Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations

A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H-theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard -- linear Fokker-Planck equation -- may be also related to a family of nonlinear Fokker-Planck equations.Comment: 19 pages, no figure

Chaos edges of zz-logistic maps: Connection between the relaxation and sensitivity entropic indices

Chaos thresholds of the $z$-logistic maps $x_{t+1}=1-a|x_t|^z$ $(z>1; t=0,1,2,...)$ are numerically analysed at accumulation points of cycles 2, 3 and 5. We verify that the nonextensive $q$-generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More precisely, we computationally verify $\lim_{t \to\infty}< S_{q_{sen}^{av}} >(t)/t= \lim_{t \to\infty}(t)/t \equiv \lambda_{q_{sen}^{av}}^{av}$, where the entropy $S_{q} \equiv (1- \sum_i p_i^q)/ (q-1)$ ($S_1=-\sum_ip_i \ln p_i$), the sensitivity to the initial conditions $\xi \equiv \lim_{\Delta x(0) \to 0} \Delta x(t)/\Delta x(0)$, and $\ln_q x \equiv (x^{1-q}-1)/ (1-q)$ ($\ln_1 x=\ln x$). The entropic index $q_{sen}^{av}0$ depend on both $z$ and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases as $1/t^{1/(q_{rel}-1)}$ ($q_{rel}>1$). These results led to (i) the first illustration of the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely $q_{rel}-1 \simeq A (1-q_{sen}^{av})^\alpha$, where the positive numbers $(A,\alpha)$ depend on the cycle; (ii) an unexpected and new scaling, namely $q_{sen}^{av}(cycle n)=2.5 q_{sen}^{av}(cycle 2)+ \epsilon$ ($\epsilon=-0.03$ for $n=3$, and $\epsilon = 0.03$ for $n=5$).Comment: 5 pages, 5 figure

Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos

We consider nonequilibrium probabilistic dynamics in logistic-like maps $x_{t+1}=1-a|x_t|^z$, $(z>1)$ at their chaos threshold: We first introduce many initial conditions within one among $W>>1$ intervals partitioning the phase space and focus on the unique value $q_{sen}<1$ for which the entropic form $S_q \equiv \frac{1-\sum_{i=1}^{W} p_i^q}{q-1}$ {\it linearly} increases with time. We then verify that $S_{q_{sen}}(t) - S_{q_{sen}}(\infty)$ vanishes like $t^{-1/[q_{rel}(W)-1]}$ [$q_{rel}(W)>1$]. We finally exhibit a new finite-size scaling, $q_{rel}(\infty) - q_{rel}(W) \propto W^{-|q_{sen}|}$. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.Comment: Final version with new Title and small modifications. REVTeX, 8 pages and 4 eps figure

Generalized Heisenberg Algebras and Fibonacci Series

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliar operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two previous levels. This happens, for example, for systems having the energy spectrum given by Fibonacci sequence. Moreover, the algebraic structure depends on two functions f(x) and g(x). When these two functions are linear we classify, analysing the stability of the fixed points of the functions, the possible representations for this algebra.Comment: 24 pages, 2 figures, subfigure.st

On a generalised model for time-dependent variance with long-term memory

The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of stochastic time series with time-dependent variance like it appears on a wide broad of systems besides economics in which ARCH was born. Although the ARCH process captures the so-called "volatility clustering" and the asymptotic power-law probability density distribution of the random variable, it is not capable to reproduce further statistical properties of many of these time series such as: the strong persistence of the instantaneous variance characterised by large values of the Hurst exponent (H > 0.8), and asymptotic power-law decay of the absolute values self-correlation function. By means of considering an effective return obtained from a correlation of past returns that has a q-exponential form we are able to fix the limitations of the original model. Moreover, this improvement can be obtained through the correct choice of a sole additional parameter, $q_{m}$. The assessment of its validity and usefulness is made by mimicking daily fluctuations of SP500 financial index.Comment: 6 pages, 4 figure

The effect of pre- and postharvest calcium applications on 'Hayward' kiwifruit storage ability

The benefits of calcium applications pre and postharvest on fruit storage ability have been mentioned in the bibliography. It was objective of this work to study the effect of calcium preharvest application in two different forms and calcium chloride application postharvest on 'Hayward' kiwifruit storage ability. Kiwifruit vines were sprayed with 0.03% CaCl2 or 0.03% CaO at one, three and four months before harvest. The control did not have any treatment. After harvest, half fruits were dipped for 2 min in a solution of 1% CaCl2, left to dry and stored at 0 degrees C. The other half was stored at the same temperature without any treatment. The commercial yield was not affected by treatments. During storage, fruits dipped in 1% CaCl2 softened slower and than fruits not treated. Weight loss was higher in fruits treated with CaO preharvest. SSC showed a significant decrease in fruits sprayed with CaO from 4 to 6 months storage. This work suggests that immersion of kiwifruit in 1% CaCl2 postharvest benefits storage life capacity; preharvest spraying with CaCl2 seems to be better than with CaO. However, we have to try higher calcium concentrations in order to get better results in storage ability but, without causing toxicity on the vines

Aging in Models of Non-linear Diffusion

We show that for a family of problems described by non-linear diffusion equations an exact calculation of the two time correlation function gives C(t,t')=f(t-t')g(t'), t>t', exhibiting normal and anomalous diffusions, as well as aging effects, depending on the degree of non-linearity. We discuss also the form in which FDT is violated in this class of systems. Finally we argue that in this type of models aging may be consequence of the non conservation of the "total mass".Comment: 4 pages, 1 figure, to appear in Phys.Rev.

Nonextensivity of the cyclic Lattice Lotka Volterra model

We numerically show that the Lattice Lotka-Volterra model, when realized on a square lattice support, gives rise to a {\it finite} production, per unit time, of the nonextensive entropy $S_q= \frac{1- \sum_ip_i^q}{q-1}$ $(S_1=-\sum_i p_i \ln p_i)$. This finiteness only occurs for $q=0.5$ for the $d=2$ growth mode (growing droplet), and for $q=0$ for the $d=1$ one (growing stripe). This strong evidence of nonextensivity is consistent with the spontaneous emergence of local domains of identical particles with fractal boundaries and competing interactions. Such direct evidence is for the first time exhibited for a many-body system which, at the mean field level, is conservative.Comment: Latex, 6 pages, 5 figure

Taste panel quality evaluation of 'Hayward' kiwifruit of different origins

In Portugal, the regions of Entre-Douro e Minho and Beira Litoral have exceptional conditions for the production of good quality kiwifruit. However, demand exceeds local supply resulting in importation of kiwifruit from Italy, Chile and New Zealand. Taste panels were run in December, May and June to study the preferences of Portuguese consumers for 'Hayward' kiwifruit. In December, a comparison was made between kiwifruit from Portugal and Italy, in May between kiwifruit from Portugal and Chile and in June between kiwifruit from Portugal, Chile and New Zealand. Attributes assessed by panellists were: appearance, texture, flavour, sweetness and acidity. The same samples were analyzed for: weight, equatorial and longitudinal diameter, firmness, soluble solids (as measured by refractometer) and titratable acidity. In May, panellists preferred Portuguese kiwifruit, but in December and June the preference was not so clear. In December, Portuguese kiwifruit were not at the eating ripe stage and in June they were at the end of their storage life (bad appearance but better flavour, sweetness and texture)