Consider 8 gas of Nb noninterscting molecules enclosed in a...

Consider 8 gas of $N_{b}$ noninterscting molecules enclosed in a container of volume $\boldsymbol{V}_{0}$ Focus attention on any bubvoluine $\boldsymbol{V}$ of this container and denote by $\mathbf{N}$ the number of molecules located within this bubvolume. Esch molecule is equally likely to be located anywere within the container; hence the probsbility that s given molecule is located within the subvolume $\boldsymbol{V}$ is simply equsl to $\boldsymbol{V} / \boldsymbol{V}_{0}$ (a) What is the mean number $\bar{N}$ of molecules locsted witbin $V ?$ Express your answer in terms $N_{b}, V_{0}$, and $V$. (b) Find the relative dispersion $\overline{(N-\bar{N})^{2}} / \bar{N}^{*}$ in the number of molecules located within $\boldsymbol{V}$. Expreas your answer in terms of $\vec{N}, V$, snd $V_{\mathrm{E}}$ (c) What does the answer to part (b) becone wben $V \ll V_{0} ?$ (d) What value should the dispersion $\overline{(N-\bar{N})^{2}}$ sssume when $V \rightarrow V_{a} ?$ Does the auswer to part (b) sqree with this expectation?

Consider 8 gas of $N_{b}$ noninterscting molecules enclosed in a container of volume $\boldsymbol{V}_{0}$ Focus attention on any bubvoluine $\boldsymbol{V}$ of this container and denote by $\mathbf{N}$ the number of molecules located within this bubvolume. Esch molecule is equally likely to be located anywere within the container; hence the probsbility that s given molecule is located within the subvolume $\boldsymbol{V}$ is simply equsl to $\boldsymbol{V} / \boldsymbol{V}_{0}$
(a) What is the mean number $\bar{N}$ of molecules locsted witbin $V ?$ Express your answer in terms $N_{b}, V_{0}$, and $V$.
(b) Find the relative dispersion $\overline{(N-\bar{N})^{2}} / \bar{N}^{*}$ in the number of molecules located within $\boldsymbol{V}$. Expreas your answer in terms of $\vec{N}, V$, snd $V_{\mathrm{E}}$
(c) What does the answer to part (b) becone wben $V \ll V_{0} ?$
(d) What value should the dispersion $\overline{(N-\bar{N})^{2}}$ sssume when $V \rightarrow V_{a} ?$ Does the auswer to part (b) sqree with this expectation?

Key Concepts

Limit Conditions in Statistical Mechanics

Analyzing the behavior of statistical quantities, such as the mean and dispersion, as certain parameters approach extreme values (for instance, when the subvolume is much smaller than the total volume, or as it becomes the entire container) is crucial in understanding the consistency of probabilistic predictions with physical expectations. This includes checking that fluctuations vanish in the appropriate limits and that the relative dispersion behaves as expected.

Poisson Limit (Law of Small Numbers)

When the number of trials is large and the probability of success is very small, the binomial distribution approximates the Poisson distribution. In the context of gas molecules, this limit is relevant when considering a very small subvolume relative to the total container volume, simplifying the analysis of fluctuations in the number of molecules.

Expected Value (Mean) of a Random Variable

This concept refers to the average outcome one expects from a random process after many repetitions. For the binomial distribution applied to the number of molecules in a subvolume, the mean is computed by multiplying the total number of molecules by the probability of any individual molecule being in that subvolume.

Binomial Distribution

This concept describes the probability distribution of obtaining a fixed number of successes in a given number of independent trials, with each trial having the same probability of success. In the context of gas molecules in a container, each molecule is an independent trial, and being in the subvolume is considered a 'success' with probability equal to the ratio of the subvolume to the total volume.

Variance and Relative Dispersion

Variance measures the spread or dispersion of a set of values around their mean, quantifying the fluctuations in the number of molecules. Relative dispersion, often expressed as the variance divided by the mean, provides a normalized measure of these fluctuations. For a binomial process, the variance is determined by the product of the total number of trials, the probability of success, and the probability of failure.

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