Consider a room divided by imaginary lines into three equal parts. Sketch a two-axis plot of the number of ways of arranging particles versus ${\mathit{N}}_{\mathbf{\text{left}}}$and${\mathit{N}}_{\mathbf{\text{right}}}$for the case$\mathit{N}\mathbf{=}{\mathbf{10}}^{\mathbf{23}}$, Note that ${\mathit{N}}_{\mathbf{\text{middle}}}$is not independent, being of course$\mathit{N}\mathbf{-}{\mathit{N}}_{\mathbf{\text{nght}}}\mathbf{-}{\mathit{N}}_{\mathbf{\text{left}}}$Your axes should berole="math" localid="1658331658925" ${\mathit{N}}_{\mathbf{\text{left}}}$and${\mathit{N}}_{\mathbf{\text{right}}}$, and the number of ways should be represented by density of shading. (A form for numbers of ways applicable to a three-sided room is given in Appendix I. but the question can be answered without it.)

The number of ways of arranging particles in the room is given by:${\mathit{W}}_{{\mathbf{N}}_{\mathbf{\text{riju}}}\mathbf{,}{\mathbf{N}}_{\mathbf{\text{hat}}}}^{\mathbf{N}}\mathbf{=}\frac{\mathbf{N}\mathbf{!}}{{\mathbf{N}}_{\mathbf{\text{right}}}\mathbf{!}{\mathbf{N}}_{\mathbf{\text{left}}}\mathbf{!}\mathbf{(}\mathbf{N}\mathbf{-}{\mathbf{N}}_{\mathbf{\text{left}}}\mathbf{-}{\mathbf{N}}_{\mathbf{\text{right}}}\mathbf{)}\mathbf{!}}\mathbf{.}$

In this problem, we are to consider a room divided into three equal parts. The total number of particles in the room is given to be$N={10}^{23}$ .

The room is now divided into three equal parts, then the number of ways of arranging particles in the room is given by:

${\mathrm{W}}_{{\mathrm{N}}_{\text{riju}},{\mathrm{N}}_{\text{hat}}}^{\mathrm{N}}=\frac{\mathrm{N}!}{{\mathrm{N}}_{\text{right}}!{\mathrm{N}}_{\text{left}}!(\mathrm{N}-{\mathrm{N}}_{\text{left}}-{\mathrm{N}}_{\text{right}})!}.$ …… (1)

Since we are dealing with an enormous amount of particles $~{10}^{23}$, it will be more convenient to take the logarithm of eq. (I) and the desired surface plot in terms of the logarithm of$W$ versus$N_{\text{left}}$ and$N_{\text{right-}}$ .

Taking the logarithm of eq. (I) we obtain:

$\begin{array}{c}\ln W=\ln \left[\frac{N!}{N_{\text{right}}!N_{\text{left}}!(N-N_{\text{left}}-N_{\text{right}})!}\right]\\ =\ln N!-\ln [N_{\text{right}}!N_{\text{left}}!(N-N_{\text{left}}-N_{\text{right}})!\\ =\ln N!-\ln N_{\text{right}}!-\ln N_{\text{left}}!-\ln (N-N_{\text{left}}-N_{\text{right}})!\end{array}$

Now, we make use of Stirling's approximation,

$\mathrm{lnx}!\approx \mathrm{xlnx}-\mathrm{x}$

$\begin{array}{c}\mathrm{lnW}\approx \mathrm{NlnN}-\mathrm{N}-{\mathrm{N}}_{\text{right}}{\mathrm{lnN}}_{\text{right}}+{\mathrm{N}}_{\text{right}}-{\mathrm{N}}_{\text{left}}{\mathrm{lnN}}_{\text{left}}+{\mathrm{N}}_{\text{left}}\\ -(\mathrm{N}-{\mathrm{N}}_{\text{left}}-{\mathrm{N}}_{\text{right}})\ln (\mathrm{N}-{\mathrm{N}}_{\text{left}}-{\mathrm{N}}_{\text{right}})+(\mathrm{N}-{\mathrm{N}}_{\text{left}}-{\mathrm{N}}_{\text{right}})\mathrm{n}\\ \approx \mathrm{NlnN}-{\mathrm{N}}_{\text{right}}{\mathrm{lnN}}_{\text{right}}-{\mathrm{N}}_{\text{left}}{\mathrm{lnN}}_{\text{left}}-(\mathrm{N}-{\mathrm{N}}_{\text{left}}-{\mathrm{N}}_{\text{right}})\ln (\mathrm{N}-{\mathrm{N}}_{\text{left}}-{\mathrm{N}}_{\text{right}})\end{array}$

We have a convenient form for$W$ which takes into account large values of $N$, we write a script using an appropriate software to represent as a surface plot the dependence of $\ln W$onto$N_{\text{left}}$ and$N_{\text{right}}$ :

$\text{N}=1\text{e}23$;

$[\text{Nr},\text{Nl}]=$meshgrid;$(1:1\text{e}21:5\text{e}22)$

$W=N^{*}\log \left(N\right)-Nr.*\log \left(Nr\right)-N1.*\log \left(N1\right)-(N-N1-Nr).*\log (N-N1-Nr);$

$\mathrm{surf}(\text{Nr},\text{N1},\text{W})$

caxis([5e22 1.1e23])

colorbar;

xlabel('N_{right \}'); ylabel('N_\{left \}');

The surface plot is