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**From**: <vera.horvath AT eli-alps.hu>**To**: fizinfo AT lists.kfki.hu**Subject**: [Fizinfo] Meghívó az ELI ALPS szemináriumára

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

**[Fizinfo] Meghívó az ELI ALPS szemináriumára**,*vera.horvath, 10/07/2022*- <Possible follow-up(s)>
**[Fizinfo] Meghívó az ELI ALPS szemináriumára**,*vera.horvath, 10/14/2022*

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