Planck Distribution Research Articles

Time distribution of stimulated transition probabilities

In recent years, an increasing number of experimental results have been published for a wide variety of systems in which high-precision, ultra-fast, and ultra-short laser pulses techniques have been used to study time-resolved dynamical processes. In the absence of a theoretical prediction of the time distribution of the transition probabilities, the lifetimes τ are generally extracted by fitting the decay time with an exponential, bi-exponential or three-exponential function. In fitting the data, the short-time behavior is generally neglected. The purpose of this study is to show that an explicit formula for the time distribution of the transition probability can be determined rigorously using time-dependent perturbation theory. A formula that perfectly fits an important subset of the experimental results from t = 0 to t=∞ . We show that by following a route different from the usual procedure for deriving transition amplitudes and Einstein coefficients in time-dependent perturbation theory, one ends up with a time-dependent factor, that is, a temporal distribution function of the transition probabilities, and for the Einstein coefficients. The time distribution reported here looks in the time domain similar to the Planck distribution in the frequency domain. In fact, the behavior of the time distribution, with a maximum at 2τln⁡2 , and with τ playing the role of temperature, resembles the behavior of the Planck frequency distribution. We present several examples to demonstrate that the theoretical formula allows for easier fitting of the experimental results reported in the literature. We also show that the time distribution explains the difference in resonance heights between the experimental and theoretical blue laser spectra.

Scalar QED Model for Polarizable Particles in Thermal Equilibrium or in Hyperbolic Motion in Vacuum

We consider a scalar QED (quantum electrodynamics) model for the frictional force and the momentum fluctuations of a polarizable particle in thermal equilibrium with radiation or in hyperbolic motion in a vacuum. In the former case the loss of particle kinetic energy due to the frictional force is compensated by the increase in kinetic energy associated with the momentum diffusion, resulting in the Planck distribution when it is assumed that the average kinetic energy satisfies the equipartition theorem. For hyperbolic motion in vacuum the frictional force and the momentum diffusion are similarly consistent with an equilibrium with a Planckian distribution at the temperature T=ℏa/2πkBc. The quantum fluctuations of the momentum imply that it is only the average acceleration a that is constant when the particle is subject to a constant applied force.

Missing Data Statistics Provide Causal Insights into Data Loss in Diabetes Health Monitoring by Wearable Sensors.

Data loss in wearable sensors is an inevitable problem that leads to misrepresentation during diabetes health monitoring. We systematically investigated missing wearable sensors data to get causal insight into the mechanisms leading to missing data. Two-week-long data from a continuous glucose monitor and a Fitbit activity tracker recording heart rate (HR) and step count in free-living patients with type 2 diabetes mellitus were used. The gap size distribution was fitted with a Planck distribution to test for missing not at random (MNAR) and a difference between distributions was tested with a Chi-squared test. Significant missing data dispersion over time was tested with the Kruskal-Wallis test and Dunn post hoc analysis. Data from 77 subjects resulted in 73 cleaned glucose, 70 HR and 68 step count recordings. The glucose gap sizes followed a Planck distribution. HR and step count gap frequency differed significantly (p < 0.001), and the missing data were therefore MNAR. In glucose, more missing data were found in the night (23:00-01:00), and in step count, more at measurement days 6 and 7 (p < 0.001). In both cases, missing data were caused by insufficient frequency of data synchronization. Our novel approach of investigating missing data statistics revealed the mechanisms for missing data in Fitbit and CGM data.

Effect of non-Planckian spectrum on shock velocities and subsonic to transonic transition in mid-Z witness plate elements

In this work, we obtain the shock velocities in mid-Z elements, namely, Al, Ti, and Cu by performing radiation hydrodynamic simulations using constant radiation drives consisting of equilibrium Planckian distribution along with high-energy Gaussian profiles. The drive temperatures have been varied from 80 to 270 eV, and the fraction of total energy density due to Gaussian distribution (α) changes from 0 to 0.4. Though the shock velocity in all the three elements rise with the strength of the drive temperature, its variation with α is quite complicated. Using simulations, we have shown that rear surface expansion is not responsible for the observed variation in shock velocity with the fraction of hard x rays at various temperatures and explained it in terms of distribution of total extinction coefficient over the spectral form of incident drive source leading to change in albedos. The percentage variations in shock velocities are found to be commensurate with those in albedos as expected for x-ray driven ablation process. Accurate scaling laws relating the drive temperatures with the shock velocities and α have also been obtained. In Al, subsonic to transonic transition temperature is found to increase with α, whereas in Ti it is found to decrease. We have explained the variation in transition temperature in terms of Mach number. As this transition temperature in Cu is anticipated to be high due to its higher atomic number and density, we did not observe any transition up to the temperature (500 eV) considered in this paper.

