File:NegativeTemperature.webm
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}];
Licensing
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