Entropy production in the cyclic lattice Lotka-Volterra model

Anteneodo, Celia

doi:10.1140/epjb/e2004-00379-2

Cited by 6 publications

(7 citation statements)

References 22 publications

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“…Equation ( 17) can be re-written as q lim = 1 − 1/(ν − 1), which reminds many analogous relations existing in the literature, such as q entropy = 1 − 1/d in [1,32,33].…”

Section: Let Us Focus Now On the Random Variablementioning

confidence: 62%

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

2008

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The celebrated Leibnitz triangle has a remarkable property, namely that each of its elements equals the sum of its South-West and South-East neighbors. In probabilistic terms, this corresponds to a specific form of correlation of N equally probable binary variables which satisfy scaleinvariance. Indeed, the marginal probabilities of the N -system precisely coincide with the joint probabilities of the (N − 1)-system. On the other hand, the nonadditive entropy, which grounds nonextensive statistical mechanics, is, under appropriate constraints, extremized by the). These distributions also result, as attractors, from a generalized central limit theorem for random variables which have a finite generalized variance, and are correlated in a specific way called q-independence. In order to physically enlighten this concept, we introduce here three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index ν = 1, 2, 3, . . . , unifying the Leibnitz triangle (ν = 1) and the case of independent variables (ν → ∞); (ii) two slightly different discretizations of q-Gaussians; (iii) a special family, characterized by the parameter χ, which generalizes the usual case of independent variables (recovered for χ = 1/2). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the N → ∞ probability distribution is a q-Gaussian with q = (ν − 2)/(ν − 1). Models (ii) approach q-Gaussians by construction, and we numerically show that they do so with asymptotic scale-invariance. Models (iii), like two other strictly scale-invariant models recently discussed by Hilhorst and Schehr (2007), approach instead limiting distributions which are not q-Gaussians. The scenario which emerges is that asymptotic (or even strict) scale-invariance is not sufficient but it might be necessary for having strict (or asymptotic) q-independence, which, in turn, mandates q-Gaussian attractors.

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“…Equation ( 17) can be re-written as q lim = 1 − 1/(ν − 1), which reminds many analogous relations existing in the literature, such as q entropy = 1 − 1/d in [1,32,33].…”

Section: Let Us Focus Now On the Random Variablementioning

confidence: 62%

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

2008

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“…(98). A second example is the µ-dimensional lattice Lotka-Volterra, for which it has precisely been verified [34,35] [36]. Its Hamiltonian-like behavior leads to µ = d/2, which, from Eq.…”

Section: The Nonlinear Dynamical Hintmentioning

confidence: 86%

Nonextensive Statistical Mechanics, Anomalous Diffusion and Central Limit Theorems

Tsallis

2005

Milan j. math.

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We briefly review Boltzmann-Gibbs and nonextensive statistical mechanics as well as their connections with Fokker-Planck equations and with existing central limit theorems. We then provide some hints that might pave the road to the proof of a new central limit theorem, which would play a fundamental role in the foundations and ubiquity of nonextensive statistical mechanics. The basic novelty introduced within this conjectural theorem is the generalization of the hypothesis of independence of the N random variables being summed. In addition to this, we also advance some nonlinear dynamical (possibly exact) relations which generalize the concepts of Lyapunov exponents, entropy production per unit time, and their interconnection as first proved by Pesin for chaotic systems.

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“…A large literature is devoted to its study. Here we focus on the time dependence of its configurational entropy by following [579,580] the two constituents, and the white color indicates that the cell is empty. The entropy S q of the D-dimensional model with very localized initial conditions asymptotically increases linearly with time only for…”

Section: Lattice Lotka-volterra Model For Chemical Reactions and Growthmentioning

confidence: 99%

“…The D = 2 snapshots correspond to the square lattice itself. The D = 3, 4 snapshots correspond to a two-dimensional section of the D-dimensional lattice (from[580]). …”

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confidence: 99%