Dataset Information

Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive.

ABSTRACT: Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S(BG) identical with -k Sigma(i) p(i) ln p(i) to be extensive, i.e., S(BG)(N) proportional, variant N for N --> infinity. In particular, if they are independent, S(BG) is strictly additive, i.e., S(BG)(N) = NS(BG)(1), for allN. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S(q) identical with k[1 - Sigma(i) p(q)(i)]/(q - 1) (with S(1) = S(BG)) for some special value of q not equal 1 to be the one which is extensive [i.e., S(q)(N) proportional, variant N for N --> infinity]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance), defined as the situation for which all marginal probabilities of the N-system coincide or asymptotically approach (for N --> infinity) the joint probabilities of the (N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule, i.e., the sum of two successive joint probabilities of the N-system coincides or asymptotically approaches the corresponding joint probability of the (N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is S(q) with q not equal 1, and not S(BG), the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.

SUBMITTER: Tsallis C

PROVIDER: S-EPMC1266086 | biostudies-literature | 2005 Oct

REPOSITORIES: biostudies-literature

ACCESS DATA

Publications

Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive.

Tsallis Constantino C Gell-Mann Murray M Sato Yuzuru Y

Proceedings of the National Academy of Sciences of the United States of America 20051017 43

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S(BG) identical with -k Sigma(i) p(i) ln p(i) to be extensive, i.e., S(BG)(N) proportional, variant N for N --> infinity. In particular, if they are independent, S(BG) is strictly additive, i.e., S(BG)(N) = NS(BG)(1), for allN. However, if the subsystems are globally correla ...[more]

PMID: 16230624

Similar Datasets

Project description:Appropriate large-scale citizen-science data present important new opportunities for biodiversity modelling, due in part to the wide spatial coverage of information. Recently proposed occupancy modelling approaches naturally incorporate random effects in order to account for annual variation in the composition of sites surveyed. In turn this leads to Bayesian analysis and model fitting, which are typically extremely time consuming. Motivated by presence-only records of occurrence from the UK Butterflies for the New Millennium data base, we present an alternative approach, in which site variation is described in a standard way through logistic regression on relevant environmental covariates. This allows efficient occupancy model-fitting using classical inference, which is easily achieved using standard computers. This is especially important when models need to be fitted each year, typically for many different species, as with British butterflies for example. Using both real and simulated data we demonstrate that the two approaches, with and without random effects, can result in similar conclusions regarding trends. There are many advantages to classical model-fitting, including the ability to compare a range of alternative models, identify appropriate covariates and assess model fit, using standard tools of maximum likelihood. In addition, modelling in terms of covariates provides opportunities for understanding the ecological processes that are in operation. We show that there is even greater potential; the classical approach allows us to construct regional indices simply, which indicate how changes in occupancy typically vary over a species' range. In addition we are also able to construct dynamic occupancy maps, which provide a novel, modern tool for examining temporal changes in species distribution. These new developments may be applied to a wide range of taxa, and are valuable at a time of climate change. They also have the potential to motivate citizen scientists.

Project description:Delimiting the habitat characteristics describing the environmental conditions required by a species has become a critical tool for predicting organismal responses to environmental change. Grinnell emphasized the effects of environmental factors on the ability of a population to maintain a positive growth rate, yet few studies have included demographic or reproductive data in analyses of the Grinnellian niche. Identifying differences in habitat exploitation patterns in response to structural variation in the environment presents an incomplete description of the ability of species to adapt to changing habitats if demographic traits are not included. We estimated the vegetation characteristics used by individuals within a population of hooded warblers (Setophaga citrina) across a spatial transect that includes three structurally different forest habitats. We predicted individuals should select similar structural characteristics within each habitat and have similar reproductive success across sample sites. In the two years post burn, adults were present but no young fledged indicating the habitat requirements necessary for reproduction were absent in this habitat. We found significant differences in habitat space occupied by individuals in unaltered and harvested habitats. Nesting habitats used by female warblers differed from available habitat. Fledging success was lower in the harvested habitat 10 to 12 years post-harvest. In the harvested habitat, fledging success was greater on mesic slopes but decreased along a habitat gradient to xeric ridgetops, suggesting compensation in habitat use does not ameliorate fitness costs. In contrast, there was no difference in the number of fledged young along this gradient in the unaltered habitat. Based solely on occupancy data, traditional ecological niche models would incorrectly conclude the environmental characteristics found across the three forested habitats are included in the Grinnellian niche of the hooded warbler. However, examination of demographic and environmental data simultaneously allows differentiation between occupied habitat space and niche space.

Dataset Information

Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive.

Publications

Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive.

Similar Datasets

OmicsDI is part of the ELIXIR infrastructure

Tweets