91影视

(a) Suppose two big, similar Einstein spheres, every with $N$clocks and $q=2N$amount of energy. We had (from issue $2.22$) for the state during which all business can make are able:

$\begin{array}{c}{\mathrm{惟}}_{\text{total}}鈮�\frac{2^{4N}}{\sqrt{8蟺N}}\end{array}$

This complete agency's volatility is:

$\begin{array}{c}{S}_{\text{total}}=k\ln 鈦�\left({\mathrm{惟}}_{\text{total}}\right)\\ \end{array}$

If we modify ${\mathrm{惟}}_{\text{total}}$with , we get:

$S_{\text{total}}=k\ln 鈦�\left(\frac{2^{4N}}{\sqrt{8蟺N}}\right)$

$\ln 鈦�\left(ab\right)=\ln 鈦�\left(a\right)+\ln 鈦�\left(b\right),\ln 鈦�\left(\frac{a}{b}\right)=\ln 鈦�\left(a\right)鈭�\ln 鈦�\left(b\right)\text{and}\ln 鈦�\left(a^b\right)=b\ln 鈦�\left(a\right)$

$S_{\text{total}}=k\left[4N\ln 鈦�\left(2\right)鈭�\frac{1}{2}\ln 鈦�\left(8蟺N\right)\right]$

$k=1.38脳{10}^{鈭�23}\mathrm{J}鈰�{\mathrm{K}}^{鈭�1}\text{and}N={10}^{23}\text{:}$

$\begin{array}{c}{S}_{\text{total}}=\left(1.38脳{10}^{鈭�23}\right)\left[4脳{10}^{23}脳\ln 鈦�\left(2\right)鈭�\frac{1}{2}脳\ln 鈦�\left(8蟺脳{10}^{23}\right)\right]\\ \end{array}$

$S_{\text{total}}=3.826\mathrm{J}鈰�{\mathrm{K}}^{鈭�1}$

(b) Its presumably situation, that the energy is spread evenly between the $2$ solids, had a redundancy (from problem $2.22$):

$\begin{array}{c}{\mathrm{惟}}_{mp}鈮�\frac{2^{4N}}{4蟺N}\end{array}$

The volatility of its most frequent state was even as follows:

$\begin{array}{c}{S}_{mp}=k\ln 鈦�\left({\mathrm{惟}}_{mp}\right)\\ \end{array}$

Assuming we swap ${\mathrm{惟}}_{mp}$, we get:

$S_{mp}=k\ln 鈦�\left(\frac{2^{4N}}{4蟺N}\right)$

$S_{\text{total}}=k[4N\ln 鈦�(2)鈭�\ln 鈦�(4蟺N\left)\right]$

$k=1.38脳{10}^{鈭�23}\mathrm{J}鈰�{\mathrm{K}}^{鈭�1}\text{and}N={10}^{23}$

$\begin{array}{c}{S}_{mp}=\left(1.38脳{10}^{鈭�23}\right)\left[4脳{10}^{23}脳\ln 鈦�\left(2\right)鈭�脳\ln 鈦�\left(4蟺脳{10}^{23}\right)\right]\\ \end{array}$

$S_{mp}=3.826\mathrm{J}鈰�{\mathrm{K}}^{鈭�1}$

Hence, whether object is viewed as either an unified location with all feasible microstates or as $2$ independent systems for his or her most plausible shape, the energy is virtually defined solely more by $2^{4N}$variable.

(c) The entity is liberal to pursue each one of the microstates included in $S_{\text{total}}$, over longer timeframes, yet it's extremely likely to be in or near its most frequent state at any given time. Thus, we may claim that the entropy is supplied by $S_{mp}$ over small scales, while it's given by $S_{\text{total}}$ very long time scales, which has become so marginally bigger than $S_{mp}$.

(d) Think about a situation within which we elect some extent t so when system is in its possibly efficient than one (i.e., the entropy is $S_{mp}$) and so place a wonderful resistance between two surfaces, keeping them from transmitting energy. In effect, this shows that the model is stuck in its current condition.