Backreaction issue for the black hole in de Sitter spacetime

We consider quantum real massive scalar field in the de Sitter-Schwarzschild space-time backround. To have an analytic head way we study in detail the two-dimensional case, assuming that the situation in four dimensions will not be much different conceptually. It is assumed, that quantum field is in a thermal state i.e. described by the planckian distribution for the exact modes in the geometry under consideration. We calculate approximately the expectation value of stress-energy tensor near the cosmological and black hole horizons. It is shown that for a generic temperature backreaction from quantum fields on the geometry cannot be neglected. Thus, de Sitter-Schwarzschild space-time geometry inevitably is strongly modified by the quantum fluctuations of the matter fields.

Particle creation and entanglement in a dispersive model with step velocity profile

We investigate particle creation and entanglement structure in a dispersive model with subliminal dispersion relation. Assuming the step function spatial velocity profile of the background flow, mode functions for a massless scalar field is exactly obtained by the matching method. Power spectrums of created particles are calculated for the subsonic and the transsonic flow cases. For the transsonic case, the sonic horizon exists and created particles show the Planckian distribution for low frequency region but the thermal property disappears for high frequency region near the cutoff frequency introduced by the non-linear dispersion. For the subsonic case, although the sonic horizon does not exist, the effective group velocity horizon appears due to the non-linear dispersion for high frequency region and approximate thermal property of the power spectrum arises. Relation between particle creation and entanglement between each mode is also discussed.

What is Hidden in the Planck Distribution Function and the Wien´s Peaks? III. Fission of Solar Photons into Thermons (“Dark Heat”)

There were derived many forms of the Planck distribution function (PDF) since its discovery by Planck in 1900 and formulae for the positions of Wien´s peaks in those distributions. There were published many attempts searching for the hidden carriers of heat because the existing known heat effects of photons cannot interpret all observed data. In this presented model we work with concept of fission of one Solar photon into two thermons – the missing carriers of heat. Properties of Solar photons and Solar thermons are compared for the surface of the Sun and in the vicinity of the Earth. Solar thermons obey the Stefan-Boltzmann law and their heat action can be experimentally analyzed in the whole volume of the Solar System. One of those effects can explain the microwave background radiation as the reflection of thermons on the surface of the Termination shock. The measure of the quantity of heat S for thermons is constant in agreement with the Carnot´s model. The specific heat of solids was newly interpreted as the joint action of three types of thermons with frequency ν/2, ν, and 2ν. The ratio of these three thermons can be experimentally determined from infrared spectra of those studied solids. This model could bring a new way to better describe the old, predicted concept of “dark heat” as appeared many times in the historical literature. We have summarized the known forms of the PDF and positions of Wien´s peaks in order to search some hidden properties in those mathematical structures. It will be shown that these very well-known formulae to all scholars might still keep some hidden surprising properties.

What is Hidden in the Planck Distribution Function and the Wien´s Peaks? II. Do Atoms Fuse Solar Photons into Gravitons?

There were derived many forms of the Planck distribution function (PDF) since its discovery by Planck in 1900 and formulae for the positions of Wien´s peaks in those distributions. There are published many attempts searching for the quantum gravity model. In this presented model we work with concept of fusion of two Solar photons into one graviton inside of atoms. The PDF and Wien´s peak for graviton number distribution was presented. The formula for the description of the graviton momentum distribution was derived. Three tests are proposed to estimate the reality of this model. The first test searches for the dependence of the Solar gravitational constant on the value of the Rydberg constant of atoms used in the source masses. There were collected experimental data for the big G value during the last decade and the confirmation of this prediction is promising. The second test should analyze the influence of temperature of other central stars on the gravitation events in those surroundings. The third test should explore the effect of the magnitude of the graviton momentum in other Stellar Systems on gravitational effects in those systems. This could be a new way to remove fitting data with the introduction of models with “dark matter”. We have summarized the known forms of the PDF and positions of Wien´s peaks in order to search some hidden properties in those mathematical structures. It will be shown that these very well-known formulae to all scholars might still keep some hidden surprising properties.

