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NegativeTemperature.webm ‎(WebM audio/video file, VP9, length 43 s, 480 × 480 pixels, 1.6 Mbps overall, file size: 8.27 MB)

Summary

Description
English: In classical statistical mechanics temperature tells you how likely it is to occupy a given energy level. At T=0 all particles occupy the ground state. For T>0 higher energy levels become accessible, with a probability given by the Boltzmann distribution, which is essentially en exponential, so states with energies higher than ~kB T (where "kB" is the Boltzmann constant) are unlikely to be occupied. For T=∞ all states are equiprobable. On the other hand there are cases where you have more population in higher energy levels than in the lower ones (e.g. in laser's population inversion). How to describe this case?

Enter negative temperatures.

If we use a negative temperature we "flip" the Boltzmann distribution, making higher energy levels more likely to be populated. Oddly, a large negative temperature lead to a distribution not too dissimilar from the one from a large positive temperature. But a small negative temperature means that almost all particles will populate the highest energy levels. In a sense a negative temperature is MUCH hotter than a positive one!
Date
Source https://twitter.com/j_bertolotti/status/1366420591247560707
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929


Mathematica 12.0 code

n = 1000;
Emax = 100;
kb = 1;
T = -(Emax/kb)*0.01;
\[Delta] = 1;
tsteps = Table[Emax/kb E^t, {t, Log[0.01], Log[5], (Log[5] - Log[0.01])/100}];
frames = Table[
   \[ScriptCapitalD] = ProbabilityDistribution[E^(-(x/(kb T)))/Sum[E^(-(j/(kb T))), {j, 0, Emax, 1}], {x, 0, Emax, 1}];
   occupation = Transpose[{Range[Emax], BinCounts[RandomVariate[\[ScriptCapitalD], n], {1, Emax + 1, 1} ]
      }];
   Graphics[{
     Blue, 
     Table[Line[{{0, j}, {Emax*\[Delta], j}}], {j, 0, Emax*\[Delta], \[Delta]}],
     Black, 
     Table[Disk[{(Emax*\[Delta]*j)/(occupation[[k, 2]] + 1) + RandomReal[{-5, 5}], occupation[[k, 1]]}, \[Delta]], {k, 1, Emax}, {j, 1, occupation[[k, 2]]}]
     ,
     Text[ Style["\!\(\*SubscriptBox[\(E\), \(0\)]\)", Black, Bold, FontSize -> 16], {-5, 0}],
     Text[ Style["\!\(\*SubscriptBox[\(E\), \(max\)]\)", Black, Bold, FontSize -> 16], {-5, Emax}],
     Text[ Style[StringForm["T=`` \!\(\*SubscriptBox[\(E\), \\(max\)]\)/\!\(\*SubscriptBox[\(k\), \(b\)]\)", NumberForm[T/Emax*kb, {3, 2}]], Black, Bold, FontSize -> 16], {Emax/2, Emax + 7}]
     }, PlotRange -> {{-(Emax/10), Emax + Emax/10}, {-(Emax/10), Emax + Emax/10}}]
   , {T, tsteps}];
(**)
occupation = Table[{j, n/Emax}, {j, 1, Emax}];
framesInf = Table[
   Graphics[{
     Blue, 
     Table[Line[{{0, j}, {Emax*\[Delta], j}}], {j, 0, Emax*\[Delta], \[Delta]}],
     Black, 
     Table[Disk[{(Emax*\[Delta]*j)/(occupation[[k, 2]] + 1) + RandomReal[{-5, 5}], occupation[[k, 1]]}, \[Delta]], {k, 1, Emax}, {j, 1, occupation[[k, 2]]}]
     ,
     Text[Style["\!\(\*SubscriptBox[\(E\), \(0\)]\)", Black, Bold, FontSize -> 16], {-5, 0}],
