# 2. (a) *D1* Let A be a fluctuating quantity like the position, velocity, or energy of...

###### Question:

2. (a) *D1* Let A be a fluctuating quantity like the position, velocity, or energy of a molecule. At any given time, we can quantify the fluctuation of A away from the average (A) by the quantity δ = A-(A). Show that the mean square fluctuation in A can be written in terms of its mean and mean square values. «6A)2-(A2>-(A)2. The following properties of averages might be useful (X and Y are fluctuating quantities, c is a constant) · 〈XY〉 = 〈X) 〈Y) only if X and Y are not correlated. .e., the outcome of X does not depend on Yand vice versa (2 pts) (b) *D1* Generalize this result to the case of two different fluctuating quantities A and B, i.e., show that 〈6A8B〉 = 〈AB) _ 〈A)(B). (2 pts) *D1* quantities A and B. To see this, consider tossing a pair of coins A and B. We will assign values of-1 and 1 for "heads" and "tails", respectively. The coins are fair, so 〈A)-〈B) 0. Calculate (6ASB) for the following cases: (c) The quantity 46B) is often used to measure the correlation between two fluctuating i. A and B are statistically independent, i.e., they behave like normal coins ii. By some dark magic, B always shows the same result as A iii. By some even darker magic, B always shows a result opposite to A Explain in what sense the values of 〈5.45B) can be interpreted as a measure of correlation. (2 ts

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