Classical Statistical Model of Black Body Radiation: Emission Spectra and Particle Interactions

Introduction

The classical statistical model of black body radiation offers a unique perspective on understanding the emission spectra of blackbodies by focusing on particle interactions and energy distributions. This model provides a classical solution to the ultraviolet catastrophe, a challenge faced by classical physics, without relying on quantum mechanics. In this approach, the energy levels of particles are intimately tied to their interactions and collisions, leading to a deeper understanding of the observed emission spectra.

Key Concepts

Maxwell-Boltzmann Distribution and Particle Collisions

The foundation of this model lies in the Maxwell-Boltzmann distribution, a cornerstone of classical statistical mechanics. This distribution describes the statistical spread of particle energies in a system at thermal equilibrium. Crucially, it is the particle collisions and interactions that give rise to this distribution. When particles collide, they exchange energy, leading to a redistribution of energies within the system.

The Maxwell-Boltzmann distribution predicts that most particles possess moderate energies, while fewer particles have very high or very low energies. This distribution is not a result of individual particle properties but rather emerges from the collective behavior of particles and their interactions.

The following code shows how this energy level of a specific portion of the Maxwell-Boltzmann distribution is exactly the same as what Planck's law says. 

import numpy as np

from scipy import constants

from scipy.optimize import fsolve

def energy_at_percentage(T, percentage):

    k = constants.Boltzmann

    def equation(E):

        return np.exp(-E / (k * T)) - percentage

    return fsolve(equation, k*T)[0]

def frequency_from_energy(E):

    return E / constants.h 

def planck_peak_frequency(T):

    return 2.821 * constants.k * T / constants.h

temperatures = [1000, 3000, 5000, 7000, 10000, 20000]

percentage = 0.0595463665592699

print("Temperature (K) | predicted (Hz) | Planck's Law (Hz) | ratio (%)")

print("---------------------------------------------------------------------------")

for T in temperatures:

    E = energy_at_percentage(T, percentage)

    refined_freq = frequency_from_energy(E)

    planck_freq = planck_peak_frequency(T)

    ratio = refined_freq / planck_freq * 100

    print(f"{T:14d} | {refined_freq:15.6e} | {planck_freq:15.6e} | {ratio:20.17f}")

The small variations from 100% are most likely due to floating point errors.

Peak Emission Determination

This model determines peak emission frequency by identifying a specific percentage (precisely 5.95463665592699%) of particles contributing to peak emission, derived from their statistical behavior and energy distribution.

Energy Levels and Collision Events

In this model, the energy levels of particles are not solely determined by their individual kinetic energies. Instead, energy levels are directly related to collision events. When particles collide, they can gain or lose energy, and these energy exchanges contribute to the overall energy distribution of the system.

The temperature of the system, a macroscopic property, is a measure of the average kinetic energy of particles due to their random motions and interactions. It is not a property of individual atoms or particles in isolation. This distinction is essential for understanding the relationship between particle collisions and observed energy levels in blackbody emissions.

Emission Spectra and Collisions

Blackbody radiation is a macroscopic phenomenon that arises from the collective behavior of a vast number of particles. When particles collide, they may experience deceleration or acceleration, leading to changes in their kinetic energies. These energy exchanges can result in the emission or absorption of photons, as described by this classical theory. 

Avoiding the Ultraviolet Catastrophe

The Rayleigh-Jeans law predicted infinite energy at high frequencies, contradicting observations.

The classical statistical model successfully addresses the ultraviolet catastrophe by considering particle interactions and energy distributions:

Limited High-Energy Collisions: The Maxwell-Boltzmann distribution naturally limits the number of particles with extremely high energies. As a result, there are fewer high-energy collision events, leading to a decrease in the intensity of emitted radiation at higher frequencies.

Continuous Spectrum: The model explains the smooth and continuous spectrum of blackbody radiation. The continuous range of particle energies, resulting from collisions, ensures a gradual transition in intensity across frequencies.

Mathematical Framework

Calculating Energy Levels from Collisions

The energy levels of particles are determined by their collision events. The probability of a particle having a specific energy is given by the Maxwell-Boltzmann distribution. The energy level corresponding to a certain percentage of particles can be calculated using:

[ E = kT \cdot \ln\left(\frac{1}{\text{percentage}}\right) ]

Where:

( k ) is Boltzmann's constant.

( T ) is the absolute temperature in Kelvin.

"percentage" refers to the fraction of particles contributing to the energy level.

Emission Spectra and Peak Frequency

The emission spectra of blackbodies can be analyzed by considering the energy levels of particles. The peak frequency, corresponding to the maximum intensity of emitted radiation, can be determined using the energy levels calculated from collision events.

Particle Interactions and Observed Energy Levels

The relationship between particle collisions and observed energy levels in blackbody emissions is fundamental to this model. When particles collide, they exchange energy, leading to a redistribution of energies within the system. This process is directly related to the observed emission spectra:

Collision-Induced Emission: When particles collide, they may emit photons with energies corresponding to the energy exchanges during the collision. This results in the emission of radiation at specific frequencies, contributing to the overall emission spectrum.

Temperature and Collision Frequency: The temperature of the system influences the frequency and intensity of particle collisions. Higher temperatures lead to more frequent and energetic collisions, resulting in a broader range of emitted frequencies and higher intensities.  This will be explored in a different paper because of the complexity involved. 

Conclusion

The classical statistical model of black body radiation provides a compelling explanation for the emission spectra of blackbodies by considering particle interactions and energy distributions. This model highlights the importance of collision events in determining energy levels and the resulting emission spectra. By avoiding the ultraviolet catastrophe and providing a classical solution, it demonstrates the power of classical statistical mechanics in understanding complex physical phenomena.

Moreover, this model underscores the role of particle interactions in shaping the macroscopic properties of blackbody radiation. It offers a bridge between the microscopic world of particle collisions and the macroscopic observation of emission spectra, providing valuable insights into the behavior of matter and radiation.