Aside: Quiz Solutions

1.

For some system the dependence of the probability of micro-state $\Prob(E)$ on energy was obtained for different temperatures:

Check all statements that apply:

(a)

T₁>T₂>T₃

(b)

T₃>T₂>T₁

(c)

T₂>T₁

(d)

T₃>0

(e)

T₃<0

Solution:

Since $\Prob(E)\sim\exp(-E/kT)$, the log of probability depends on E linearly:

$$\ln\Prob(E) = \mathit{const}- \frac{E}{kT}$$

• If T>0 this is a decreasing function, so T₁>0, T₂>0, T₃<0
• The greater is T, the less steep is the line (what happens if $T=\infty$?), so T₁>T₂>T₃

2.

Prove that $\sqrt{\left\langle (x- \overline{x})^2\right\rangle} /\overline{x} \sim 1/\sqrt{V}$. What happens with fluctuations in the thermodynamic limit?

Solution:

A is an extensive value , X is an intensive value. Therefore at $V\to\infty$ the scaling is $A\sim V$, $\partial A/\partial X\sim V$, $\partial^2 A/\partial X^2\sim V$,...

We obtain:

$$\left\langle (x- \overline{x})^2\right\rangle = \left\langle... ...ght\rangle^2 \sim V, \quad \left\langle x\right\rangle\sim V$$

and

$$\frac{\sqrt{\left\langle (x- \overline{x})\right\rangle}}{\... ...le x\right\rangle}\sim \frac{\sqrt V}{V} = \frac{1}{\sqrt V}$$