What is Hidden in the Planck Distribution Function and the Wien´s Peaks? I. Three Features of the Solar Photons

There were derived many forms of the Planck distribution function (PDF) since its discovery by Planck in 1900 and formulae for the positions of Wien´s peaks in those distributions. There are three features of the Solar photons – their refrangibility, their heat effect, and their chemical effect. The first feature – their refrangibility – can be modelled using the linear-wavelength dispersion rule for spectrometers with diffraction gratings and using the frequency-square rule for spectrometers with prisms. However, there are no accepted forms of the PDF to model heat effect of the Solar photons on atoms and molecules, and to model the effect of the PDF on chemical reactions where the Solar photons play very significant role. We have summarized the known forms of the PDF and positions of Wien´s peaks in order to search some hidden properties in those mathematical structures. It will be shown that these very well-known formulae to all scholars might still keep some hidden surprising properties.

White hole cosmology and Hawking radiation from quantum cosmological perturbations

The spacetime inside the white hole is like an anisotropic cosmological background with the past singularity playing the role of a big bang singularity. The scale factor along the extended spatial direction is contracting while the scale factor along the two-sphere is expanding. We consider an eternal Schwarzschild manifold and study quantum cosmological perturbations generated near the white hole singularity which propagate towards the past event horizon and to the exterior of the black hole. It is shown that an observer deep inside the white hole and an observer far outside the black hole both share the same vacuum. We calculate the Hawking radiation associated to these quantum white hole perturbations as measured by an observer in the exterior of the black hole. Furthermore, we also consider the Hawking radiation for the general case where the initial cosmological perturbations deep inside the white hole are in a ``nonvacuum'' state yielding to a deviation from Planck distribution. This analysis suggests that if the black hole is not entirely black (due to Hawking radiation) then the white hole is not entirely white either.

Quantum Explosions of Black Holes and Thermal Coordinates

The Hawking temperature for a Schwarzschild black hole is T=1/8πM, where M is the black hole mass. This formula is derived for a fixed Schwarzschild background metric, where the mass M could be arbitrary small. Note that, for vanishing M→0, the temperature T becomes infinite. However, the Schwarzschild metric itself is regular when the black hole mass M tends to zero; it is reduced to the Minkowski metric, and there are no reasons to believe that the temperature becomes infinite. We point out that this discrepancy may be due to the fact that the Kruskal coordinates are singular in the limit of the vanishing mass of the black hole. To elucidate the situation, new coordinates for the Schwarzschild metric are introduced, called thermal coordinates, which depend on the black hole mass M and the parameter b. The parameter b specifies the motion of the observer along a special trajectory. The thermal coordinates are regular in the limit of vanishing black hole mass M. In this limit, the Schwarzschild metric is reduced to the Minkowski metric, written in coordinates dual to the Rindler coordinates. Using the thermal coordinates, the Schwarzschild black hole radiation is reconsidered, and it is found that the Hawking formula for temperature is valid only for large black holes, while for small black holes, the temperature is T=1/2π(4M+b). The thermal observer in Minkowski space sees radiation with temperature T=1/2πb, similar to the Unruh effect with non-constant acceleration. The thermal coordinates for more general spherically symmetric metrics, including the Reissner–Nordstrom, de Sitter, and anti-de Sitter, are also considered. In these coordinates, one sees a Planck distribution with constant temperature. One obtains that the thermal Planck distribution of massless particles is not restricted to the cases of black holes or constant acceleration, but is valid for any spherically symmetric metric written in thermal coordinates.

