ELFT HÍRADÓ

[Szeminárium koordinátor <sem-admin AT szfki.hu>]

SZFI SeminarVarró SándorWigner FK SZFI; ELI-ALPS‘Grey photons’ and Hausdorff 
dimension of binary excitationsTuesday, 27 September 2022, 10:00, KFKI 
Campus, Bldg. 1, 2nd floor, Conference RoomIn 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] Minden érdeklődőt szívesen látunk! - Everyone is welcome 
to attend.Attila Nagysem-admin AT szfki.hu