91影视

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

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:

${\mathrm{W}}_{{\mathrm{N}}_{\text{riju}},{\mathrm{N}}_{\text{hat}}}^{\mathrm{N}}=\frac{\mathrm{N}!}{{\mathrm{N}}_{\text{right}}!{\mathrm{N}}_{\text{left}}!(\mathrm{N}鈭�{\mathrm{N}}_{\text{left}}鈭�{\mathrm{N}}_{\text{right}})!}.$ 鈥︹�� (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:

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

Now, we make use of Stirling's approximation,

$\mathrm{lnx}!鈮�\mathrm{xlnx}鈭�\mathrm{x}$

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

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;$(1:1\text{e}21:5\text{e}22)$

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

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

caxis([5e22 1.1e23])

colorbar;

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

The surface plot is