaluation of such a family of information I (b) on the invariant set [21] because the information of b < 0 extracts the characteristics of lower probability region. The methods to resolve the problem, e.g.…”