Maximal Diversity and Zipf’s Law

Emergent Ecological Patterns and Modelling of Gut Microbiomes in Health and in Disease

“…, which describes the abundance fluctuations of a species among different samples, without considering compositionality [36, 19] and sampling effects [17]. Following a recent proposal [37], we refer to it as the bare distribution. If we now consider compositionality and work with the relative populations , then the latter are distributed following the SDD [31], which describes the “dressed” joint probability distribution of S − 1 species relative abundance, i.e., where , δ is the Dirac delta and is the normalization constant.…”

Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model

“…In Ref. [21] it was argued that an upper bound for H is obtained when N is divided into smaller domains in a deterministic way: one cluster of each size, starting from 1, until the system is fully covered. Obviously, for stochastic systems this bound is never approached either because of repeated sizes or too large clusters that decrease their total number.…”

“…The diversity of the clusters may be quantified by the so-called cluster size heterogeneity H, the average number of distinct cluster sizes occurring in a finite sample, irrespective of the number of domains that are equally sized. This observable was studied in percolation [14][15][16], spin equilibrium models [17][18][19] and recently extended to out-of-equilibrium [20] and more general contexts [21]. The rich behavior of the time-dependent heterogeneity, H (L, t ), confirms that it may be useful to understand the interplay between percolation and coarsening dynamics.…”

Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model

“…The non homogeneity of the clusters may be quantified by the so-called cluster size heterogeneity H, the average number of distinct cluster sizes occurring in a finite sample, irrespective of the number of domains that are equally large. This observable was studied in percolation [14][15][16], spin equilibrium models [17][18][19] and recently extended to out-ofequilibrium [20] and more general contexts [21]. The rich behavior of the time-dependent heterogeneity, H(L, t), confirms that it may be useful to understand the interplay between percolation and coarsening dynamics.…”