File:Fermi Gas.gif
Summary
| Description |
English: Fermions can't occupy the same energy state, so (at zero temperature) they just pile up until the last one has an energy known as the "Fermi energy". At nonzero temperature the lower energy fermions can't do anything because there is no free state to excite them too, but the ones close to the Fermi energy can be excited, and thus (e.g.) contribute to conductivity. |
| Date | |
| Source | https://mathstodon.xyz/@j_bertolotti/112365117189864463 |
| Author | Berto |
| Permission (Reusing this file) |
https://mathstodon.xyz/@j_bertolotti/111363365323269417 |
Mathematica 14.0 code
pfd[\[CapitalEpsilon]_, \[Mu]_, T_] := 1/(Exp[(\[CapitalEpsilon] - \[Mu])/(T)] + 1);
nparticles = 100;
plot[T_] := Module[{occupation, excited, j, n},
occupation = Boole[RandomReal[] <= #] & /@ Table[pfd[en, nparticles, T], {en, 1, nparticles}];
dim = Dimensions[occupation][[1]];
excited = {};
If[Dimensions[Position[occupation, 0]][[1]] > 0,
For[i = 1, i <= Dimensions[Position[occupation, 0]][[1]], i++,
j = 0;
Until[n == 1 && (Not@MemberQ[excited, j]),
j++;
If[j > 3*T, Break[];];
n = Boole[RandomReal[] <= Exp[-(nparticles + j - Position[occupation, 0][[i, 1]])/T] ];
];
AppendTo[excited, j];
];
occupation =
Join[occupation, Table[If[MemberQ[excited, j], 1, 0], {j, 1, Max[excited]}]];
];
Plot[pfd[e, nparticles, T], {e, 0, 1.7*nparticles}, PlotStyle -> Directive[Thickness[0.01], Purple], Axes -> False, PlotRange -> {{-10, 1.9*nparticles}, {-0.1, 1.1}},
Prolog -> {Orange, Rectangle[{nparticles - T, 0}, {nparticles + T, 1}], Black, Arrow[{{0, 0}, {1.9*nparticles, 0}}], Arrow[{{0, 0}, {0, 1.1}}], Text[Style["Fermi Energy", Medium, Bold], {nparticles, 1.05}], Text[Style["\!\(\*SubscriptBox[\(k\), \(B\)]\)T", Medium, Bold, Orange], {nparticles + T + 10, 0.8}]},
Epilog -> {Table[ Disk[{j, 0}, 1*occupation[[j]]*{1, 1/nparticles}], {j, 1, Dimensions[occupation][[1]]}], Thick, Dashed, Line[{{nparticles, 0}, {nparticles, 1}}], Rotate[Text[Style["Fermi Function", Medium, Bold], {-5, 0.5}], \[Pi]/2], Text[Style["Energy", Medium, Bold], {0.8*nparticles, -0.06}]}]
]
sinstep[t_] := Sin[\[Pi]/2 t]^2;
frames = Flatten@Table[Table[plot[20*sinstep[t] + 0.001], {3}], {t, 0, 1, 0.01}];
ListAnimate[frames]
Licensing
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