Question

-e- Consider a room full of air molecules. Each molecule can be considered to be in one of two states: in the right half of the room or in the left half. The probability of each of these states is $1 / 2$. If there are $N$ distinguishable molecules in the room, then the total number of ways of arranging the molecules is $2^{N}$. (a) Argue that the number of ways of arranging the $N$ distinguishable molecules with $N_{\mathrm{L}}$ molecules on the left is $$ P_{\mathrm{L}}=\frac{N !}{\left(N-N_{\mathrm{L}}\right) ! N_{\mathrm{L}} !} $$ (b) If there are 100 molecules in the room, what is the ratio $\left(P_{\mathrm{L}}=0.5\right) /\left(P_{\mathrm{L}}=0.6\right) ;$ that is, how much more likely is it that the molecules are evenly distributed compared to having 60 on the left and 40 on the right? (c) If there are $10^{25}$ molecules in the room, what is the ratio $\left(P_{\mathrm{L}}=0.500\right) /\left(P_{\mathrm{L}}=0.501\right)$ ? To compute the factorial of large numbers, use Stirling's approximation: $\ln N ! \approx N \ln N-N$

          -e- Consider a room full of air molecules. Each molecule can be considered to be in one of two states: in the right half of the room or in the left half. The probability of each of these states is $1 / 2$. If there are $N$ distinguishable molecules in the room, then the total number of ways of arranging the molecules is $2^{N}$. (a) Argue that the number of ways of arranging the $N$ distinguishable molecules with $N_{\mathrm{L}}$ molecules on the left is
$$
P_{\mathrm{L}}=\frac{N !}{\left(N-N_{\mathrm{L}}\right) ! N_{\mathrm{L}} !}
$$ (b) If there are 100 molecules in the room, what is the ratio $\left(P_{\mathrm{L}}=0.5\right) /\left(P_{\mathrm{L}}=0.6\right) ;$ that is, how much more likely is it that the molecules are evenly distributed compared to having 60 on the left and 40 on the right? (c) If there are $10^{25}$ molecules in the room, what is the ratio $\left(P_{\mathrm{L}}=0.500\right) /\left(P_{\mathrm{L}}=0.501\right)$ ? To compute the factorial of large numbers, use Stirling's approximation: $\ln N ! \approx N \ln N-N$

Added by Alvaro P.

Question

Please give Ace some feedback