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Register a new userNon-distributive algebraic structures derived from nonextensive statistical mechanics
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Added by
Ernesto Pinheiro Borges
Publication date
2008
fields
Physics
and research's language is
English
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We propose a two-parametric non-distributive algebraic structure that follows from $(q,q)$-logarithm and $(q,q)$-exponential functions. Properties of generalized $(q,q)$-operators are analyzed. We also generalize the proposal into a multi-parametric structure (generalization of logarithm and exponential functions and the corresponding algebraic operators). All $n$-parameter expressions recover $(n-1)$-generalization when the corresponding $q_nto1$. Nonextensive statistical mechanics has been the source of successive generalizations of entropic forms and mathematical structures, in which this work is a consequence.
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We introduce a two-dimensional growth model where every new site is located, at a distance $r$ from the barycenter of the pre-existing graph, according to the probability law $1/r^{2+alpha_G} (alpha_G ge 0)$, and is attached to (only) one pre-existing site with a probability $propto k_i/r^{alpha_A}_i (alpha_A ge 0$; $k_i$ is the number of links of the $i^{th}$ site of the pre-existing graph, and $r_i$ its distance to the new site). Then we numerically determine that the probability distribution for a site to have $k$ links is asymptotically given, for all values of $alpha_G$, by $P(k) propto e_q^{-k/kappa}$, where $e_q^x equiv [1+(1-q)x]^{1/(1-q)}$ is the function naturally emerging within nonextensive statistical mechanics. The entropic index is numerically given (at least for $alpha_A$ not too large) by $q = 1+(1/3) e^{-0.526 alpha_A}$, and the characteristic number of links by $kappa simeq 0.1+0.08 alpha_A$. The $alpha_A=0$ particular case belongs to the same universality class to which the Barabasi-Albert model belongs. In addition to this, we have numerically studied the rate at which the average number of links $<k_i>$ increases with the scaled time $t/i$; asymptotically, $<k_i > propto (t/i)^beta$, the exponent being close to $beta={1/2}(1-alpha_A)$ for $0 le alpha_A le 1$, and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs $Gamma$-space for Hamiltonian systems) a scale-free network.
After a brief review of the present status of nonextensive statistical mechanics, we present a conjectural scenario where mixing (characterized by the entropic index $q_{mix} le 1$) and equilibration (characterized by the entropic index $q_{eq} ge 1$) play central and inter-related roles, and appear to determine {it a priori} the values of the relevant indices of the formalism. Boltzmann-Gibbs statistical mechanics is recovered as the $q_{mix}=q_{eq}=1$ particular case.
In this article we give an in depth overview of the recent advances in the field of equilibrium networks. After outlining this topic, we provide a novel way of defining equilibrium graph (network) ensembles. We illustrate this concept on the classical random graph model and then survey a large variety of recently studied network models. Next, we analyze the structural properties of the graphs in these ensembles in terms of both local and global characteristics, such as degrees, degree-degree correlations, component sizes, and spectral properties. We conclude with topological phase transitions and show examples for both continuous and discontinuous transitions.
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.
The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, using the maximization of entropy. In this article we investigate the canonical ensemble in the non-extensive statistical mechanics using a more traditional way, by considering a small system interacting with a large reservoir via short-range forces. The reservoir is characterized by generalized entropy instead of the Boltzmann-Gibbs entropy. Assuming equal probabilities for all available microstates we derive the equations of the non-extensive statistical mechanics. Such a procedure can provide deeper insight into applicability of the non-extensive statistics.
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