Suppose that in the preceding problem the volume V under...

Suppose that in the preceding problem the volume $V$ under consideration is such that $0 \ll V / V_{0} \ll 1 .$ What is the probability that the number of molecules in this volume is between $N$ and $N+d N ?$

Key Concepts

Probability Distribution in Statistical Mechanics

This concept refers to the use of statistical methods to describe the likelihood of various microstates of a system, such as the number of molecules in a given volume. It captures the inherent fluctuations that occur in a system at the microscopic scale due to the random behavior of its constituent particles.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events (in this case, molecules entering a small volume) occurring in a fixed interval of space or time, under the assumption that these events occur with a known constant mean rate and independently of the time since the last event. This distribution naturally arises in the context of a small subvolume of a larger system where the average number of particles is low.

Limiting Case of the Binomial Distribution

In statistical mechanics, when analyzing particle distribution in a small volume relative to a large system, the problem can initially be framed in terms of a binomial distribution where each particle has a small probability to be found in the subvolume. However, when the number of particles is large and the probability for each is small, the binomial distribution tends to the Poisson distribution, simplifying the analysis of fluctuations in particle number.

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