91Ó°ÊÓ

(a) Assume we get a paramagnet has $N={10}^{23}$ magnets, and shut to half them are as in spin-up state, giving in $N↑=$$N↓=\frac{1}{2}N\text{,}$, whereby $N↓$ signifies the energy units..This technique provides quantity of microstates:

$\mathrm{Î\copyright }=\left(\begin{array}{c}N↓+N↑+1\\ N↓\end{array}\right)=\left(\begin{array}{c}\frac{N}{2}+\frac{N}{2}+1\\ \frac{N}{2}\end{array}\right)$

when N is simply a good number

$\mathrm{Î\copyright }≈\left(\begin{array}{c}N\\ \frac{N}{2}\end{array}\right)=\frac{N!}{\left(\frac{N}{2}\right)!\left(\frac{N}{2}\right)!}$

For such design variables, we will employ Stirling's approximation:

$n!≈\sqrt{2\mathrm{π}n}{n}^n{e}^{−n}$

$\begin{array}{c}\mathrm{Î\copyright }≈\frac{\sqrt{2\mathrm{Ï€}N}{N}^N{e}^{−N}}{\left(\sqrt{2\mathrm{Ï€}\left(\frac{N}{2}\right)}{\left(\frac{N}{2}\right)}^{\left(\frac{N}{2}\right)}{e}^{{\left.−\left(\frac{N}{2}\right)\right)}^2}\right.}\\ \end{array}$

$\mathrm{Î\copyright }≈\frac{\sqrt{2\mathrm{Ï€}N}{N}^N{e}^{−N}}{\left(2\mathrm{Ï€}\left(\frac{N}{2}\right){\left(\frac{N}{2}\right)}^N{e}^{−N}\right)}$

localid="1650343548696" $\begin{array}{r}\mathrm{Î\copyright }≈\frac{\sqrt{2\mathrm{Ï€}N}{N}^N}{\mathrm{Ï€}N{\left(\frac{N}{2}\right)}^N}\\ \end{array}$

$\begin{array}{r}\mathrm{Î\copyright }≈\frac{\sqrt{2\mathrm{Ï€}N}{2}^N}{\mathrm{Ï€}N}\end{array}$$\begin{array}{r}\mathrm{Î\copyright }≈\frac{2^{N+1}}{\sqrt{2\mathrm{Ï€}N}}\\ \end{array}$

$\mathrm{Î\copyright }≈2^N=2^{N^{23}}$

where every other line probably applies for $N$amounts about ${10}^{23}$, although this makes $2N$ a really vast concentration, but dividing by the denominator won't make a big difference.

(b) Above a ten-billion-year interval, if the machine was during a changing microstate a billion views a second, it'd examine:

${10}^9×{10}^{10}×365.25×24×3600=3.15576×{10}^{26}$microstates

The variability of microstates allowed is:

$2^{{10}^{23}}≈{10}^{3×{10}^{22}}$

As both a result, the quantity of independent states accessed over 10 billion centuriesis simply alittle fraction among those possible.