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Sabir Umarov

Publications and source records attributed to Sabir Umarov.

29 records · Page 2Linked to original sources
arXiv2009

Functional-differential equations for $F_q$%-transforms of $q$-Gaussians

In the paper the question - Is the q-Fourier transform of a q-Gaussian a q'-Gaussian (with some q') up to a constant factor? - is studied for the whole range of $q\in (-\infty, 3)$. This question is connected with applicability of the q-Fourier transform in the study of limit processes in nonextensive statistical mechanics. We prove that the answer is affirmative if and only if q > 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, which are also out of the theory valid for q \ge 1. We also discuss some applications of the q-Fourier transform to nonlinear partial differential equations such as the porous medium equation.

cond-mat.stat-mech↗
arXiv2009

Variable order differential equations with piecewise constant order-function and diffusion with changing modes

In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous in the nature. Examples include diffusion in a heterogeneous media and protein movement in cell biology.

math-ph↗
arXiv2008

The Relationship between Tsallis Statistics, the Fourier Transform, and Nonlinear Coupling

Tsallis statistics (or q-statistics) in nonextensive statistical mechanics is a one-parameter description of correlated states. In this paper we use a translated entropic index: $1 - q \to q$ . The essence of this translation is to improve the mathematical symmetry of the q-algebra and make q directly proportional to the nonlinear coupling. A conjugate transformation is defined $\hat q \equiv \frac{- 2q}{2 + q}$ which provides a dual mapping between the heavy-tail q-Gaussian distributions, whose translated q parameter is between $ - 2 < q < 0$, and the compact-support q-Gaussians, between $0 < q < \infty $ . This conjugate transformation is used to extend the definition of the q-Fourier transform to the domain of compact support. A conjugate q-Fourier transform is proposed which transforms a q-Gaussian into a conjugate $\hat q$ -Gaussian, which has the same exponential decay as the Fourier transform of a power-law function. The nonlinear statistical coupling is defined such that the conjugate pair of q-Gaussians have equal strength but either couple (compact-support) or decouple (heavy-tail) the statistical states. Many of the nonextensive entropy applications can be shown to have physical parameters proportional to the nonlinear statistical coupling.

cs.IT↗
arXiv2008

Symmetric $(q,α)$-Stable Distributions. Part II: Second Representation

This paper is a continuation of papers \cite{UmarovTsallisSteinberg,UmarovTsallisGellmannSteinberg}. In Part I \cite{UmarovTsallisGellmannSteinberg} a description (representation) of $(q,α)$-stable distributions based on a $F_q$-transform was given. Here, in Part II, we present another description of these distributions. This approach generalizes results of \cite{UmarovTsallisSteinberg} (which corresponds to $α=2, Q\in [1,3)$) to the whole range of stability and nonextensivity parameters $α\in (0,2]$ and $Q \in [1,3),$ respectively. The present case $α=2$ recovers the $q$-Gaussian distributions. Similar to what is discussed in \cite{UmarovTsallisSteinberg}, a triplet $(q^{\ast},q,q_{\ast})$ arises for which the mapping $F_{q^{\ast}}: \mathcal{G}_{q} \to \mathcal{G}_{q_{\ast}}$ holds. Moreover, by unifying the two preceding descriptions, further possible extensions are discussed and some conjectures are formulated.

cond-mat.stat-mech↗
arXiv2008

Symmetric $(q,α)$-Stable Distributions. Part I: First Representation

The classic central limit theorem and $α$-stable distributions play a key role in probability theory, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the random variables that are being summed. A generalization of the BG theory, usually referred to as nonextensive statistical mechanics and characterized by the index $q$ ($q=1$ recovers the BG theory), introduces special (long range) correlations between the random variables, and recovers independence for $q=1$. Recently, a $q$-central limit theorem consistent with nonextensive statistical mechanics was established \cite{UmarovTsallisSteinberg} which generalizes the classic Central Limit Theorem. In the present paper we introduce and study symmetric $(q,α)$-stable distributions. The case $q=1$ recovers the Lévy $α$-stable distributions.

cond-mat.stat-mech↗
arXiv2008

On a representation of the inverse Fq transform

A recent generalization of the Central Limit Theorem consistent with nonextensive statistical mechanics has been recently achieved through a generalized Fourier transform, noted $q$-Fourier transform. A representation formula for the inverse $q$-Fourier transform is here obtained in the class of functions $\mathcal{G}=\bigcup_{1\le q<3}\mathcal{G}_q,$ where $\mathcal{G}_{q}=\{f = a e_{q}^{-βx2}, \, a>0, \, β>0 \}$. This constitutes a first step towards a general representation of the inverse $q$-Fourier operation, which would enable interesting physical and other applications.

cond-mat.stat-mech↗
arXiv2007

A generalization of the central limit theorem consistent with nonextensive statistical mechanics

The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized \cite{Tsallis1988} in 1988 by using the entropy $S_q = \frac{1-\sum_i p_i^q}{q-1}$ (with $q \in \mathcal{R}$) instead of its particular BG case $S_1=S_{BG}= -\sum_i p_i \ln p_i$. The theory which emerges is usually referred to as {\it nonextensive statistical mechanics} and recovers the standard theory for $q=1$. During the last two decades, this $q$-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture\cite{Tsallis2005} and numerical indications available in the literature have been, for a few years, suggesting the possibility of $q$-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case $q=1$ corresponding to standard probabilistic independence. This is what we prove in the present paper for $1 \leq q<3$. The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form $p(x) =C_q [1-(1-q) βx^2]^{1/(1-q)}$ with $β>0$, and normalizing constant $C_q$. These distributions, sometimes referred to as $q$-Gaussians, are known to make, under appropriate constraints, extremal the functional $S_q$ (in its continuous version). Their $q=1$ and $q=2$ particular cases recover respectively Gaussian and Cauchy distributions.

cond-mat.stat-mech↗
arXiv2007

Multivariate Generalizations of the q--Central Limit Theorem

We study multivariate generalizations of the $q$-central limit theorem, a generalization of the classical central limit theorem consistent with nonextensive statistical mechanics. Two types of generalizations are addressed, more precisely the {\it direct} and {\it sequential} $q$-central limit theorems are proved. Their relevance to the asymptotic scale invariance of some specially correlated systems is studied. A $q$-analog of the classic weak convergence is introduced and its equivalence to the $q$-convergence is proved for $q>1$.

cond-mat.stat-mech↗
arXiv2006

Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations

In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a diffusion process in the sense of distributions is proved. Simulations based upon multi-term fractional order differential equations are performed.

math.DS↗
arXiv2006

A fractional analog of Duhamel's principle

The well known Duhamel's principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for the corresponding homogeneous equation. In the paper one of the possible generalizations of the classical Duhamel's principle to the time-fractional pseudo-differential equations is established.

math-ph↗