Effect of soft and hard x-rays on shock propagation, preheating, and ablation characteristics in pure and doped Be ablators

Detailed investigations are carried out on shock, preheat, and ablation characteristics in x-ray driven beryllium based targets, a candidate ablator material for many inertial confinement fusion studies due to its high mass ablation rate. The study involves extensive radiation hydrodynamic simulations performed on pure and 1% copper doped beryllium foils irradiated by a temperature drive source consisting of both Planckian and Gaussian distributions with peaks lying in soft and hard x-ray regions, respectively. The results of steady state x-ray driven ablation and radiant heat exchange in a sub-critical shock are extended to a non-Planckian source. Based on that, new scaling relations are proposed for shock velocity, shock breakout temperature, maximum preheat temperature, and mass ablation rate with the temperature (120−200 eV) and the fraction of total energy density due to Gaussian distribution (0−0.25) of the incident drive. All parameters increase with drive temperature strength, but the presence of hard x rays does not affect them uniformly. Among all, preheat and shock breakout temperature exhibit a strong dependence on fraction of hard x rays present in the drive spectrum. The effect of doping translates into a pronounced decrease in preheat and shock breakout temperature, while mass ablation rate reduces marginally. The resulting variations in different parameters are explained on the basis of distribution of total extinction coefficient over the spectral form of an incident drive source.

Monte Carlo simulation of COVID-19 pandemic using Planck’s probability distribution

We present a Monte Carlo simulation model of an epidemic spread inspired on physics variables such as temperature, cross section and interaction range, which considers the Plank distribution of photons in the black body radiation to describe the mobility of individuals. The model consists of a lattice of cells that can be in four different states: susceptible, infected, recovered or death. An infected cell can transmit the disease to any other susceptible cell within some random range R. The transmission mechanism follows the physics laws for the interaction between a particle and a target. Each infected particle affects the interaction region a number n of times, according to its energy. The number of interactions is proportional to the interaction cross section σ and to the target surface density ρ. The discrete energy follows a Planck distribution law, which depends on the temperature T of the system. For any interaction, infection, recovery and death probabilities are applied. We investigate the results of viral transmission for different sets of parameters and compare them with available COVID-19 data. The parameters of the model can be made time dependent in order to consider, for instance, the effects of lockdown in the middle of the pandemic.

Radiative thermalization in semiclassical simulations of light-matter interaction

Prediction of the equilibrium populations in quantum dynamics simulations of molecules exposed to black-body radiation has proved challenging for semiclassical treatments, with the usual Ehrenfest and Maxwell-Bloch methods exhibiting serious failures. In this context, we explore the behavior of a recently introduced semiclassical model of light-matter interaction derived from a dissipative Lagrangian [C. M. Bustamante, E. D. Gadea, A. Horsfield, T. N. Todorov, M. C. Gonz\'alez Lebrero, and D. A. Scherlis, Phys. Rev. Lett. 126, 087401 (2021)]. It is shown that this model reproduces the Boltzmann populations for two-level systems, predicting the black-body spectra in approximate agreement with Planck's distribution. In multilevel systems, small deviations from the expected occupations are seen beyond the first excited level. By averaging over fast oscillations, a rate equation is derived from the dissipative equation of motion that makes it possible to rationalize these deviations. Importantly, it enables us to conclude that this model will produce the correct equilibrium populations provided the occupations of the lowest levels remain close to unity, a condition satisfied at low temperature or small excitations.

Jordan in The Church of The Higher Hilbert Space: Entanglement and Thermal Fluctuations

I revisit Jordan's derivation of Einstein's formula for energy fluctuations in the black body in thermal equilibrium. This formula is usually taken to represent the unification of the wave and the particle aspects of the electromagnetic field since the fluctuations can be shown to be the sum of wave-like and particle-like contributions. However, in Jordan's treatment there is no mention of the Planck distribution and all averages are performed with respect to pure number states of radiation (mixed states had not yet been discovered!). The chief reason why Jordan does reproduce Einstein's result despite not using thermal states of radiation is that he focuses on fluctuations in a small (compared to the whole) volume of the black body. The state of radiation in a small volume is highly entangled to the rest of the black body which leads to the correct fluctuations even though the overall state might, in fact, be assumed to be pure (i.e. at zero temperature). I present a simple derivation of the fluctuations formula as an instance of mixed states being reductions of higher level pure states, a representation that is affectionately known as "Church of the Higher Hilbert Space". According to this view of mixed states, temperature is nothing but the amount of entanglement between the system and its environment.Quanta 2022; 11: 1–4.

