Theoretical Physics VI - Thermodynamics Sheet 11

<< Graphics`Graphics`

Problem 11.1

b)

mu = 0.5; T1 = 1; T2 =  .1; T3 = 0.05; T4 =  .01; 
    T5 =  .0001;

Bose-Einstein

DisplayTogether[
        Plot[1/(Exp[(x - mu)/T1] - 1), {x, 0.5, 0.9}, 
          PlotStyle -> Hue[0.1]], 
        Plot[1/(Exp[(x - mu)/T2] - 1), {x, 0.5, 0.9}, 
          PlotStyle -> Hue[0.3]], 
        Plot[1/(Exp[(x - mu)/T3] - 1), {x, 0.5, 0.9}, 
          PlotStyle -> Hue[0.5]], 
        Plot[1/(Exp[(x - mu)/T4] - 1), {x, 0.5, 0.9}, 
          PlotStyle -> Hue[1]], 
        AxesLabel -> {"epsilon/kb", "n(epsilon/kb)"}, 
        ImageSize -> 550];

orange=T1 green=T2 lightblue=T3 red=T4, the graphs will have the same area underneath, meaning n will get high for low range of energy states. This is especially interesting, since the number of states is getting very high for low temperature, while the range of energy states is very small, this phenomenon for T->0 is known as Bose-Einstein-Condensation, while the ground state becomes a singularity.

Fermi-Dirac

DisplayTogether[
        Plot[1/(Exp[(x - mu)/T1] + 1), {x, 0, 1.1}, 
          PlotStyle -> Hue[0.1]], 
        Plot[1/(Exp[(x - mu)/T2] + 1), {x, 0, 1.1}, 
          PlotStyle -> Hue[0.3]], 
        Plot[1/(Exp[(x - mu)/T3] + 1), {x, 0, 1.1}, 
          PlotStyle -> Hue[0.5]], 
        Plot[1/(Exp[(x - mu)/T4] + 1), {x, 0, 1.1}, 
          PlotStyle -> Hue[1]], 
        Plot[1/(Exp[(x - mu)/T5] + 1), {x, 0, 1.1}], 
        AxesOrigin -> {0, 0}, PlotRange -> All, 
        AxesLabel -> {"epsilon/kb", "n(epsilon/kb)"}, 
        ImageSize -> 550];

orange=T1 green=T2 lightblue=T3 red=T4 black=T5

Boltzmann

DisplayTogether[
        Plot[1/(Exp[(x - mu)/T1]), {x, 0,  .05}, 
          PlotStyle -> Hue[0.1]], 
        Plot[1/(Exp[(x - mu)/T2]), {x, 0,  .05}, 
          PlotStyle -> Hue[0.3]], 
        Plot[1/(Exp[(x - mu)/T3]), {x, 0,  .05}, 
          PlotStyle -> Hue[0.5]], 
        Plot[1/(Exp[(x - mu)/T4]), {x, 0,  .05}, 
          PlotStyle -> Hue[1]], AxesOrigin -> {0, 0}, 
        PlotRange -> All, 
        AxesLabel -> {"epsilon/kb", "n(epsilon/kb)"}, 
        ImageSize -> 550];

orange=T1 green=T2 lightblue=T3 red=T4, but while the high temperatures lead to very small values compared to T4 you can only see the lightblue colour which is the highest of the three high temperatures.