     Text[Style["\!\(\*SubscriptBox[\(E\), \(max\)]\)", Black, Bold, FontSize -> 16], {-5, Emax}],
     Text[Style[StringForm["T=\[Infinity] \!\(\*SubscriptBox[\(E\), \\(max\)]\)/\!\(\*SubscriptBox[\(k\), \(b\)]\)", NumberForm[T/Emax*kb, {3, 2}]], Black, Bold, FontSize -> 16], {Emax/2, Emax + 7}]
     }, PlotRange -> {{-(Emax/10), Emax + Emax/10}, {-(Emax/10), Emax + Emax/10}}]
   , {10}];
(**)
occupation = Table[{j, If[j == 1, n, 0]}, {j, 1, Emax}];
frames0 = Table[
   Graphics[{
     Blue, 
     Table[Line[{{0, j}, {Emax*\[Delta], j}}], {j, 0, Emax*\[Delta], \[Delta]}],
     Black, 
     Table[Disk[{(Emax*\[Delta]*j)/(occupation[[k, 2]] + 1), occupation[[k, 1]]}, \[Delta]], {k, 1, Emax}, {j, 1, occupation[[k, 2]]}]
     ,
     Text[Style["\!\(\*SubscriptBox[\(E\), \(0\)]\)", Black, Bold, FontSize -> 16], {-5, 0}],
     Text[Style["\!\(\*SubscriptBox[\(E\), \(max\)]\)", Black, Bold, FontSize -> 16], {-5, Emax}],
     Text[Style[StringForm["T=0 \!\(\*SubscriptBox[\(E\), \(max\)]\)/\!\(\*SubscriptBox[\\(k\), \(b\)]\)", NumberForm[T/Emax*kb, {3, 2}]], Black, Bold, FontSize -> 16], {Emax/2, Emax + 7}]
     }, PlotRange -> {{-(Emax/10), Emax + Emax/10}, {-(Emax/10), Emax + Emax/10}}]
   , {1}];
(**)
tsteps = Table[-(Emax/kb) E^t, {t, Log[0.01], Log[5], (Log[5] - Log[0.01])/100}];
framesNeg = Table[
   \[ScriptCapitalD] = ProbabilityDistribution[E^(-(x/(kb T)))/Sum[E^(-(j/(kb T))), {j, 0, Emax, 1}], {x, 0, Emax, 1}];
   occupation = Transpose[{Range[Emax], BinCounts[RandomVariate[\[ScriptCapitalD], n], {1, Emax + 1, 1} ]
      }];
   Graphics[{
     Blue, 
     Table[Line[{{0, j}, {Emax*\[Delta], j}}], {j, 0, Emax*\[Delta], \[Delta]}],
     Black, 
     Table[Disk[{(Emax*\[Delta]*j)/(occupation[[k, 2]] + 1) + RandomReal[{-5, 5}], occupation[[k, 1]]}, \[Delta]], {k, 1, Emax}, {j, 1, occupation[[k, 2]]}]
     ,
     Text[Style["\!\(\*SubscriptBox[\(E\), \(0\)]\)", Black, Bold, FontSize -> 16], {-5, 0}],
     Text[Style["\!\(\*SubscriptBox[\(E\), \(max\)]\)", Black, Bold, FontSize -> 16], {-5, Emax}],
     Text[Style[StringForm["T=`` \!\(\*SubscriptBox[\(E\), \\(max\)]\)/\!\(\*SubscriptBox[\(k\), \(b\)]\)", NumberForm[T/Emax*kb, {3, 2}]], Black, Bold, FontSize -> 16], {Emax/2, Emax + 7}]
     }, PlotRange -> {{-(Emax/10), Emax + Emax/10}, {-(Emax/10), Emax + Emax/10}}]
   , {T, tsteps}];

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The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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URL: https://commons.wikimedia.org/wiki/user:Berto
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author name string: Jacopo Bertolotti

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inception

1 March 2021

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Date/TimeThumbnailDimensionsUserComment
current09:20, 2 March 202143 s, 480 × 480 (8.27 MB)BertoImported media from uploads:30128ef2-7b38-11eb-8389-0a7fb64cb320
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