Translational Kinetic Energy — Question 8

Question 8

Finite samples fluctuate even at fixed equilibrium temperature

For independent classical equilibrium molecules, use the supplied moments $\langle K\rangle=\tfrac32k_BT$ and $\operatorname{Var}(K)=\tfrac32(k_BT)^2$. A temperature estimate from $N$ independently sampled molecular energies is $\widehat T=2\overline K/(3k_B)$.

1. Derive the relative standard deviation of $\widehat T$ and evaluate it for $N=600$.

2. Find the minimum independent sample count needed for a relative standard deviation at most 1.00%. Explain why repeated readings of strongly correlated particles do not automatically count as independent samples.

Question 8 - Solution

Finite samples fluctuate even at fixed equilibrium temperature

(a) Method and calculation

$$\operatorname{Var}(\overline K)=\frac{\operatorname{Var}(K)}N,\quad\operatorname{Var}(\widehat T)=\frac{4}{9k_B^2}\frac{3(k_BT)^2}{2N}=\frac{2T^2}{3N}.$$$$\frac{\sigma_{\widehat T}}T=\sqrt{\frac2{3N}}=\frac1{30}\approx3.333\%\quad(N=600).$$

(b) Interpretation and further calculation

$$\sqrt{2/(3N)}\le.0100\ \Rightarrow\ N\ge\frac2{3(.0100)^2}=6666.666\ldots.$$$$N_{\min}=6667.$$With correlations, covariance terms enter the variance of the mean; the $1/N$ law no longer follows from the raw reading count.

Physical check

This is a statistical standard deviation, not a guaranteed hard error bound.