-e- Consider a room full of air molecules. Each molecule can...

-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: $\ln N ! \approx N \ln N-N$

Key Concepts

Combinatorial Analysis

This concept involves counting the number of distinct ways to choose a subset from a larger set. When arranging distinguishable objects into two groups, the number of arrangements with a fixed number in one group is given by the binomial coefficient. This coefficient, expressed as N!/(N-N_L)!N_L!, counts the number of ways to choose N_L objects from a total of N and is crucial in problems that involve partitioning objects between two states.

Binomial Distribution

The binomial distribution reflects the probability of achieving a certain number of successes in a series of independent trials where each trial has two possible outcomes with fixed probabilities. In contexts where each object or event can independently fall into one of two categories (for instance, left or right), the probability for a specific configuration is determined by the number of ways this configuration can occur, weighted by the individual probabilities of each outcome.

Stirling’s Approximation

Stirling’s approximation is a method used to estimate factorials when dealing with very large numbers where direct calculation is impractical. It approximates the natural logarithm of a factorial and is widely used in statistical mechanics and combinatorics to simplify expressions involving large factorials, making it easier to analyze the probabilistic behavior of systems with an enormous number of components.

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