Fractal von Neumann entropy

Abstract

We consider the fractal von Neumann entropy associated with the fractal distribution function and we obtain for some universal classes h of fractons their entropies. We obtain also for each of these classes a fractal-deformed Heisenberg algebra. This one takes into account the braid group structure of these objects which live in two-dimensional multiply connected space.

On non-extensive statistics, chaos and fractal strings

“…Not surprsingly this should have important applications to the work on fractal strings and sprays ( branes ). Closely related to this is its relation with the fractal spin statistics , the fractal index and the Knot invariants associated with the quantum paths of quasi-particles ( fractons ) in spacetime with fractal Hausdorff dimensions [ 15 ].…”

“…In particular, the statistics for quasi-particles of continous spin based on the generalized Fibonnaci series has been studied in [ 51 ] ; the statistics for quasi-particles coined f ractons with fractal spin has been developed by da Cruz [ 15 ] ; the so-called detailed-balanced statistics models associated with particles of intermediate statistics between bosons and fermions was presented in [ 49 ]; and the fractal inspired statistics model within the context of Tsallis statistics studied earlier by Oprisal [46 ] . It is warranted to find a unifying picture, if possible, of all these different statistical descriptions associated with quasi-particles of continous spin ( fractal spin ) and with intermediate statistics.…”

A fractal-like structure for the fractional quantum Hall effect

“…The foundational basis of our theoretical formulation is free of any empirical formula and this characteristic constitutes the great difference between our insight and others of the literature. Besides this, we have obtained other results, such as [3][4][5][6][7][8][9][10]: a relation between the fractal parameter and the Rogers dilogarithm function, through the concept of fractal index, which is defined in terms of the partition function associated with each universal class of particles; a connection between the fractal parameter h and the Farey sequences of rational numbers. Farey series F n of order n is the increasing sequence of irreducible fractions in the range 0 − 1 whose denominators do not exceed n. We have the following Theorem [8]: The elements of the Farey series F n of the order n, belong to the fractal sets, whose Hausdorff dimensions are the second fractions of the fractal sets.…”

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

“…We can check that all experimental data for the occurrence of FQHE satisfies a symmetry principle discovered by us, that is, the duality symmetry between universal classes h of particles or quasiparticles with any value of spin [1][2][3][4][5]. For example, we have the dual filling factors (f,f ) =…”