Statistical mechanics characterization of spatio-compositional inhomogeneity

On the basis of a model system of pillars built of unit cubes, a two-component entropic measure for the multiscale analysis of spatio-compositional inhomogeneity is proposed. It quantifies the statistical dissimilarity per cell of the actual configurational macrostate and the theoretical reference one that maximizes entropy. Two kinds of disorder compete: (i) the spatial one connected with possible positions of pillars inside a cell (the first component of the measure), (ii) the compositional one linked to compositions of each local sum of their integer heights into a number of pillars occupying the cell (the second component). As both the number of pillars and sum of their heights are conserved, an upper limit for a pillar height hmax occurs. If due to a further constraint there is the more demanding limit h⩽h∗<hmax, the exact number of restricted compositions can be then obtained only through the generating function. However, at least for systems with exclusive composition degrees of freedom, we show that neglecting the h∗ is not destructive yet for a nice correlation of the h∗-constrained entropic measure and its less demanding counterpart, which is much easier to compute. Given examples illustrate a broad applicability of the measure and its ability to quantify some of the subtleties of a fractional Brownian motion, time evolution of a quasipattern [A.M. Rucklidge, M. Silber, SIAM J. Appl. Dyn. Syst. 8 (2009) 298 http://www.maths.leeds.ac.uk/~alastair/papers/RS_qp_siads_abs.html; A.M. Rucklidge, M. Silber, Phys. Rev. E 75 (2007) 055203] and reconstruction of a laser-speckle pattern [Y. Jiao, F.H. Stillinger, S. Torquato, Phys. Rev. E 77 031135, (the II part) (2008); Phys. Rev. E 76 031110 (the I part) (2007)], which are hard to discern.