Fluctuations in the ideal gas

The well-known equation of state for an ideal gas, \(PV = Nk_BT,\) is only true on average. In reality, the pressure fluctuates from moment to moment as more or fewer gas particles hit the wall in a given instant.

Suppose we measure the pressure on the walls of the container at \(m\) different times \(\mathcal{P} = \{P_{t_1}, P_{t_2},\ldots, P_{t_m}\}.\) We can measure the magnitude of the pressure fluctuations by finding the standard deviation \(\sigma_P = \sqrt{\langle P^2\rangle - \langle P\rangle^2}.\)

How will the size of the fluctuations shrink relative to the mean as we increase the number of gas particles?

In other words, if \(N\) denotes the number of particles, what is \(\dfrac{\sigma_P}{\langle P\rangle}\) proportional to?

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