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

Answer 1

(a)

$\left \langle \left ( \delta A \right )^2 \right \rangle = \left \langle \left ( A - \left \langle A \right \rangle \right )^2 \right \rangle = \left \langle A^2 + \left \langle A \right \rangle^2 - 2A\left \langle A \right \rangle \right \rangle$

$= \left \langle A^2 \right \rangle + \left \langle \left \langle A \right \rangle^2 \right \rangle - \left \langle 2A\left \langle A \right \rangle \right \rangle$

$= \left \langle A^2 \right \rangle + \left \langle A \right \rangle^2 - 2 \left \langle A \right \rangle\left \langle A \right \rangle$ (Note that $\left \langle A \right \rangle^2$ is square of mean value and is a constant)

$= \left \langle A^2 \right \rangle + \left \langle A \right \rangle^2 - 2 \left \langle A \right \rangle^2$

$= \left \langle A^2 \right \rangle - \left \langle A \right \rangle^2$

(b)

$\left \langle \delta A \delta B \right \rangle = \left \langle \left ( A - \left \langle A \right \rangle \right )\left ( B - \left \langle B \right \rangle \right ) \right \rangle$

$= \left \langle AB + \left \langle A\right \rangle \left \langle B \right \rangle - A\left \langle B \right \rangle - \left \langle A \right \rangle B \right \rangle$

$= \left \langle AB \right \rangle + \left \langle \left \langle A \right \rangle\left \langle B \right \rangle \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle$

$= \left \langle AB \right \rangle + \left \langle A \right \rangle\left \langle B \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle$

$= \left \langle AB \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle$

(c)

i. For independent events, $\left \langle AB \right \rangle = \left \langle A \right \rangle\left \langle B \right \rangle$

$\left \langle \delta A \delta B \right \rangle = \left \langle AB \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle = 0$

ii.

$\left \langle \delta A \delta B \right \rangle = \left \langle AB \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle = \left \langle AB \right \rangle$ (Given $\left \langle A \right \rangle = \left \langle B \right \rangle = 0$ )

If B always shows the same result as A. then A = B

$\left \langle \delta A \delta B \right \rangle = \left \langle AB \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle = \left \langle AB \right \rangle = \left \langle A^2 \right \rangle = \left \langle B^2 \right \rangle$

iii.

If B always shows the opposite result as A. then A = -B or B = -A

$\left \langle \delta A \delta B \right \rangle = \left \langle AB \right \rangle - \left \langle A \right \rangle\left \langle B \right \rangle = \left \langle AB \right \rangle = \left \langle -A^2 \right \rangle$

$= - \left \langle A^2 \right \rangle = -\left \langle B^2 \right \rangle$

If A and B are independent events, then correlation is 0.  $\left \langle \delta A \delta B \right \rangle = 0$.

If B always shows the same result as A, then correlation is positive.

If B always shows the opposite result as A, then correlation is negative.