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The Dynamic Foundation of Fractal Operators.

Description: The fractal operators discussed in this dissertation are introduced in the form originally proposed in an earlier book of the candidate, which proves to be very convenient for physicists, due to its heuristic and intuitive nature. This dissertation proves that these fractal operators are the most convenient tools to address a number of problems in condensed matter, in accordance with the point of view of many other authors, and with the earlier book of the candidate. The microscopic foundation of the fractal calculus on the basis of either classical or quantum mechanics is still unknown, and the second part of this dissertation aims at this important task. This dissertation proves that the adoption of a master equation approach, and so of probabilistic as well as dynamical argument yields a satisfactory solution of the problem, as shown in a work by the candidate already published. At the same time, this dissertation shows that the foundation of Levy statistics is compatible with ordinary statistical mechanics and thermodynamics. The problem of the connection with the Kolmogorov-Sinai entropy is a delicate problem that, however, can be successfully solved. The derivation from a microscopic Liouville-like approach based on densities, however, is shown to be impossible. This dissertation, in fact, establishes the existence of a striking conflict between densities and trajectories. The third part of this dissertation is devoted to establishing the consequences of the conflict between trajectories and densities in quantum mechanics, and triggers a search for the experimental assessment of spontaneous wave-function collapses. The research work of this dissertation has been the object of several papers and two books.
Date: May 2003
Creator: Bologna, Mauro
Partner: UNT Libraries

Linear response at criticality

Description: In this article, the authors study a set of cooperatively interacting units at criticality, and prove with analytical and numerical arguments that they generate the same renewal non-Poisson intermittency as that produced by blinking quantum dots.
Date: October 24, 2012
Creator: Svenkeson, Adam; Bologna, Mauro & Grigolini, Paolo
Partner: UNT College of Arts and Sciences

From power law intermittence to macroscopic coherent regime

Description: Article addressing the problem of establishing which is the proper form of quantum master equation generating a survival probability identical to that corresponding to the nonergodic sequence of "light on" and "light off" fluorescence fluctuations in blinking quantum dots.
Date: June 25, 2009
Creator: Bologna, Mauro; Budini, Adrián A.; Giraldi, Filippo & Grigolini, Paolo
Partner: UNT College of Arts and Sciences

Trajectory versus probability density entropy

Description: In this article, the authors show that the widely accepted conviction that a connection can be established between the probability density entropy and the Kolmogorov-Sinai (KS) entropy is questionable.
Date: 2001
Creator: Bologna, Mauro; Grigolini, Paolo; Karagiorgis, Markos & Rosa, Angelo
Partner: UNT College of Arts and Sciences

Renewal aging and linear response

Description: Paper discussing the linear response to an external perturbation of a renewal process, in an aging condition that, with no perturbation, would yield super-diffusion.
Date: February 6, 2008
Creator: Allegrini, Paolo; Ascolani, Gianluca; Bologna, Mauro & Grigolini, Paolo
Partner: UNT College of Arts and Sciences

Ergodicity breaking and localization

Description: This article studies the joint action of the non-Poisson renewal events (NPR) yielding Continuous-time random walk (CTRW) with index α < 1 and two different generators of Hurst coefficient H ≠ 0.5, one generating fractional Brownian motion (FBM) and another scaled Brownian motion (SBM).
Date: July 25, 2016
Creator: Geneston, Elvis L.; Tuladhar, Rohisha; Beig, Mirza Tanweer A.; Bologna, Mauro & Grigolini, Paolo
Partner: UNT College of Arts and Sciences