**Speaker:** Varró Sándor (Wigner FK SZFI; ELI-ALPS)

**Title:** 'Grey photons' and Hausdorff dimension of binary excitations

**Date:** Tuesday, 27 September 2022, 10:00,

**Place:** KFKI Campus, Bldg. 1, 2nd floor, Conference Room

**Abstract:**

In our earlier studies, we have shown that the Planck-Bose distribution can be derived from the exponential distribution by splitting the continuous random energy into its integer and fractional (dark) parts [1]. The binary digits (0 and 1) of the fractional part inherit the randomness, and they are independent random variables [2]. In the meantime, we have realized that the variance of the fractional part (which may also be considered as a sort of rounding-off error in energy measurements) can be brought to the sum of a particle-like and a wave-like fluctuation. In the first part of the talk, we discuss some features of the associated 'particles', which may be called 'dark quanta' or 'grey photons', because, 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 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's fractal dimensions has first been worked out by Besicovitch [3], and generalized later by others. By applying the mathematical results in [3], it turns out that the entropy of the two-level system is k(log2) times the Hausdorff 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 S of the corresponding spectral component of the black-body radiation can also be expressed by the Hausdorff dimension [4]. Our results may be useful in describing some physical systems generating random numbers.

[1] Varró S, Irreducible decomposition of Gaussian distributions and the spectrum of black-body radiation. Physica Scripta 75, 160-169 (2007). arXiv: quant-ph/0610184 .

[2] Varró S, The digital randomness of black-body radiation. Journal of Physics Conference Series 414, 012041 (2013). arXiv:1301.1997 [quant-ph] .

[3] 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).

[4] 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]