ELFT HÍRADÓ

[<vera.horvath AT eli-alps.hu>]

Előadó: Varró Sándor (ELI-ALPS, Szeged; ELKH Wigner FK , Budapest)
Cím: Digital noise, and entropy as Hausdorff fractional dimension

Időpont: 2022. október 14. 9 óra
Helyszín: ELI ALPS konferenciaterme (Szeged, Wolfgang Sandner utca 3.)

Az előadás rövid összefoglalója:

In our earlier studies [1-3] we have shown that the Planck-Bose distribution
of black-body radiation can be derived from the exponential  distribution, by
splitting the continuous random energy into its integer and fractional part
[3]. The binary digits (0 and 1) of the fractional part (which may also be
considered as a sort of rounding-off error in energy measurements) inherit the
randomness, and they are independent random variables [4]. According to our
recent investigations, the variance of the  fractional part is the sum of
particle-like and a wave-like fluctuations [1-2]. In the first part of the
talk we discuss some features of the associated ‘particles’ which may be
called ‘dark quanta’ or ‘grey photons’, since at large temperatures their
energy is 2kT, where k is the Boltzmann constant and T is the absolute
temperature.
In the second part of the talk we shall discuss the statistics of a two-level
system being in thermal equilibrium with the black-body radiation. By
associating the numbers 0 and 1 to the ground state and to the excited state,
respectively, the outcomes of a series of measurements of the population can
be mapped to the continuum of numbers (like x = 0.10010110010...) of the unit
interval. The relative frequencies of digits 0 and 1 tend to the corresponding
probabilities, namely to 1 – b and b, respectively, where b is the Boltzmann
factor of the upper state.  If b = 1/2, then the points corresponding to the
realizations in the measurements visit the whole unit interval, except for a
set of (Lebesgue) measure zero. In order to compare the sizes of sets of
measure zero, the use of Hausdorff fractional dimensions has first been worked
out by Besicovitch [5], and generalized later by others. By applying the
mathematical results in [5], it turns out that the entropy of the two-level
system is k(log2) times the Hausdorff fractional dimension d of the set of
average populations in the unit interval. For instance,  in cases of b=1/2 and
b=1/5 we have d=1 and d=0.721928, respectively. The Planck entropy of the
corresponding spectral component of the black-body radiation can also be
expressed by the Hausdorff dimension [6]. Our results contribute to the
mathematics of digital processing measurement results, and may also be useful
in describing some physical systems generating random numbers.

[1] Varró S, Einstein's fluctuation formula. A historical overview.
Fluctuation and Noise Letters, 6, R11-R46 (2006). arXiv: quant-ph/0611023 .
[2] Varró S, A study on black-body radiation: classical and binary photons.
Acta Physica Hungarica B 26, 365-389 (2006). arXiv: quant-ph/0611010 .
[3] Varró S, Irreducible decomposition of Gaussian distributions and the
spectrum of black-body radiation. Physica Scripta 75, 160-169 (2007). arXiv:
quant-ph/0610184 .
[4] Varró S, The digital randomness of black-body radiation. Journal of
Physics Conference Series 414, 012041 (2013). arXiv:1301.1997 [quant-ph] .
[5] Besicovitch A S, On the sum of digits of real numbers represented in the
dyadic system. (On sets of fractional dimensions II.) Mathematische Annalen
110, 321-330 (1935).
[6] Varró S, Planck entropy expressed by the Hausdorff dimension of the set of
average excitation degrees of a two-level atom in thermal equilibrium. Talk
S7.4.1.  presented at LPHYS’18  [27th International Laser Physics Workshop,
16-20 July 2018., Nottingham, UK]

A szeminárium angol nyelvű. Minden érdeklődőt szívesen látunk.

Horváth Vera