File:BlackbodyEnergies.svg
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Summary
| Description |
English: This shows how Planck's model predicts that oscillators with "photon" constrained-energies ε ≡ hν = hc/λ greater than kT get significantly frozen out of the black body mix, thus solving the ultraviolet catastrophe to which the Rayleigh-Jeans model gave birth. This freeze-out may be explained if (for whatever reason) "quantum oscillators" exist only in packets with energy/frequency & momentum/spatial-frequency linked by the constant h = E/ν = p/g = 2πL.
 This plot was created with Gnuplot by v. |
| Date | |
| Source |
https://systemreboot.net/post/black-body-energy-as-a-sum-of-photon-oscillator-energies The probability of an oscillator having photons is given by The energy contribution of an oscillator with photons is given by The average black body energy is the sum of contributions from all oscillators. |
| Author | Arun Isaac |
| gnuplot source | click to expand
#! /usr/bin/env gnuplot
# This work is in the public domain.
#
# It was completed by Arun I in 2016.
#
# Though you are not legally obliged to do so, I would appreciate it
# if you credit me for this work by not removing this notice, and
# hopefully linking to my blog post at
# https://systemreboot.net/post/black-body-energy-as-a-sum-of-photon-oscillator-energies
# Base filename for output files
filename='blackbody-energies'
# Set terminal and output file
set term epslatex color colortext standalone
set output sprintf('%s.tex', filename)
# Axes' properties, title and key
set xrange [-0.01:5]
set yrange [0:1.05]
set mxtics 5
set ytics 0.2
set mytics 4
set xlabel '$ h\nu/kT $'
set ylabel 'Average oscillator energy (in units of $ kT $)'
set title "Planck's solution as a freeze-out of\nphoton oscillator-energies as $ h\\nu/kT \\to \\infty $"
unset key
# Line styles
set style line 1 lc 'blue' dt 4
set style line 2 lc 'red' dt 4
set style line 3 lc 'dark-yellow' dt 4
set style line 4 lc 'forest-green' dt 4
# Labels
set arrow from graph 0, first 1 to graph 1, first 1 nohead lc 'black' dt 6
set label '$ E_{\textrm{\tiny avg}}^{\textrm{\tiny Rayleigh}} = kT $' at graph 1, first 1 right offset graph -0.05, -0.05
set label '\footnotesize $ h\nu $' at 1.4, 0.15 tc ls 1
set label '\footnotesize $ 2h\nu $' at 1, 0.33 tc ls 2
set label '\footnotesize $ 3h\nu $' at 0.8, 0.45 tc ls 3
set label '\footnotesize $ 4h\nu $' at 0.65, 0.53 tc ls 4
set label '$ E_{\textrm{\tiny avg}}^{\textrm{\tiny Planck}} $' at 0.6, 0.8
# Expressions to plot
En(x,n) = n*x*exp(-n*x)*(1-exp(-x))
yN(x,N) = sum [n=1:N] En(x,n)
plot x/(exp(x) - 1) with lines lc 'black', \
for [N=1:4] yN(x,N) with lines ls N
# Convert tex -> pdf -> svg
set output
system(sprintf('pdflatex %s.tex && pdf2svg %s.pdf %s.svg', filename, filename, filename))
|
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 22:58, 31 October 2016 | | 450 × 315 (109 KB) | Arunisaac | User created page with UploadWizard |
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