Planck's law is one of the outstanding laws of physics, it describes the energy of the radiation emitted by a black body at temperature T as a function of the frequency. Derived in 1900 by Max Planck 1 , with the idea of energy quantization, marked the beginning of Quantum Mechanics. It is impressive that the electromagnetic radiation emitted about 400,000 years after the Big Bang is today observed as Cosmic Microwave Background (CMB) radiation, that is the cooled remnant of the Big Bang radiation, and is well described as blackbody radiation at 2.728 K (Fig. 1A) . Furthermore, Wien law, a consequence of the Planck law, is used to determine the temperature of a star by measuring the peak frequency of the electromagnetic radiation emitted by the star. The Planck law expressed as number of photons per unit area, unit time and unit frequency,  emitted by a black body at thermal equilibrium with temperature T, is: , where h, c and kB are respectively Planck constant, speed of light and Boltzmann constant 1 . In a previous paper 2 , we analyzed the evolution of the pandemic in Italy using various distributions of the type of the Gauss (normal), Gamma, Beta, Weibull and Lognormal distributions and of the type of the Planck blackbody law (with two or three parameters). Since the number of daily swabs was changing and increasing with time, we analyzed the experimental data consisting in the number of new daily cases per daily swabs. It turned out that the best distribution to both fit the experimental data of the pandemic and to predict its future All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 28, 2020. ; evolution in Italy is a Planck distribution. In this paper we analyze the evolution of this second wave of the pandemic in Italy, using the Planck distribution with three parameters to make predictions on the number of new daily cases per swab in particular during the next Christmas holidays.
To describe the evolution of the pandemic, we analyzed the evolution of the new daily cases per daily swab using a Planck distribution with three parameters. The parameters of the Plank's law can take into account some of the parameter considered for instance in the SIR model of the pandemic 3 such as the daily number of infected persons, the transmissibility, that is the probability of infection due to the contact between a susceptible and infected individual, the average rate of contacts between susceptible and infected individuals (more persons the infected individual will contact, more persons will be infected) and the duration of infectiousness (more will last the infection of a contagious individual, more persons will be infected). In Fig. 1B , one can observe the impressive potentiality of the Planck's distribution to fit the experimental data of the pandemic, we first took the moving average of over four weeks of the ratios of daily cases per swab of Italy and we then fitted it with the Planck distribution Eq. (1).
Microwave Radiation fitted with a Planck distribution 4 at 2.728 K. B. First wave of Covid-19, moving average over four weeks of the ratios of daily cases per swab for Italy and its best fit curve using a Planck's distribution with three parameters. The red dots are the moving average of the ratios of new daily cases per swab and the solid black line is the best fit Planck's distribution.
In a previous paper 2 , using a Planck's distribution with three parameters, we fitted the experimental data of new daily cases per swab in Italy and we predicted, a few weeks ahead of time, that such ratio perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 28, 2020. ; would decrease below 0.0043 on May 27 with a three-sigma uncertainty of +/-9 days calculated using a Monte Carlo simulation with 25000 runs. Indeed, in agreement with our mathematical estimate, the first day in which such low ratio was below 0.0043 was June 4 when the ratio reached the value of 0.0035 (its historical minimum in Italy from February 25 to June 4). We also attempted to estimate the uncertainty of our prediction, possibly taking into account the various systematic errors, by calculating the spread of the pandemic among the twenty Italian regions and we obtained a one-sigma uncertainty of about 11.4 days. In Fig. 2 we reproduce our fit and mathematical prediction 2 for the number of new daily cases per swab, obtained on May 16. It agrees with a relative error of only 39% with the number of new daily cases per swab measured in June 6, i.e., three weeks later. The red dots are the new daily cases per swab in Italy from February 25 to May 16, the black solid line is the best fit Planck's function which fits these cases per swab from February 25 to May 16, it is extrapolated up to June 13. The blue star represents the real value of cases per swab measured on June 6. The difference between such measured value and that extrapolated from the Planck's fitting function is about 39%.
In Fig. 3 the daily cases-to-swab ratios smoothed with 7-days moving average are reported. The beginning of the curve corresponds to the first day in which data on nasopharyngeal swabs are available i.e. the 25 February 2020 and the last point corresponds to the 22 November 2020. The two waves of the pandemic diffusion in Italy are clearly visible. 
We now show the results of this second wave of Covid-19 analyzed using the Planck's law. In Fig. 4 are reported the daily cases-to-swab ratios for 36 days starting from 17 October 2020, The continuous curve is the best fit of the data using a three parameter Planck's law. Christmas is indicated by the vertical blue line. perpetuity.
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The copyright holder for this this version posted November 28, 2020. ;
In spite of the fact that the fluctuations of the data are very limited, those can be reduced using the moving average technique. In Fig. 5 are reported the 7-days moving average ratios of this second wave of the pandemic. perpetuity.
preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in
The copyright holder for this this version posted November 28, 2020. ; Starting from mid-October the Italian government has enacted four decrees that gradually introduced containment measures finalized to reduce the pandemic. Specifically the four DPCM (Decreto del Presidente del Consiglio dei Ministri) were in effects on 13, 18, 24 October 2020 and 3 November 2020. Many parameters, as mentioned earlier, should be taken into account for a better prediction of the effects of the decrees on the pandemic evolution, however a simpler approach can be followed as we suggest here. First a number of consecutive days has to be chosen such that they can produce a significant curve fitting. Taking 30 days as a reasonably large number of days you then perform the relevant curve fitting by sweeping the 45 days of data available. The values of the positive daily cases-to-swab ratios at Christamas holyday is then compared for the different time interval used. The results are reported in Table 1 and Fig. 8 . We observe that the values of the ratios reduce while getting closer to the last days. This is what one would expect if the measures and the individual behaviour are correctly applied. The first interval is reported for completeness although its value seems too large, even larger than the maximum value obtained in the first wave of Covid-19. Anyway the curve fitting, and consequently the prediction at Christmas, in this case has been obtained with data in the period in which the effects of the DPCM were practically absent because the effects of the restrictions show the effects with a delay of at least a couple of weeks. So the first interval will have at least 25 out of 30 days with no restrictions in effect and that may explain the high ratio obtained. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20238139 doi: medRxiv preprint 
Planck's blackbody law is one of the outstanding achievements of the physical sciences. It marked the beginning of Quantum Mechanics and remarkably described the cosmic electromagnetic radiation emitted shortly after the Big Bang observed today as cosmic microwave radiation pervading the universe. It turns out that, in addition to the various applications in physics and astronomy, it can well mathematically describe the evolution of the pandemic and well predict its future evolution, which however depends on a number of social parameters that cannot be predicted. We applied the Planck's law to analyze the experimental data of the new daily cases per swab of Italy during this second wave of the pandemic, and showed that the Planck's distribution confirms its remarkable good fitting capability and we believe a good potential for predicting the pandemic evolution. The parameters appearing in the Planck's distribution to fit the pandemic are related to the relevant parameters of the pandemic such as the daily number of infected persons, the probability of infection due to contact between a susceptible and an infected individual, the average rate of contacts between susceptible and infected individuals and the duration of infectiousness. Of course any unpredictable change in these parameters would affect the mathematical prediction with the Planck's distribution that assumes constant parameters. perpetuity.
preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in
The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20238139 doi: medRxiv preprint