Scaling Planck's law: a unified approach to the Casimir effect and radiative heat-conductance in nanogaps.

Using Planck's law from an innovative point of view brings about the possibility to understand the common origin of the repulsive Casimir thermal pressure and the heat exchange in nanogaps. Based on a scale transformation, a procedure that removes divergences of the energy density, we prove the validity of Planck's law to describe confined thermal radiation properties in nanoscale gaps. This scaling involves a configurational temperature obtained from Wien's displacement law and having an entropic origin. We derive analytical expressions for the Casimir thermal pressure as well as for the heat conductance. Comparison of our results with experimental data shows a remarkable agreement.

Analysis of the prediction of the 2021 time-evolution of the Covid-19 pandemic in Italy using a Planck's distribution.

In a previous paper, we studied the time-evolution of the Covid-19 pandemic in Italy during the first wave of 2020 using a number of distribution laws. We concluded that the best distribution law to predict the evolution of the pandemic is a distribution of the type of Planck’s law with three parameters, provided that the basic conditions of the pandemic (such as social distancing, vaccination campaigns and new Covid variants) do not appreciably change the spread of the pandemic. In our 2020 study, we did not use the number of daily positive cases in Italy but the ratio of daily positive cases per number of daily tests, ratio today sometimes referred to as: “positivity rate.” We showed that if basic conditions do not change, the Planck’s distribution with three parameters provides very good predictions of the positivity rate about 1 month in advance. In a second paper, using the Planck’s distribution with three parameters, we predicted, about 1 month in advance, the spread of the pandemic in Italy during the Christmas 2020 holidays. Here we show that indeed in our second paper the spread of the pandemic in Italy, after 1 month, was well predicted using the Planck’s distribution with an error of a few percent only. We then studied the present (October 2021) evolution of the pandemic in Italy, and we showed that the Planck’s distribution, based on the data of July and August, predicted well the evolution of the pandemic. We then show that the peak of the positivity rate was approximately reached during the middle of August. However, the end of the Italian holidays and the start of all the activities including schools, intensive use of public transportation and further distancing measures may change again the trend of the positivity rate of the pandemic.

Dual-Temperature Acceleration Radiation

We solve for a system that emits acceleration radiation at two different temperatures. The equilibrium states occur asymptotically in Planck distributions and transition non-thermally. The model is simple enough to obtain a global solution for the radiation spectrum analytically. We present it as a potentially useful model for investigation of non-thermal vacuum acceleration radiation in the presence of final(initial) asymptotic thermodynamic horizon(less) states in equilibrium.

On the derivation of the Kompaneets equation

The relaxation of a photon bath to thermal equilibrium via Compton scattering with electrons is described in the Kompaneets equation (1957). The equation is mostly known from studies of astrophysical plasmas, for its convergence to the Planck distribution and for possible corrections to that Planck law in the cosmic microwave background, most notably from the Sunyaev–Zeldovich effect. We revisit its derivation emphasizing its structure as a Kramers–Moyal diffusion approximation to the quantum Boltzmann equation or Master equation with stimulated emission. We do not assume that the Planck law is stationary in performing the continuum approximation but we emphasize the necessity of the flux or Møller factor to arrive at a continuity equation. On the other hand, the structure allows more general assumptions than originally envisioned by Kompaneets.

Statistical Mechanics and Thermodynamics: Boltzmann's versus Planck's State Definitions and Counting †.

During the physical foundation of his radiation formula in his December 1900 talk and subsequent 1901 article, Planck refers to Boltzmann’s 1877 combinatorial-probabilistic treatment and obtains his quantum distribution function, while Boltzmann did not. For this, Boltzmann’s memoirs are usually ascribed to classical statistical mechanics. Agreeing with Bach, it is shown that Boltzmann’s 1868 and 1877 calculations can lead to a Planckian distribution function, where those of 1868 are even closer to Planck than that of 1877. Boltzmann’s and Planck’s calculations are compared based on Bach’s three-level scheme ‘configuration–occupation–occupancy’. Special attention is paid to the concepts of interchangeability and the indistinguishability of particles and states. In contrast to Bach, the level of exposition is most elementary. I hope to make Boltzmann’s work better known in English and to remove misunderstandings in the literature.