Form Of Generalized Entropy Research Articles

Kinetic theory of collisionless relaxation for systems with long-range interactions

We develop the kinetic theory of collisionless relaxation for systems with long-range interactions in relation to the statistical theory of Lynden-Bell. We treat the multi-level case. We make the connection between the kinetic equation obtained from the quasilinear theory of the Vlasov equation and the relaxation equation obtained from a maximum entropy production principle. We propose a method to close the infinite hierarchy of kinetic equations for the phase level moments and obtain a kinetic equation for the coarse-grained distribution function in the form of a generalized Landau, Lenard–Balescu or Kramers equation associated with a generalized form of entropy (Chavanis, 2004). This allows us to go beyond the two-level case associated with a Fermi–Dirac-type entropy. We discuss the numerous analogies with two-dimensional turbulence. We also mention possible applications of the present formalism to fermionic and bosonic dark matter halos.

The Generalized Stochastic Smoluchowski Equation

We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids.

Application of relative entropy theory to streamwise velocity profile in open-channel flow: effect of prior probability distributions

Applying the concept of relative or cross-entropy and the principle of minimum cross-entropy, this study derives the velocity distribution in a wide open channel. Previous studies have employed Shannon entropy and the principle of maximum entropy to derive distributions of various flow variables, including velocity. Relative entropy is a generalized form of entropy and can specialize into Shannon entropy if the prior probability distribution is taken to be a uniform distribution. The prior distribution is often formulated, based on intuition, experience, or thought experiment. When deriving the velocity distribution in wide open channels, this study assumes four prior probability distributions and analyzes the effect of these assumed priors. It is found that a normal-type and a gamma-type prior can significantly influence the velocity profile, especially near the channel bed, and their prediction accuracies are superior to the previously obtained velocity distribution based on Shannon entropy. Furthermore, closed-form explicit analytical solutions are obtained for the nonlinear differential equations that arise while incorporating these priors. Experimental and field data are used to verify the derived velocity distributions, and an error analysis is carried out to evaluate their performance.

The generalized maximum Tsallis entropy estimators and applications to the Portland cement dataset

ABSTRACTTsallis entropy is a generalized form of entropy and tends to be Shannon entropy when q → 1. Using Tsallis entropy, an alternative estimation methodology (generalized maximum Tsallis entropy) is introduced and used to estimate the parameters in a linear regression model when the basic data are ill-conditioned. We describe the generalized maximum Tsallis entropy and for q = 2 we call that GMET2 estimator. We apply the GMET2 estimator for estimating the linear regression model Y = Xβ + e where the design matrix X is subject to severe multicollinearity. We compared the GMET2, generalized maximum entropy (GME), ordinary least-square (OLS), and inequality restricted least-square (IRLS) estimators on the analyzed dataset on Portland cement.

232. Generalized Entropy Based Clonal Diversity Estimation of TCR Repertoire

In depth study of T-cell receptor (TCR) repertoire, specifically clonal expansion of T cells and diversity of repertoire, can provide invaluable information in various health and diseases states including infection, cancer and genetic disorders etc. Currently, Shannon and Simpson diversity indices used to estimate TCR clonality have certain limitations. These indices are biased either towards total number of different TCR receptor types (richness) or distribution of each clonal type (evenness). We introduced here a new framework for accurate and precise quantification of clonal diversity of TCR cell population that overcomes shortcomings of already available indices. Our concept is based on the generalized form of entropy known as Renyi numbers and two components of diversity, richness and evenness. Based on the Renyi equation, we can prove that at a specific richness value the highest possible evenness obtained cannot go beyond the stated richness value. Plotting richness versus evenness helps to define maximal theoretical polyclonality and monoclonality regions and diversity of sample can be measured by estimating distances of sample from these theoretical bounds. We have validated our method on in-silico data sets and TCR repertoire study. We conclude that our idea of clonal plane and diversity index measurement provides more reliable and robust estimation of immune population diversity. It can prove to be a useful tool for quantitative characterization of TCR repertoire in immunological studies.

Emergence of cosmic space and minimal length in quantum gravity

An emergence of cosmic space has been suggested by Padmanabhan in [1]. This new interesting approach argues that the expansion of the universe is due to the difference between the number of degrees of freedom on a holographic surface and the one in the emerged bulk. In this paper, we derive, using emergence of cosmic space framework, the general dynamical equation of FRW universe filled with a perfect fluid by considering a generic form of the entropy as a function of area. Our derivation is considered as a generalization of emergence of cosmic space with a general form of entropy. We apply our equation with higher dimensional spacetime and derive modified Friedmann equation in Gauss–Bonnet gravity. We then apply our derived equation with the corrected entropy-area law that follows from Generalized Uncertainty Principle (GUP) and derive a modified Friedmann equations due to the GUP. We then derive the modified Raychaudhuri equation due to GUP in emergence of cosmic space framework and investigate it using fixed point method. Studying this modified Raychaudhuri equation leads to nonsingular solutions which may resolve singularities in FRW universe.

Generalized relative entropy in functional magnetic resonance imaging

The generalized Kullback–Leibler distance D q ( q is the Tsallis parameter) is shown to be an useful measure for analysis of functional magnetic resonance imaging (fMRI) data series. This generalized form of entropy is used to evaluate the “distance” between the probability functions p 1 and p 2 of the signal levels related to periods of stimulus and non-stimulus in event-related fMRI experiments. The probability densities of the mean distance D ̄ q (averaged over the N epochs of the entire experiment) are obtained through numerical simulations for different values of signal-to-noise ratio (SNR) and found to be fitted very well by Gamma distributions ( χ 2 < 0.0008 ) for small values of N ( N < 30 ). These distributions allow us to determine the sensitivity and specificity of the method by construction of the receiver operating characteristic (ROC) curves. The performance of the method is also investigated in terms of the parameters q and L (number of signal levels) and our results indicate that the optimum choice is q = 0.8 and L = 3 . The entropic index q is found to exert control on both sensitivity and specificity of the method. As q ( q > 0 ) is raised, sensitivity increases but specificity decreases. Finally, the method is applied in the analysis of a real event-related fMRI motor stimulus experiment and the resulting maps show activation in primary and secondary motor brain areas.